TPTP Problem File: ITP218^4.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP218^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_List_Assn 00019_000444
% 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 : 0062_VEBT_List_Assn_00019_000444 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 7442 (2217 unt; 731 typ; 0 def)
% Number of atoms : 20190 (7944 equ; 9 cnn)
% Maximal formula atoms : 28 ( 3 avg)
% Number of connectives : 131415 (2092 ~; 225 |;1369 &;118347 @)
% ( 0 <=>;9382 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 8 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 5906 (5906 >; 0 *; 0 +; 0 <<)
% Number of symbols : 724 ( 720 usr; 9 con; 0-9 aty)
% Number of variables : 25675 (1960 ^;22355 !; 530 ?;25675 :)
% ( 830 !>; 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 16:49:12.481
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
thf(ty_t_Heap__Time__Monad_OHeap,type,
heap_Time_Heap: $tType > $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_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 ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (708)
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_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Otimes,type,
times:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oinf,type,
inf:
!>[A: $tType] : $o ).
thf(sy_cl_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_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
semiri6843258321239162965malize:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde4346867609351753570nf_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde4967611905675639751up_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_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_Ofreeze,type,
array_freeze:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
thf(sy_c_Array__Time_Oget,type,
array_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > ( list @ A ) ) ).
thf(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
thf(sy_c_Array__Time_Omake,type,
array_make:
!>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onew,type,
array_new:
!>[A: $tType] : ( nat > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onth,type,
array_nth:
!>[A: $tType] : ( ( array @ A ) > nat > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oof__list,type,
array_of_list:
!>[A: $tType] : ( ( list @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Oset,type,
array_set:
!>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Array__Time_Oupd,type,
array_upd:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Assertions_Oassn_OAbs__assn,type,
abs_assn: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > assn ).
thf(sy_c_Assertions_Oassn_ORep__assn,type,
rep_assn: assn > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oentails,type,
entails: assn > assn > $o ).
thf(sy_c_Assertions_Oentailst,type,
entailst: assn > assn > $o ).
thf(sy_c_Assertions_Oex__assn,type,
ex_assn:
!>[A: $tType] : ( ( A > assn ) > assn ) ).
thf(sy_c_Assertions_Oin__range,type,
in_range: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oin__range__rel,type,
in_range_rel: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Ois__pure__assn,type,
is_pure_assn: assn > $o ).
thf(sy_c_Assertions_Oone__assn__raw,type,
one_assn_raw: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oone__assn__raw__rel,type,
one_assn_raw_rel: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oprecise,type,
precise:
!>[A: $tType,B: $tType] : ( ( A > B > assn ) > $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_Automation_OFI,type,
fi: ( list @ ( product_prod @ assn @ assn ) ) > assn > assn > assn > assn > assn > $o ).
thf(sy_c_Automation_OFI__QUERY,type,
fI_QUERY: assn > assn > assn > $o ).
thf(sy_c_Automation_OFI__RESULT,type,
fI_RESULT: ( list @ ( product_prod @ assn @ assn ) ) > assn > assn > assn > $o ).
thf(sy_c_Automation_OSLN,type,
sln: assn ).
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_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_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > 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__Greatest__Fixpoint_OShift,type,
bNF_Greatest_Shift:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > A > ( set @ ( list @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
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_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__Embedding_Oembed,type,
bNF_Wellorder_embed:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_OembedS,type,
bNF_Wellorder_embedS:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
bNF_We4791949203932849705sMinim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
bNF_We1388413361240627857o_max2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > A > A ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
bNF_We6954850376910717587_minim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
bNF_Wellorder_wo_suc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
thf(sy_c_Basic__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_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_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_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__axioms,type,
boolea6902313364301356556axioms:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms,type,
boolea5476839437570043046axioms:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > ( A > A > A ) > $o ) ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
comple1908693960933563346ssible:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
comple187402453842119260l_fixp:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
comple115746919287870866o_fixp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates,type,
comple6359979572994053840erates:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) ) ).
thf(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
condit8047198070973881523_above:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
condit8119078960628432327_below:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_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_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_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_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_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__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_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_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__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set,type,
groups778175481326437816id_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
groups_comm_monoid_F:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
groups_comm_monoid_G:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list,type,
groups1828464146339083142d_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set,type,
groups4802862169904069756st_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__list,type,
groups_monoid_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Omonoid__list_OF,type,
groups_monoid_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
groups5270119922927024881d_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_HOL_OEx1,type,
ex1:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Heap_Oaddr__of__array,type,
addr_of_array:
!>[A: $tType] : ( ( array @ A ) > nat ) ).
thf(sy_c_Heap_Oaddr__of__ref,type,
addr_of_ref:
!>[A: $tType] : ( ( ref @ A ) > nat ) ).
thf(sy_c_Heap_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_Heap__Time__Monad_Obind,type,
heap_Time_bind:
!>[A: $tType,B: $tType] : ( ( heap_Time_Heap @ A ) > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ B ) ) ).
thf(sy_c_Heap__Time__Monad_Oraise,type,
heap_Time_raise:
!>[A: $tType] : ( ( list @ char ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oreturn,type,
heap_Time_return:
!>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Owait,type,
heap_Time_wait: nat > ( heap_Time_Heap @ product_unit ) ).
thf(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Hoare__Triple_Ohoare__triple,type,
hoare_hoare_triple:
!>[A: $tType] : ( assn > ( heap_Time_Heap @ A ) > ( A > assn ) > $o ) ).
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_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( A > int > A ) ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice,type,
semilattice:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__axioms,type,
semilattice_axioms:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__order,type,
semilattice_order:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__order__axioms,type,
semila6385135966242565138axioms:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
lattic501386751177426532rg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__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__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_List_OBleast,type,
bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oabort__Bleast,type,
abort_Bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oall__interval__int,type,
all_interval_int: ( int > $o ) > int > int > $o ).
thf(sy_c_List_Oall__interval__nat,type,
all_interval_nat: ( nat > $o ) > nat > nat > $o ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Oarg__min__list__rel,type,
arg_min_list_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ocan__select,type,
can_select:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
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_Odistinct__adj,type,
distinct_adj:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofolding__insort__key__axioms,type,
foldin3648464208017769352axioms:
!>[B: $tType,A: $tType] : ( ( 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_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
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_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type,
linord144544945434240204of_set:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olinorder__class_Ostable__sort__key,type,
linord3483353639454293061rt_key:
!>[B: $tType,A: $tType] : ( ( ( B > A ) > ( list @ B ) > ( list @ B ) ) > $o ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Olist__all,type,
list_all:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : ( C > ( A > ( list @ A ) > C > C ) > ( list @ A ) > C ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Olistsp,type,
listsp:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__project,type,
map_project:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_List_Omap__tailrec,type,
map_tailrec:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev__rel,type,
map_tailrec_rev_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ 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_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_Omin__list,type,
min_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Omin__list__rel,type,
min_list_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Onull,type,
null:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Oord_Olexordp,type,
lexordp2:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj__rel,type,
remdups_adj_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osorted__wrt__rel,type,
sorted_wrt_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Osuccessively,type,
successively:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osuccessively__rel,type,
successively_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Othose,type,
those:
!>[A: $tType] : ( ( list @ ( option @ A ) ) > ( option @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Map_Ograph,type,
graph:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Omap__add,type,
map_add:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__le,type,
map_le:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > $o ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
thf(sy_c_Misc_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_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset,type,
linord6283353356039996273ltiset:
!>[A: $tType] : ( ( multiset @ A ) > ( list @ A ) ) ).
thf(sy_c_Multiset_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_Omultiset_Ocount,type,
count:
!>[A: $tType] : ( ( multiset @ A ) > A > nat ) ).
thf(sy_c_Multiset_Oreplicate__mset,type,
replicate_mset:
!>[A: $tType] : ( nat > A > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
thf(sy_c_Multiset_Osubset__eq__mset__impl,type,
subset_eq_mset_impl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( option @ $o ) ) ).
thf(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
subset751672762298770561pl_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Multiset_Osubseteq__mset,type,
subseteq_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Ounion__mset,type,
union_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Owcount,type,
wcount:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > A > nat ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_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_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
neg_numeral_is_num:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
thf(sy_c_Option_Ooption_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_OAboveS,type,
order_AboveS:
!>[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_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_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder_Omono,type,
mono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering,type,
ordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Oordering__axioms,type,
ordering_axioms:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Oordering__top__axioms,type,
ordering_top_axioms:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Opartial__preordering,type,
partial_preordering:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Opreordering,type,
preordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Opreordering__axioms,type,
preordering_axioms:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( A > ( set @ A ) > A ) ).
thf(sy_c_Partial__Function_Ofun__lub,type,
partial_fun_lub:
!>[C: $tType,B: $tType,A: $tType] : ( ( ( set @ C ) > B ) > ( set @ ( A > C ) ) > A > B ) ).
thf(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
thf(sy_c_Partial__Function_Opartial__function__definitions,type,
partia7178651479351089652itions:
!>[A: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > A ) > $o ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_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_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > 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_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_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
linorder_quicksort:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
linord6200660962353139674rt_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( ( list @ A ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
thf(sy_c_Ref__Time_Oget,type,
ref_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > A ) ).
thf(sy_c_Ref__Time_Olookup,type,
ref_lookup:
!>[A: $tType] : ( ( ref @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Ref__Time_Oref,type,
ref_ref:
!>[A: $tType] : ( A > ( heap_Time_Heap @ ( ref @ A ) ) ) ).
thf(sy_c_Ref__Time_Oset,type,
ref_set:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Ref__Time_Oupdate,type,
ref_update:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_Time_Heap @ product_unit ) ) ).
thf(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Relation_ORange,type,
range:
!>[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_Oasym,type,
asym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $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_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $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_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Relation_Osym,type,
sym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $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,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Obind,type,
bind2:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert2:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_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__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
thf(sy_c_Syntax__Match_Oac__operator,type,
syntax_ac_operator:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
syntax7388354845996824322omatch:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oright__total,type,
right_total:
!>[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_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_VEBT__List__Assn_Olist__assn,type,
vEBT_List_list_assn:
!>[A: $tType,C: $tType] : ( ( A > C > assn ) > ( list @ A ) > ( list @ C ) > assn ) ).
thf(sy_c_VEBT__List__Assn_Olist__assn__rel,type,
vEBT_L4249061453398456502sn_rel:
!>[A: $tType,C: $tType] : ( ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) > ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) > $o ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ 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_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_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,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: a > b > assn ).
thf(sy_v_l_H,type,
l: list @ b ).
% Relevant facts (6141)
thf(fact_0_pure__assn__eq__conv,axiom,
! [P: $o,Q: $o] :
( ( ( pure_assn @ P )
= ( pure_assn @ Q ) )
= ( P = Q ) ) ).
% pure_assn_eq_conv
thf(fact_1_list__assn_Osimps_I1_J,axiom,
! [C: $tType,A: $tType,P: A > C > assn] :
( ( vEBT_List_list_assn @ A @ C @ P @ ( nil @ A ) @ ( nil @ C ) )
= ( one_one @ assn ) ) ).
% list_assn.simps(1)
thf(fact_2_is__pure__assn__pure,axiom,
! [P: $o] : ( is_pure_assn @ ( pure_assn @ P ) ) ).
% is_pure_assn_pure
thf(fact_3_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_4_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_5_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_6_is__pure__assnE,axiom,
! [A3: assn] :
( ( is_pure_assn @ A3 )
=> ~ ! [P2: $o] :
( A3
!= ( pure_assn @ P2 ) ) ) ).
% is_pure_assnE
thf(fact_7_is__pure__assn__def,axiom,
( is_pure_assn
= ( ^ [A4: assn] :
? [P3: $o] :
( A4
= ( pure_assn @ P3 ) ) ) ) ).
% is_pure_assn_def
thf(fact_8_gen__length__code_I1_J,axiom,
! [A: $tType,N: nat] :
( ( gen_length @ A @ N @ ( nil @ A ) )
= N ) ).
% gen_length_code(1)
thf(fact_9_list__assn_Osimps_I4_J,axiom,
! [C: $tType,A: $tType,Uu: A > C > assn,V: C,Va: list @ C] :
( ( vEBT_List_list_assn @ A @ C @ Uu @ ( nil @ A ) @ ( cons @ C @ V @ Va ) )
= ( bot_bot @ assn ) ) ).
% list_assn.simps(4)
thf(fact_10_list__assn_Osimps_I3_J,axiom,
! [C: $tType,A: $tType,Uu: A > C > assn,V: A,Va: list @ A] :
( ( vEBT_List_list_assn @ A @ C @ Uu @ ( cons @ A @ V @ Va ) @ ( nil @ C ) )
= ( bot_bot @ assn ) ) ).
% list_assn.simps(3)
thf(fact_11_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_12_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( one_one @ assn ) )
= P ) ).
% pure_assn_eq_emp_iff
thf(fact_13_pure__true,axiom,
( ( pure_assn @ $true )
= ( one_one @ assn ) ) ).
% pure_true
thf(fact_14_pure__assn__eq__false__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( bot_bot @ assn ) )
= ~ P ) ).
% pure_assn_eq_false_iff
thf(fact_15_pure__false,axiom,
( ( pure_assn @ $false )
= ( bot_bot @ assn ) ) ).
% pure_false
thf(fact_16_assn__basic__inequalities_I3_J,axiom,
( ( bot_bot @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(3)
thf(fact_17_member__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_18_is__pure__assn__basic__simps_I2_J,axiom,
is_pure_assn @ ( one_one @ assn ) ).
% is_pure_assn_basic_simps(2)
thf(fact_19_is__pure__assn__basic__simps_I1_J,axiom,
is_pure_assn @ ( bot_bot @ assn ) ).
% is_pure_assn_basic_simps(1)
thf(fact_20_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_21_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_22_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_23_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Xs2: list @ A] : ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys2 ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_24_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_25_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_26_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ Xs2 ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_27_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_28_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_29_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_30_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X3: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_31_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ B ) )
=> ( ! [X_1: list @ B] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X2: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( R @ X2 @ Y2 )
=> ( ( P @ Xs2 @ ( cons @ B @ Y2 @ Ys2 ) )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ~ ( R @ X2 @ Y2 )
=> ( ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_32_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member2 @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_33_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_34_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_35_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_36_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X1: A,X23: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_37_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W2: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [E: A,W12: list @ A,W22: list @ B] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons @ A @ E @ W12 ) @ W22 ) )
=> ( ! [E: B,W13: list @ A,W23: list @ B] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons @ B @ E @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_38_neq__NilE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( L
!= ( cons @ A @ X2 @ Xs2 ) ) ) ).
% neq_NilE
thf(fact_39_quicksort_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ Xs2 ) ) ) ) ).
% quicksort.cases
thf(fact_40_list__assn_Oelims,axiom,
! [C: $tType,A: $tType,X: A > C > assn,Xa: list @ A,Xb: list @ C,Y: assn] :
( ( ( vEBT_List_list_assn @ A @ C @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ C ) )
=> ( Y
!= ( one_one @ assn ) ) ) )
=> ( ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [C2: C,Cs: list @ C] :
( ( Xb
= ( cons @ C @ C2 @ Cs ) )
=> ( Y
!= ( times_times @ assn @ ( X @ A6 @ C2 ) @ ( vEBT_List_list_assn @ A @ C @ X @ As @ Cs ) ) ) ) )
=> ( ( ? [V2: A,Va2: list @ A] :
( Xa
= ( cons @ A @ V2 @ Va2 ) )
=> ( ( Xb
= ( nil @ C ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V2: C,Va2: list @ C] :
( Xb
= ( cons @ C @ V2 @ Va2 ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) ) ) ) ) ) ).
% list_assn.elims
thf(fact_41_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_42_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_43_merge__pure__star,axiom,
! [A3: $o,B2: $o] :
( ( times_times @ assn @ ( pure_assn @ A3 ) @ ( pure_assn @ B2 ) )
= ( pure_assn
@ ( A3
& B2 ) ) ) ).
% merge_pure_star
thf(fact_44_star__false__right,axiom,
! [P: assn] :
( ( times_times @ assn @ P @ ( bot_bot @ assn ) )
= ( bot_bot @ assn ) ) ).
% star_false_right
thf(fact_45_star__false__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( bot_bot @ assn ) @ P )
= ( bot_bot @ assn ) ) ).
% star_false_left
thf(fact_46_is__pure__assn__starI,axiom,
! [A3: assn,B2: assn] :
( ( is_pure_assn @ A3 )
=> ( ( is_pure_assn @ B2 )
=> ( is_pure_assn @ ( times_times @ assn @ A3 @ B2 ) ) ) ) ).
% is_pure_assn_starI
thf(fact_47_assn__times__comm,axiom,
( ( times_times @ assn )
= ( ^ [P3: assn,Q2: assn] : ( times_times @ assn @ Q2 @ P3 ) ) ) ).
% assn_times_comm
thf(fact_48_assn__times__assoc,axiom,
! [P: assn,Q: assn,R: assn] :
( ( times_times @ assn @ ( times_times @ assn @ P @ Q ) @ R )
= ( times_times @ assn @ P @ ( times_times @ assn @ Q @ R ) ) ) ).
% assn_times_assoc
thf(fact_49_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_50_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_51_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% mult.assoc
thf(fact_52_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.comm_neutral
thf(fact_53_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_54_assn__one__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ P )
= P ) ).
% assn_one_left
thf(fact_55_list__assn_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,P: A > C > assn,A3: A,As2: list @ A,C3: C,Cs2: list @ C] :
( ( vEBT_List_list_assn @ A @ C @ P @ ( cons @ A @ A3 @ As2 ) @ ( cons @ C @ C3 @ Cs2 ) )
= ( times_times @ assn @ ( P @ A3 @ C3 ) @ ( vEBT_List_list_assn @ A @ C @ P @ As2 @ Cs2 ) ) ) ).
% list_assn.simps(2)
thf(fact_56_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_57_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_58_list__tail__coinc,axiom,
! [A: $tType,N1: A,R1: list @ A,N2: A,R2: list @ A] :
( ( ( cons @ A @ N1 @ R1 )
= ( cons @ A @ N2 @ R2 ) )
=> ( ( N1 = N2 )
& ( R1 = R2 ) ) ) ).
% list_tail_coinc
thf(fact_59_norm__assertion__simps_I1_J,axiom,
! [A3: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ A3 )
= A3 ) ).
% norm_assertion_simps(1)
thf(fact_60_norm__assertion__simps_I2_J,axiom,
! [A3: assn] :
( ( times_times @ assn @ A3 @ ( one_one @ assn ) )
= A3 ) ).
% norm_assertion_simps(2)
thf(fact_61_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_62_sngr__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: ref @ A,X: A,Y: A] :
( ( times_times @ assn @ ( sngr_assn @ A @ P4 @ X ) @ ( sngr_assn @ A @ P4 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% sngr_same_false
thf(fact_63_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_64_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_65_snga__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: array @ A,X: list @ A,Y: list @ A] :
( ( times_times @ assn @ ( snga_assn @ A @ P4 @ X ) @ ( snga_assn @ A @ P4 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% snga_same_false
thf(fact_66_revg_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [A6: A,As: list @ A] :
( ( X
= ( cons @ A @ A6 @ As ) )
=> ( Y
!= ( revg @ A @ As @ ( cons @ A @ A6 @ Xa ) ) ) ) ) ) ).
% revg.elims
thf(fact_67_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),A3: B] :
( ( list_collect_set @ B @ A @ F @ ( cons @ B @ A3 @ ( nil @ B ) ) )
= ( F @ A3 ) ) ).
% list_collect_set_simps(2)
thf(fact_68_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y != Xb ) )
=> ~ ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X @ As @ ( cons @ B @ ( X @ A6 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_69_arg__min__list_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F: A > B,X: A] :
( ( arg_min_list @ A @ B @ F @ ( cons @ A @ X @ ( nil @ A ) ) )
= X ) ) ).
% arg_min_list.simps(1)
thf(fact_70_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_71_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_72_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_73_ord_Olexordp__eq__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ( lexordp_eq @ A @ Less @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_74_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( set @ A )] :
( ( list_collect_set @ B @ A @ F @ ( nil @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_simps(1)
thf(fact_75_ord_Olexordp__eq__refl,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_76_ord_Olexordp__eq_Ocong,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( lexordp_eq @ A ) ) ).
% ord.lexordp_eq.cong
thf(fact_77_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_78_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq @ A @ Less @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_79_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_80_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F: A > B,A3: A,As2: list @ A,Bs: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( cons @ A @ A3 @ As2 ) @ Bs )
= ( map_tailrec_rev @ A @ B @ F @ As2 @ ( cons @ B @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_81_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F: A > B,Bs: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( nil @ A ) @ Bs )
= Bs ) ).
% map_tailrec_rev.simps(1)
thf(fact_82_revg_Osimps_I2_J,axiom,
! [A: $tType,A3: A,As2: list @ A,B2: list @ A] :
( ( revg @ A @ ( cons @ A @ A3 @ As2 ) @ B2 )
= ( revg @ A @ As2 @ ( cons @ A @ A3 @ B2 ) ) ) ).
% revg.simps(2)
thf(fact_83_revg_Osimps_I1_J,axiom,
! [A: $tType,B2: list @ A] :
( ( revg @ A @ ( nil @ A ) @ B2 )
= B2 ) ).
% revg.simps(1)
thf(fact_84_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp_eq @ A @ Less @ A1 @ A22 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X2: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( Less @ X2 @ Y2 ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_85_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_86_assn__aci_I10_J,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( times_times @ assn @ ( times_times @ assn @ A3 @ B2 ) @ C3 )
= ( times_times @ assn @ ( times_times @ assn @ A3 @ C3 ) @ B2 ) ) ).
% assn_aci(10)
thf(fact_87_star__aci_I3_J,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( times_times @ assn @ A3 @ ( times_times @ assn @ B2 @ C3 ) )
= ( times_times @ assn @ B2 @ ( times_times @ assn @ A3 @ C3 ) ) ) ).
% star_aci(3)
thf(fact_88_star__aci_I2_J,axiom,
( ( times_times @ assn )
= ( ^ [A4: assn,B3: assn] : ( times_times @ assn @ B3 @ A4 ) ) ) ).
% star_aci(2)
thf(fact_89_star__assoc,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( times_times @ assn @ ( times_times @ assn @ A3 @ B2 ) @ C3 )
= ( times_times @ assn @ A3 @ ( times_times @ assn @ B2 @ C3 ) ) ) ).
% star_assoc
thf(fact_90_and__extract__pure__left__iff,axiom,
! [B2: $o,Q: assn] :
( ( inf_inf @ assn @ ( pure_assn @ B2 ) @ Q )
= ( times_times @ assn @ ( inf_inf @ assn @ ( one_one @ assn ) @ Q ) @ ( pure_assn @ B2 ) ) ) ).
% and_extract_pure_left_iff
thf(fact_91_and__extract__pure__right__iff,axiom,
! [P: assn,B2: $o] :
( ( inf_inf @ assn @ P @ ( pure_assn @ B2 ) )
= ( times_times @ assn @ ( inf_inf @ assn @ ( one_one @ assn ) @ P ) @ ( pure_assn @ B2 ) ) ) ).
% and_extract_pure_right_iff
thf(fact_92_list__collect__set__map__simps_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,A3: C] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( cons @ C @ A3 @ ( nil @ C ) ) ) )
= ( F @ ( X @ A3 ) ) ) ).
% list_collect_set_map_simps(2)
thf(fact_93_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,A1: list @ A,A22: list @ B] :
( ( listrelp @ A @ B @ R3 @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X2: A,Y2: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ B] :
( ( A22
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( R3 @ X2 @ Y2 )
=> ~ ( listrelp @ A @ B @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_94_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R4: A > B > $o,A12: list @ A,A23: list @ B] :
( ( ( A12
= ( nil @ A ) )
& ( A23
= ( nil @ B ) ) )
| ? [X3: A,Y3: B,Xs3: list @ A,Ys3: list @ B] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ B @ Y3 @ Ys3 ) )
& ( R4 @ X3 @ Y3 )
& ( listrelp @ A @ B @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_95_mergesort__by__rel__simps_I2_J,axiom,
! [A: $tType,R: A > A > $o,X: A] :
( ( mergesort_by_rel @ A @ R @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% mergesort_by_rel_simps(2)
thf(fact_96_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A] :
~ ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ) ).
% lexordp_eq_simps(3)
thf(fact_97_entails__solve__finalize_I2_J,axiom,
! [M: list @ ( product_prod @ assn @ assn )] : ( fI_RESULT @ M @ ( one_one @ assn ) @ ( one_one @ assn ) @ ( one_one @ assn ) ) ).
% entails_solve_finalize(2)
thf(fact_98_frame__inference__finalize,axiom,
! [M: list @ ( product_prod @ assn @ assn ),F2: assn] : ( fI_RESULT @ M @ F2 @ ( one_one @ assn ) @ F2 ) ).
% frame_inference_finalize
thf(fact_99_mult_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% mult.right_assoc
thf(fact_100_mult_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( times_times @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% mult.right_commute
thf(fact_101_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: list @ A] :
( ( ( map @ A @ B @ F @ A3 )
= ( nil @ B ) )
= ( A3
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_102_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_103_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_104_lexordp__eq__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ) ).
% lexordp_eq_simps(2)
thf(fact_105_lexordp__eq__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq_simps(1)
thf(fact_106_mergesort__by__rel__simps_I1_J,axiom,
! [A: $tType,R: A > A > $o] :
( ( mergesort_by_rel @ A @ R @ ( nil @ A ) )
= ( nil @ A ) ) ).
% mergesort_by_rel_simps(1)
thf(fact_107_merge__pure__and,axiom,
! [A3: $o,B2: $o] :
( ( inf_inf @ assn @ ( pure_assn @ A3 ) @ ( pure_assn @ B2 ) )
= ( pure_assn
@ ( A3
& B2 ) ) ) ).
% merge_pure_and
thf(fact_108_and__extract__pure__right__ctx__iff,axiom,
! [P: assn,Q: assn,B2: $o] :
( ( inf_inf @ assn @ P @ ( times_times @ assn @ Q @ ( pure_assn @ B2 ) ) )
= ( times_times @ assn @ ( inf_inf @ assn @ P @ Q ) @ ( pure_assn @ B2 ) ) ) ).
% and_extract_pure_right_ctx_iff
thf(fact_109_and__extract__pure__left__ctx__iff,axiom,
! [P: assn,B2: $o,Q: assn] :
( ( inf_inf @ assn @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ Q )
= ( times_times @ assn @ ( inf_inf @ assn @ P @ Q ) @ ( pure_assn @ B2 ) ) ) ).
% and_extract_pure_left_ctx_iff
thf(fact_110_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > ( set @ A ),X: C > B] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( nil @ C ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_map_simps(1)
thf(fact_111_assn__aci_I4_J,axiom,
! [X: assn,Y: assn] :
( ( inf_inf @ assn @ X @ ( inf_inf @ assn @ X @ Y ) )
= ( inf_inf @ assn @ X @ Y ) ) ).
% assn_aci(4)
thf(fact_112_assn__aci_I3_J,axiom,
! [X: assn,Y: assn,Z2: assn] :
( ( inf_inf @ assn @ X @ ( inf_inf @ assn @ Y @ Z2 ) )
= ( inf_inf @ assn @ Y @ ( inf_inf @ assn @ X @ Z2 ) ) ) ).
% assn_aci(3)
thf(fact_113_assn__aci_I1_J,axiom,
( ( inf_inf @ assn )
= ( ^ [X3: assn,Y3: assn] : ( inf_inf @ assn @ Y3 @ X3 ) ) ) ).
% assn_aci(1)
thf(fact_114_norm__assertion__simps_I31_J,axiom,
! [X: assn] :
( ( inf_inf @ assn @ X @ X )
= X ) ).
% norm_assertion_simps(31)
thf(fact_115_norm__assertion__simps_I14_J,axiom,
! [X: assn,Y: assn,Z2: assn] :
( ( inf_inf @ assn @ ( inf_inf @ assn @ X @ Y ) @ Z2 )
= ( inf_inf @ assn @ X @ ( inf_inf @ assn @ Y @ Z2 ) ) ) ).
% norm_assertion_simps(14)
thf(fact_116_lexordp__eq__refl,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% lexordp_eq_refl
thf(fact_117_lexordp__eq__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ( ord_lexordp_eq @ A @ Ys @ Zs )
=> ( ord_lexordp_eq @ A @ Xs @ Zs ) ) ) ) ).
% lexordp_eq_trans
thf(fact_118_lexordp__eq__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
| ( ord_lexordp_eq @ A @ Ys @ Xs ) ) ) ).
% lexordp_eq_linear
thf(fact_119_lexordp__eq__antisym,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ( ord_lexordp_eq @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% lexordp_eq_antisym
thf(fact_120_memb__imp__not__empty,axiom,
! [A: $tType,X: A,S: set @ A] :
( ( member2 @ A @ X @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_121_set__notEmptyE,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X2: A] :
~ ( member2 @ A @ X2 @ S ) ) ).
% set_notEmptyE
thf(fact_122_map__eq__consE,axiom,
! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fa: A,Fl: list @ A] :
( ( ( map @ B @ A @ F @ Ls )
= ( cons @ A @ Fa @ Fl ) )
=> ~ ! [A6: B,L2: list @ B] :
( ( Ls
= ( cons @ B @ A6 @ L2 ) )
=> ( ( ( F @ A6 )
= Fa )
=> ( ( map @ B @ A @ F @ L2 )
!= Fl ) ) ) ) ).
% map_eq_consE
thf(fact_123_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W: list @ A,F: B > A,Ww: list @ B,A3: B] :
( ( W
= ( map @ B @ A @ F @ Ww ) )
=> ( ( cons @ A @ ( F @ A3 ) @ W )
= ( map @ B @ A @ F @ ( cons @ B @ A3 @ Ww ) ) ) ) ).
% map_consI(1)
thf(fact_124_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ? [Z3: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map @ B @ A @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_125_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F @ Ys ) )
= ( ? [Z3: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map @ B @ A @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_126_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z4: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map @ B @ A @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_127_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F @ Ys ) )
=> ? [Z4: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map @ B @ A @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_128_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X22: list @ A] :
( ( map @ A @ B @ F @ ( cons @ A @ X21 @ X22 ) )
= ( cons @ B @ ( F @ X21 ) @ ( map @ A @ B @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_129_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( map @ A @ B @ F @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_130_norm__assertion__simps_I9_J,axiom,
! [X: assn] :
( ( inf_inf @ assn @ ( bot_bot @ assn ) @ X )
= ( bot_bot @ assn ) ) ).
% norm_assertion_simps(9)
thf(fact_131_norm__assertion__simps_I10_J,axiom,
! [X: assn] :
( ( inf_inf @ assn @ X @ ( bot_bot @ assn ) )
= ( bot_bot @ assn ) ) ).
% norm_assertion_simps(10)
thf(fact_132_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq.Nil
thf(fact_133_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( R3 @ X @ Y )
=> ( ( listrelp @ A @ B @ R3 @ Xs @ Ys )
=> ( listrelp @ A @ B @ R3 @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_134_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] : ( listrelp @ A @ B @ R3 @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_135_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_136_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_137_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_138_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_139_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X3: A] : ( inf_inf @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% inf_apply
thf(fact_140_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_right_idem
thf(fact_141_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 )
= ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_142_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_left_idem
thf(fact_143_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B2 ) )
= ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_144_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% inf_idem
thf(fact_145_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ A3 )
= A3 ) ) ).
% inf.idem
thf(fact_146_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_147_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member2 @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_148_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X3: A] :
~ ( member2 @ A @ X3 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_149_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_150_Int__emptyI,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ A5 )
=> ~ ( member2 @ A @ X2 @ B4 ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_151_disjoint__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ~ ( member2 @ A @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_152_Int__empty__left,axiom,
! [A: $tType,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_153_Int__empty__right,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_154_disjoint__iff__not__equal,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ! [Y3: A] :
( ( member2 @ A @ Y3 @ B4 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_155_disjointI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ A3 )
=> ~ ( member2 @ A @ X2 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjointI
thf(fact_156_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_157_equals0D,axiom,
! [A: $tType,A5: set @ A,A3: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member2 @ A @ A3 @ A5 ) ) ).
% equals0D
thf(fact_158_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y2: A] :
~ ( member2 @ A @ Y2 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_159_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X3: A] : ( member2 @ A @ X3 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_160_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_161_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_sup_aci(4)
thf(fact_162_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_163_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z2 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_164_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_165_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C3 )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ).
% inf.assoc
thf(fact_166_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z2 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ).
% inf_assoc
thf(fact_167_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A4: A,B3: A] : ( inf_inf @ A @ B3 @ A4 ) ) ) ) ).
% inf.commute
thf(fact_168_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% inf_commute
thf(fact_169_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5
= ( inf_inf @ A @ K @ A3 ) )
=> ( ( inf_inf @ A @ A5 @ B2 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_170_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B4: A,K: A,B2: A,A3: A] :
( ( B4
= ( inf_inf @ A @ K @ B2 ) )
=> ( ( inf_inf @ A @ A3 @ B4 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_171_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A3 @ C3 ) )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ).
% inf.left_commute
thf(fact_172_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_left_commute
thf(fact_173_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X3: A] : ( inf_inf @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% inf_fun_def
thf(fact_174_FI__finalize,axiom,
! [M2: list @ ( product_prod @ assn @ assn ),P4: assn,Up: assn,Q3: assn,Uq: assn,F: assn] :
( ( fI_RESULT @ M2 @ ( times_times @ assn @ P4 @ Up ) @ ( times_times @ assn @ Q3 @ Uq ) @ F )
=> ( fi @ M2 @ P4 @ Q3 @ Up @ Uq @ F ) ) ).
% FI_finalize
thf(fact_175_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A7: set @ A] :
( A7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_176_map__eq__map__tailrec,axiom,
! [B: $tType,A: $tType] :
( ( map @ A @ B )
= ( map_tailrec @ A @ B ) ) ).
% map_eq_map_tailrec
thf(fact_177_list__collect__set__map__simps_I3_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,A3: C,L: list @ C] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( cons @ C @ A3 @ L ) ) )
= ( sup_sup @ ( set @ A ) @ ( F @ ( X @ A3 ) ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L ) ) ) ) ).
% list_collect_set_map_simps(3)
thf(fact_178_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_179_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp_eq @ A @ A1 @ A22 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X2: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( ord_less @ A @ X2 @ Y2 ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ~ ( ord_less @ A @ Y2 @ X2 )
=> ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_180_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ord_less @ A @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( ord_less @ A @ X3 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X3 )
& ( ord_lexordp_eq @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_181_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X3: A] : ( member2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_182_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_183_mult_Osafe__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [X: A,Y: A,A3: A,B2: A] :
( ( syntax7388354845996824322omatch @ A @ A @ ( times_times @ A @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ B2 )
= ( times_times @ A @ B2 @ A3 ) ) ) ) ).
% mult.safe_commute
thf(fact_184_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_185_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_186_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_187_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_188_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ B2 )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_189_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X3: A] : ( sup_sup @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% sup_apply
thf(fact_190_Int__iff,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( ( member2 @ A @ C3 @ A5 )
& ( member2 @ A @ C3 @ B4 ) ) ) ).
% Int_iff
thf(fact_191_IntI,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ A5 )
=> ( ( member2 @ A @ C3 @ B4 )
=> ( member2 @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% IntI
thf(fact_192_UnCI,axiom,
! [A: $tType,C3: A,B4: set @ A,A5: set @ A] :
( ( ~ ( member2 @ A @ C3 @ B4 )
=> ( member2 @ A @ C3 @ A5 ) )
=> ( member2 @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnCI
thf(fact_193_Un__iff,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ( member2 @ A @ C3 @ A5 )
| ( member2 @ A @ C3 @ B4 ) ) ) ).
% Un_iff
thf(fact_194_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
= A3 ) ) ).
% sup_bot.right_neutral
thf(fact_195_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A3 @ B2 ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_196_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
= A3 ) ) ).
% sup_bot.left_neutral
thf(fact_197_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( sup_sup @ A @ A3 @ B2 )
= ( bot_bot @ A ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_198_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_199_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_200_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_201_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_202_sup__inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= X ) ) ).
% sup_inf_absorb
thf(fact_203_inf__sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= X ) ) ).
% inf_sup_absorb
thf(fact_204_Un__empty,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_205_Un__Int__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_206_Un__Int__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_207_Un__Int__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T: set @ A] :
( ( inf_inf @ ( set @ A ) @ S @ ( sup_sup @ ( set @ A ) @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_208_Un__Int__eq_I4_J,axiom,
! [A: $tType,T: set @ A,S: set @ A] :
( ( inf_inf @ ( set @ A ) @ T @ ( sup_sup @ ( set @ A ) @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_209_Int__Un__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_210_Int__Un__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_211_Int__Un__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T: set @ A] :
( ( sup_sup @ ( set @ A ) @ S @ ( inf_inf @ ( set @ A ) @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_212_Int__Un__eq_I4_J,axiom,
! [A: $tType,T: set @ A,S: set @ A] :
( ( sup_sup @ ( set @ A ) @ T @ ( inf_inf @ ( set @ A ) @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_213_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( ord_less @ A @ X @ Y )
| ( ~ ( ord_less @ A @ Y @ X )
& ( ord_lexordp_eq @ A @ Xs @ Ys ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_214_list__collect__set__simps_I3_J,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),A3: B,L: list @ B] :
( ( list_collect_set @ B @ A @ F @ ( cons @ B @ A3 @ L ) )
= ( sup_sup @ ( set @ A ) @ ( F @ A3 ) @ ( list_collect_set @ B @ A @ F @ L ) ) ) ).
% list_collect_set_simps(3)
thf(fact_215_Int__left__commute,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Int_left_commute
thf(fact_216_Un__Int__distrib2,axiom,
! [A: $tType,B4: set @ A,C4: set @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) @ A5 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B4 @ A5 ) @ ( sup_sup @ ( set @ A ) @ C4 @ A5 ) ) ) ).
% Un_Int_distrib2
thf(fact_217_Int__left__absorb,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ).
% Int_left_absorb
thf(fact_218_Int__Un__distrib2,axiom,
! [A: $tType,B4: set @ A,C4: set @ A,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) @ A5 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B4 @ A5 ) @ ( inf_inf @ ( set @ A ) @ C4 @ A5 ) ) ) ).
% Int_Un_distrib2
thf(fact_219_Un__Int__distrib,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ ( sup_sup @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Un_Int_distrib
thf(fact_220_Int__Un__distrib,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Int_Un_distrib
thf(fact_221_Un__Int__crazy,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) @ ( inf_inf @ ( set @ A ) @ C4 @ A5 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) @ ( sup_sup @ ( set @ A ) @ C4 @ A5 ) ) ) ).
% Un_Int_crazy
thf(fact_222_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A7 ) ) ) ).
% Int_commute
thf(fact_223_Int__absorb,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ A5 )
= A5 ) ).
% Int_absorb
thf(fact_224_Int__assoc,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Int_assoc
thf(fact_225_IntD2,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
=> ( member2 @ A @ C3 @ B4 ) ) ).
% IntD2
thf(fact_226_IntD1,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
=> ( member2 @ A @ C3 @ A5 ) ) ).
% IntD1
thf(fact_227_IntE,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
=> ~ ( ( member2 @ A @ C3 @ A5 )
=> ~ ( member2 @ A @ C3 @ B4 ) ) ) ).
% IntE
thf(fact_228_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_229_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_230_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_231_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y3: A] : ( sup_sup @ A @ Y3 @ X3 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_232_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ C3 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C3 ) ) ) ) ).
% sup.assoc
thf(fact_233_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% sup_assoc
thf(fact_234_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_235_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less @ A @ X @ B2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_236_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb3
thf(fact_237_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb4
thf(fact_238_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A4: A,B3: A] : ( sup_sup @ A @ B3 @ A4 ) ) ) ) ).
% sup.commute
thf(fact_239_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y3: A] : ( sup_sup @ A @ Y3 @ X3 ) ) ) ) ).
% sup_commute
thf(fact_240_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5
= ( sup_sup @ A @ K @ A3 ) )
=> ( ( sup_sup @ A @ A5 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_241_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,K: A,B2: A,A3: A] :
( ( B4
= ( sup_sup @ A @ K @ B2 ) )
=> ( ( sup_sup @ A @ A3 @ B4 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_242_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A3 @ C3 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C3 ) ) ) ) ).
% sup.left_commute
thf(fact_243_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% sup_left_commute
thf(fact_244_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A3 )
=> ~ ( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_245_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A4: A] :
( ( A4
= ( sup_sup @ A @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_246_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ A3 )
=> ( ord_less @ A @ C3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_247_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_248_UnE,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( ~ ( member2 @ A @ C3 @ A5 )
=> ( member2 @ A @ C3 @ B4 ) ) ) ).
% UnE
thf(fact_249_UnI1,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ A5 )
=> ( member2 @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnI1
thf(fact_250_UnI2,axiom,
! [A: $tType,C3: A,B4: set @ A,A5: set @ A] :
( ( member2 @ A @ C3 @ B4 )
=> ( member2 @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnI2
thf(fact_251_bex__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ A5 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member2 @ A @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_252_ball__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member2 @ A @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_253_Un__assoc,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Un_assoc
thf(fact_254_Un__absorb,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ A5 )
= A5 ) ).
% Un_absorb
thf(fact_255_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A7 ) ) ) ).
% Un_commute
thf(fact_256_Un__left__absorb,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_left_absorb
thf(fact_257_Un__left__commute,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_258_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X3: A] : ( sup_sup @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% sup_fun_def
thf(fact_259_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% lt_ex
thf(fact_260_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_261_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_262_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_263_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order.asym
thf(fact_264_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3 = B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_265_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_266_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X2: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_267_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_268_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_269_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_270_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_271_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P5: A > $o] :
? [X4: A] : ( P5 @ X4 ) )
= ( ^ [P3: A > $o] :
? [N3: A] :
( ( P3 @ N3 )
& ! [M3: A] :
( ( ord_less @ A @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_272_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A6: A,B6: A] :
( ( ord_less @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A3 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_273_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_274_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_275_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_276_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_277_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_278_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_279_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_280_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_281_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order_less_asym'
thf(fact_282_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_283_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B2: B,C3: B] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less @ B @ X2 @ Y2 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_284_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F: A > B,C3: B] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_285_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_286_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C3: B] :
( ( ord_less @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less @ B @ X2 @ Y2 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_287_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C3: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F @ B2 ) @ C3 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_288_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_289_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_290_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_291_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_292_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_293_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_294_syntax__fo__nomatch__def,axiom,
! [B: $tType,A: $tType] :
( ( syntax7388354845996824322omatch @ A @ B )
= ( ^ [Pat: A,Obj: B] : $true ) ) ).
% syntax_fo_nomatch_def
thf(fact_295_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_296_boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z2 ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z2 @ X ) ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_297_boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z2 @ X ) ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_298_boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_299_boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_300_sup__inf__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z2 ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z2 @ X ) ) ) ) ).
% sup_inf_distrib2
thf(fact_301_sup__inf__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% sup_inf_distrib1
thf(fact_302_inf__sup__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z2 @ X ) ) ) ) ).
% inf_sup_distrib2
thf(fact_303_inf__sup__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_distrib1
thf(fact_304_distrib__imp2,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ! [X2: A,Y2: A,Z4: A] :
( ( sup_sup @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z4 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X2 @ Y2 ) @ ( sup_sup @ A @ X2 @ Z4 ) ) )
=> ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ) ).
% distrib_imp2
thf(fact_305_distrib__imp1,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ! [X2: A,Y2: A,Z4: A] :
( ( inf_inf @ A @ X2 @ ( sup_sup @ A @ Y2 @ Z4 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ ( inf_inf @ A @ X2 @ Z4 ) ) )
=> ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ) ).
% distrib_imp1
thf(fact_306_Un__empty__right,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Un_empty_right
thf(fact_307_Un__empty__left,axiom,
! [A: $tType,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_308_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_309_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_310_not__psubset__empty,axiom,
! [A: $tType,A5: set @ A] :
~ ( ord_less @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_311_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_312_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ A3 @ C3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_313_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B3: A] :
( ( A4
= ( inf_inf @ A @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_314_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ ( inf_inf @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_315_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb4
thf(fact_316_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb3
thf(fact_317_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A3: A] :
( ( ord_less @ A @ B2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% less_infI2
thf(fact_318_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less @ A @ A3 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% less_infI1
thf(fact_319_FI__p__nomatch,axiom,
! [M2: list @ ( product_prod @ assn @ assn ),Ps: assn,Qs: assn,Q3: assn,P4: assn,Up: assn,Uq: assn,F: assn] :
( ( fi @ M2 @ Ps @ ( times_times @ assn @ Qs @ Q3 ) @ ( times_times @ assn @ P4 @ Up ) @ Uq @ F )
=> ( fi @ M2 @ ( times_times @ assn @ Ps @ P4 ) @ ( times_times @ assn @ Qs @ Q3 ) @ Up @ Uq @ F ) ) ).
% FI_p_nomatch
thf(fact_320_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ~ ( ord_less @ A @ Y @ X )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_321_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_322_assn__aci_I11_J,axiom,
! [X: assn,Y: assn,A3: assn,B2: assn] :
( ( syntax7388354845996824322omatch @ assn @ assn @ ( times_times @ assn @ X @ Y ) @ A3 )
=> ( ( times_times @ assn @ A3 @ B2 )
= ( times_times @ assn @ B2 @ A3 ) ) ) ).
% assn_aci(11)
thf(fact_323_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M2 @ N ) ) ) ) ) ).
% less_1_mult
thf(fact_324_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_325_FI__q__nomatch,axiom,
! [M2: list @ ( product_prod @ assn @ assn ),Up: assn,Qs: assn,Q3: assn,Uq: assn,F: assn] :
( ( fi @ M2 @ ( times_times @ assn @ sln @ Up ) @ Qs @ sln @ ( times_times @ assn @ Q3 @ Uq ) @ F )
=> ( fi @ M2 @ sln @ ( times_times @ assn @ Qs @ Q3 ) @ Up @ Uq @ F ) ) ).
% FI_q_nomatch
thf(fact_326_map__tailrec__def,axiom,
! [B: $tType,A: $tType] :
( ( map_tailrec @ A @ B )
= ( ^ [F3: A > B,As3: list @ A] : ( rev @ B @ ( map_tailrec_rev @ A @ B @ F3 @ As3 @ ( nil @ B ) ) ) ) ) ).
% map_tailrec_def
thf(fact_327_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 ) )
=> ( ! [V2: A,Va2: list @ A] :
( ( X
= ( cons @ A @ V2 @ Va2 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V2 @ Va2 ) ) ) )
=> ~ ! [X1: A,L1: list @ A] :
( ( X
= ( cons @ A @ X1 @ L1 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ~ ( ( ( ord_less @ A @ X1 @ X23 )
=> ( Y
= ( cons @ A @ X1 @ ( merge @ A @ L1 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X23 )
=> ( ( ( X1 = X23 )
=> ( Y
= ( cons @ A @ X1 @ ( merge @ A @ L1 @ L22 ) ) ) )
& ( ( X1 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge.elims
thf(fact_328_list__collect__set__map__simps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,L: list @ C,L3: list @ C] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( append @ C @ L @ L3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L ) ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L3 ) ) ) ) ).
% list_collect_set_map_simps(4)
thf(fact_329_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( linord4507533701916653071of_set @ A @ A5 )
= ( nil @ A ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_330_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ! [Y2: A,Ys2: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys2 ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [X2: A,Xs2: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ X2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_331_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y2: A,Ys4: list @ A] :
( Ys
!= ( cons @ A @ Y2 @ Ys4 ) ) )
=> ( ! [X2: A] :
( ? [Xs4: list @ A] :
( Xs
= ( cons @ A @ X2 @ Xs4 ) )
=> ! [Y2: A] :
( ? [Ys4: list @ A] :
( Ys
= ( cons @ A @ Y2 @ Ys4 ) )
=> ~ ( ord_less @ A @ X2 @ Y2 ) ) )
=> ~ ! [X2: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Xs4 ) )
=> ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X2 @ Ys4 ) )
=> ~ ( ord_lexordp @ A @ Xs4 @ Ys4 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_332_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Y3: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ord_less @ A @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( ord_less @ A @ X3 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X3 )
& ( ord_lexordp @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_333_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_334_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_335_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_336_append_Oassoc,axiom,
! [A: $tType,A3: list @ A,B2: list @ A,C3: list @ A] :
( ( append @ A @ ( append @ A @ A3 @ B2 ) @ C3 )
= ( append @ A @ A3 @ ( append @ A @ B2 @ C3 ) ) ) ).
% append.assoc
thf(fact_337_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_338_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_339_merge__pure__or,axiom,
! [A3: $o,B2: $o] :
( ( sup_sup @ assn @ ( pure_assn @ A3 ) @ ( pure_assn @ B2 ) )
= ( pure_assn
@ ( A3
| B2 ) ) ) ).
% merge_pure_or
thf(fact_340_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_341_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_342_self__append__conv2,axiom,
! [A: $tType,Y: list @ A,Xs: list @ A] :
( ( Y
= ( append @ A @ Xs @ Y ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_343_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_344_self__append__conv,axiom,
! [A: $tType,Y: list @ A,Ys: list @ A] :
( ( Y
= ( append @ A @ Y @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_345_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_346_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_347_append_Oright__neutral,axiom,
! [A: $tType,A3: list @ A] :
( ( append @ A @ A3 @ ( nil @ A ) )
= A3 ) ).
% append.right_neutral
thf(fact_348_empty__append__eq__id,axiom,
! [A: $tType] :
( ( append @ A @ ( nil @ A ) )
= ( ^ [X3: list @ A] : X3 ) ) ).
% empty_append_eq_id
thf(fact_349_map__append,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F @ Xs ) @ ( map @ B @ A @ F @ Ys ) ) ) ).
% map_append
thf(fact_350_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_351_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_352_rev__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( rev @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( rev @ A @ Ys ) @ ( rev @ A @ Xs ) ) ) ).
% rev_append
thf(fact_353_lexordp__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A] :
( ( ord_lexordp @ A @ ( nil @ A ) @ Ys )
= ( Ys
!= ( nil @ A ) ) ) ) ).
% lexordp_simps(1)
thf(fact_354_lexordp__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ ( nil @ A ) ) ) ).
% lexordp_simps(2)
thf(fact_355_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_356_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L12: list @ A,E2: A,L23: list @ A,E3: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E2 @ L23 ) )
= ( cons @ A @ E3 @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E2 = E3 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_357_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E3: A,L12: list @ A,E2: A,L23: list @ A] :
( ( ( cons @ A @ E3 @ ( nil @ A ) )
= ( append @ A @ L12 @ ( cons @ A @ E2 @ L23 ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E2 = E3 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_358_list__se__match_I4_J,axiom,
! [A: $tType,L23: list @ A,A3: A,L12: list @ A] :
( ( L23
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A3 @ ( nil @ A ) )
= ( append @ A @ L12 @ L23 ) )
= ( ( L12
= ( nil @ A ) )
& ( L23
= ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_359_list__se__match_I3_J,axiom,
! [A: $tType,L12: list @ A,A3: A,L23: list @ A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A3 @ ( nil @ A ) )
= ( append @ A @ L12 @ L23 ) )
= ( ( L12
= ( cons @ A @ A3 @ ( nil @ A ) ) )
& ( L23
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_360_list__se__match_I2_J,axiom,
! [A: $tType,L23: list @ A,L12: list @ A,A3: A] :
( ( L23
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L23 )
= ( cons @ A @ A3 @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( L23
= ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_361_list__se__match_I1_J,axiom,
! [A: $tType,L12: list @ A,L23: list @ A,A3: A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L23 )
= ( cons @ A @ A3 @ ( nil @ A ) ) )
= ( ( L12
= ( cons @ A @ A3 @ ( nil @ A ) ) )
& ( L23
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_362_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L12: list @ A,E1: A,E22: A,L23: list @ A,E12: A,E23: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E1 @ ( cons @ A @ E22 @ L23 ) ) )
= ( cons @ A @ E12 @ ( cons @ A @ E23 @ ( nil @ A ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_363_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E12: A,E23: A,L12: list @ A,E1: A,E22: A,L23: list @ A] :
( ( ( cons @ A @ E12 @ ( cons @ A @ E23 @ ( nil @ A ) ) )
= ( append @ A @ L12 @ ( cons @ A @ E1 @ ( cons @ A @ E22 @ L23 ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_364_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_365_singleton__rev__conv,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( cons @ A @ X @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( ( cons @ A @ X @ ( nil @ A ) )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_366_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( ord_less @ A @ X @ Y )
| ( ~ ( ord_less @ A @ Y @ X )
& ( ord_lexordp @ A @ Xs @ Ys ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_367_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_368_list__collect__set__simps_I4_J,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),L: list @ B,L3: list @ B] :
( ( list_collect_set @ B @ A @ F @ ( append @ B @ L @ L3 ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F @ L ) @ ( list_collect_set @ B @ A @ F @ L3 ) ) ) ).
% list_collect_set_simps(4)
thf(fact_369_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs: list @ B,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X @ Xs ) @ F )
= ( append @ A @ ( F @ X ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_370_revg__fun,axiom,
! [A: $tType] :
( ( revg @ A )
= ( ^ [A4: list @ A] : ( append @ A @ ( rev @ A @ A4 ) ) ) ) ).
% revg_fun
thf(fact_371_rev__eq__Cons__iff,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( Xs
= ( append @ A @ ( rev @ A @ Ys ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_372_psubsetD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C3: A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ( member2 @ A @ C3 @ A5 )
=> ( member2 @ A @ C3 @ B4 ) ) ) ).
% psubsetD
thf(fact_373_psubset__trans,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% psubset_trans
thf(fact_374_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us ) )
& ( ( append @ A @ Us @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs )
& ( Ys
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_375_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_376_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_377_lexordp__irreflexive_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ Xs ) ) ).
% lexordp_irreflexive'
thf(fact_378_lexordp__append__leftI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Us2: list @ A,Vs: list @ A,Xs: list @ A] :
( ( ord_lexordp @ A @ Us2 @ Vs )
=> ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs ) ) ) ) ).
% lexordp_append_leftI
thf(fact_379_lexordp__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
| ( Xs = Ys )
| ( ord_lexordp @ A @ Ys @ Xs ) ) ) ).
% lexordp_linear
thf(fact_380_lexordp__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ( ord_lexordp @ A @ Ys @ Zs )
=> ( ord_lexordp @ A @ Xs @ Zs ) ) ) ) ).
% lexordp_trans
thf(fact_381_lexordp__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ~ ( ord_lexordp @ A @ Ys @ Xs ) ) ) ).
% lexordp_antisym
thf(fact_382_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Us2: list @ A,Vs: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs ) )
=> ( ! [A6: A] :
~ ( ord_less @ A @ A6 @ A6 )
=> ( ord_lexordp @ A @ Us2 @ Vs ) ) ) ) ).
% lexordp_append_leftD
thf(fact_383_lexordp__append__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ord_lexordp @ A @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ) ).
% lexordp_append_rightI
thf(fact_384_rev_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rev @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev.simps(2)
thf(fact_385_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ? [X3: A,Vs2: list @ A] :
( Ys3
= ( append @ A @ Xs3 @ ( cons @ A @ X3 @ Vs2 ) ) )
| ? [Us: list @ A,A4: A,B3: A,Vs2: list @ A,Ws: list @ A] :
( ( ord_less @ A @ A4 @ B3 )
& ( Xs3
= ( append @ A @ Us @ ( cons @ A @ A4 @ Vs2 ) ) )
& ( Ys3
= ( append @ A @ Us @ ( cons @ A @ B3 @ Ws ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_386_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Us2: list @ A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( append @ A @ Us2 @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us2 @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_387_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_388_rev__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( rev @ A @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( rev @ B @ Xs ) ) ) ).
% rev_map
thf(fact_389_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_390_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_391_list__match__lel__lel,axiom,
! [A: $tType,C1: list @ A,Qs: A,C22: list @ A,C12: list @ A,Qs2: A,C23: list @ A] :
( ( ( append @ A @ C1 @ ( cons @ A @ Qs @ C22 ) )
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C23 ) ) )
=> ( ! [C21: list @ A] :
( ( C1
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C21 ) ) )
=> ( C23
!= ( append @ A @ C21 @ ( cons @ A @ Qs @ C22 ) ) ) )
=> ( ( ( C12 = C1 )
=> ( ( Qs2 = Qs )
=> ( C23 != C22 ) ) )
=> ~ ! [C212: list @ A] :
( ( C12
= ( append @ A @ C1 @ ( cons @ A @ Qs @ C212 ) ) )
=> ( C22
!= ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C23 ) ) ) ) ) ) ) ).
% list_match_lel_lel
thf(fact_392_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_393_append_Oleft__neutral,axiom,
! [A: $tType,A3: list @ A] :
( ( append @ A @ ( nil @ A ) @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_394_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_395_map__tailrec__rev,axiom,
! [A: $tType,B: $tType] :
( ( map_tailrec_rev @ B @ A )
= ( ^ [F3: B > A,As3: list @ B] : ( append @ A @ ( rev @ A @ ( map @ B @ A @ F3 @ As3 ) ) ) ) ) ).
% map_tailrec_rev
thf(fact_396_Misc_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F @ Ls )
= ( append @ A @ Fl @ Fl2 ) )
= ( ? [L4: list @ B,L5: list @ B] :
( ( Ls
= ( append @ B @ L4 @ L5 ) )
& ( ( map @ B @ A @ F @ L4 )
= Fl )
& ( ( map @ B @ A @ F @ L5 )
= Fl2 ) ) ) ) ).
% Misc.map_eq_append_conv
thf(fact_397_Misc_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F @ Ls ) )
= ( ? [L4: list @ B,L5: list @ B] :
( ( Ls
= ( append @ B @ L4 @ L5 ) )
& ( ( map @ B @ A @ F @ L4 )
= Fl )
& ( ( map @ B @ A @ F @ L5 )
= Fl2 ) ) ) ) ).
% Misc.append_eq_map_conv
thf(fact_398_map__eq__appendE,axiom,
! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F @ Ls )
= ( append @ A @ Fl @ Fl2 ) )
=> ~ ! [L2: list @ B,L6: list @ B] :
( ( Ls
= ( append @ B @ L2 @ L6 ) )
=> ( ( ( map @ B @ A @ F @ L2 )
= Fl )
=> ( ( map @ B @ A @ F @ L6 )
!= Fl2 ) ) ) ) ).
% map_eq_appendE
thf(fact_399_append__eq__mapE,axiom,
! [B: $tType,A: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F @ Ls ) )
=> ~ ! [L2: list @ B,L6: list @ B] :
( ( Ls
= ( append @ B @ L2 @ L6 ) )
=> ( ( ( map @ B @ A @ F @ L2 )
= Fl )
=> ( ( map @ B @ A @ F @ L6 )
!= Fl2 ) ) ) ) ).
% append_eq_mapE
thf(fact_400_List_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Ys: list @ A,Zs: list @ A] :
( ( ( map @ B @ A @ F @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ? [Us: list @ B,Vs2: list @ B] :
( ( Xs
= ( append @ B @ Us @ Vs2 ) )
& ( Ys
= ( map @ B @ A @ F @ Us ) )
& ( Zs
= ( map @ B @ A @ F @ Vs2 ) ) ) ) ) ).
% List.map_eq_append_conv
thf(fact_401_List_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ A,Zs: list @ A,F: B > A,Xs: list @ B] :
( ( ( append @ A @ Ys @ Zs )
= ( map @ B @ A @ F @ Xs ) )
= ( ? [Us: list @ B,Vs2: list @ B] :
( ( Xs
= ( append @ B @ Us @ Vs2 ) )
& ( Ys
= ( map @ B @ A @ F @ Us ) )
& ( Zs
= ( map @ B @ A @ F @ Vs2 ) ) ) ) ) ).
% List.append_eq_map_conv
thf(fact_402_star__or__dist2,axiom,
! [C4: assn,A5: assn,B4: assn] :
( ( times_times @ assn @ C4 @ ( sup_sup @ assn @ A5 @ B4 ) )
= ( sup_sup @ assn @ ( times_times @ assn @ C4 @ A5 ) @ ( times_times @ assn @ C4 @ B4 ) ) ) ).
% star_or_dist2
thf(fact_403_star__or__dist1,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( times_times @ assn @ ( sup_sup @ assn @ A5 @ B4 ) @ C4 )
= ( sup_sup @ assn @ ( times_times @ assn @ A5 @ C4 ) @ ( times_times @ assn @ B4 @ C4 ) ) ) ).
% star_or_dist1
thf(fact_404_ord_Olexordp__eq__pref,axiom,
! [A: $tType,Less: A > A > $o,U: list @ A,V: list @ A] : ( lexordp_eq @ A @ Less @ U @ ( append @ A @ U @ V ) ) ).
% ord.lexordp_eq_pref
thf(fact_405_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ! [X2: A] :
~ ( ord_less @ A @ X2 @ X2 )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_406_norm__assertion__simps_I6_J,axiom,
! [X: assn] :
( ( sup_sup @ assn @ X @ ( bot_bot @ assn ) )
= X ) ).
% norm_assertion_simps(6)
thf(fact_407_norm__assertion__simps_I5_J,axiom,
! [X: assn] :
( ( sup_sup @ assn @ ( bot_bot @ assn ) @ X )
= X ) ).
% norm_assertion_simps(5)
thf(fact_408_lexordp__eq__pref,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [U: list @ A,V: list @ A] : ( ord_lexordp_eq @ A @ U @ ( append @ A @ U @ V ) ) ) ).
% lexordp_eq_pref
thf(fact_409_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( ( linord4507533701916653071of_set @ A @ A5 )
= ( linord4507533701916653071of_set @ A @ B4 ) )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( A5 = B4 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_410_merge_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L23: list @ A] :
( ( merge @ A @ ( nil @ A ) @ L23 )
= L23 ) ) ).
% merge.simps(1)
thf(fact_411_lexordp__conv__lexordp__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ord_lexordp_eq @ A @ Xs3 @ Ys3 )
& ~ ( ord_lexordp_eq @ A @ Ys3 @ Xs3 ) ) ) ) ) ).
% lexordp_conv_lexordp_eq
thf(fact_412_lexordp__eq__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( Xs3 = Ys3 )
| ( ord_lexordp @ A @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp_eq_conv_lexord
thf(fact_413_lexordp__into__lexordp__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ord_lexordp_eq @ A @ Xs @ Ys ) ) ) ).
% lexordp_into_lexordp_eq
thf(fact_414_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_415_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys2: list @ A,Y2: A] :
( Xs
!= ( append @ A @ Ys2 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_416_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys5: list @ A] :
( ( ( cons @ A @ X @ Ys5 )
= Ys )
& ( Xs
= ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_417_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
& ( ( append @ A @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_418_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_419_neq__Nil__revE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [Ll: list @ A,E: A] :
( L
!= ( append @ A @ Ll @ ( cons @ A @ E @ ( nil @ A ) ) ) ) ) ).
% neq_Nil_revE
thf(fact_420_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 ) )
=> ( ! [X2: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_421_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
= ( ? [Xs3: list @ A,X3: A] :
( L
= ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_422_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 ) )
=> ( ! [X2: A,Y2: B] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( cons @ B @ Y2 @ ( nil @ B ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B] :
( ( Xs2
!= ( nil @ A ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) )
=> ( ! [X2: A,Y2: B,Ys2: list @ B] :
( ( Ys2
!= ( nil @ B ) )
=> ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ B ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_423_list__Cons__eq__append__cases,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_424_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_425_map__consI_I2_J,axiom,
! [B: $tType,A: $tType,W: list @ A,L: list @ A,F: B > A,Ww: list @ B,A3: B] :
( ( ( append @ A @ W @ L )
= ( append @ A @ ( map @ B @ A @ F @ Ww ) @ L ) )
=> ( ( cons @ A @ ( F @ A3 ) @ ( append @ A @ W @ L ) )
= ( append @ A @ ( map @ B @ A @ F @ ( cons @ B @ A3 @ Ww ) ) @ L ) ) ) ).
% map_consI(2)
thf(fact_426_SLN__left,axiom,
! [P: assn] :
( ( times_times @ assn @ sln @ P )
= P ) ).
% SLN_left
thf(fact_427_SLN__right,axiom,
! [P: assn] :
( ( times_times @ assn @ P @ sln )
= P ) ).
% SLN_right
thf(fact_428_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% lexordp.Cons
thf(fact_429_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ~ ( ord_less @ A @ Y @ X )
=> ( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_430_lexordp_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Y: A,Ys: list @ A] : ( ord_lexordp @ A @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% lexordp.Nil
thf(fact_431_SLN__def,axiom,
( sln
= ( one_one @ assn ) ) ).
% SLN_def
thf(fact_432_merge_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X12: A,X24: A,L12: list @ A,L23: list @ A] :
( ( ( ord_less @ A @ X12 @ X24 )
=> ( ( merge @ A @ ( cons @ A @ X12 @ L12 ) @ ( cons @ A @ X24 @ L23 ) )
= ( cons @ A @ X12 @ ( merge @ A @ L12 @ ( cons @ A @ X24 @ L23 ) ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X24 )
=> ( ( ( X12 = X24 )
=> ( ( merge @ A @ ( cons @ A @ X12 @ L12 ) @ ( cons @ A @ X24 @ L23 ) )
= ( cons @ A @ X12 @ ( merge @ A @ L12 @ L23 ) ) ) )
& ( ( X12 != X24 )
=> ( ( merge @ A @ ( cons @ A @ X12 @ L12 ) @ ( cons @ A @ X24 @ L23 ) )
= ( cons @ A @ X24 @ ( merge @ A @ ( cons @ A @ X12 @ L12 ) @ L23 ) ) ) ) ) ) ) ) ).
% merge.simps(3)
thf(fact_433_merge_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va: list @ A] :
( ( merge @ A @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
= ( cons @ A @ V @ Va ) ) ) ).
% merge.simps(2)
thf(fact_434_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp @ A @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y2: A,Ys2: list @ A] :
( A22
!= ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [X2: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( ord_less @ A @ X2 @ Y2 ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ~ ( ord_less @ A @ Y2 @ X2 )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_435_finite__Int,axiom,
! [A: $tType,F2: set @ A,G3: set @ A] :
( ( ( finite_finite2 @ A @ F2 )
| ( finite_finite2 @ A @ G3 ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F2 @ G3 ) ) ) ).
% finite_Int
thf(fact_436_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member2 @ A @ X2 @ S )
& ~ ? [Xa2: A] :
( ( member2 @ A @ Xa2 @ S )
& ( ord_less @ A @ Xa2 @ X2 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_437_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X5: set @ A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ X5 )
=> ? [Xa2: A] :
( ( member2 @ A @ Xa2 @ X5 )
& ( ord_less @ A @ X2 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ A @ X5 ) ) ) ) ).
% infinite_growing
thf(fact_438_FI__init,axiom,
! [P: assn,Q: assn,F2: assn] :
( ( fi @ ( nil @ ( product_prod @ assn @ assn ) ) @ ( times_times @ assn @ sln @ P ) @ ( times_times @ assn @ sln @ Q ) @ sln @ sln @ F2 )
=> ( fI_QUERY @ P @ Q @ F2 ) ) ).
% FI_init
thf(fact_439_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_440_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_441_finite__transitivity__chain,axiom,
! [A: $tType,A5: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X2: A] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: A,Y2: A,Z4: A] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z4 )
=> ( R @ X2 @ Z4 ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ A5 )
=> ? [Y4: A] :
( ( member2 @ A @ Y4 @ A5 )
& ( R @ X2 @ Y4 ) ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_442_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 ) ) )
=> ! [L2: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L2 @ ( nil @ ( list @ A ) ) ) )
=> ( Y != L2 ) ) )
=> ( ! [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 ) )
=> ! [L2: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L2 @ ( nil @ ( list @ A ) ) ) )
=> ( Y
!= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L2 @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) ) ) ) )
=> ~ ! [L1: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) )
=> ( Y
!= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ X ) @ Ls2 ) ) ) ) ) ) ) ) ) ).
% merge_list.elims
thf(fact_443_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( list @ A ),X: B,Xs: list @ B] :
( ( maps @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
= ( append @ A @ ( F @ X ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_444_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_445_concat__eq__append__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys: list @ A,Zs: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Xss2
= ( nil @ ( list @ A ) ) )
=> ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( nil @ A ) ) ) )
& ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs3: list @ A,Xs5: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append @ A @ Xs5 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_446_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_447_concat__append,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys ) )
= ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys ) ) ) ).
% concat_append
thf(fact_448_assn__aci_I8_J,axiom,
! [X: assn,Y: assn] :
( ( sup_sup @ assn @ X @ ( sup_sup @ assn @ X @ Y ) )
= ( sup_sup @ assn @ X @ Y ) ) ).
% assn_aci(8)
thf(fact_449_assn__aci_I7_J,axiom,
! [X: assn,Y: assn,Z2: assn] :
( ( sup_sup @ assn @ X @ ( sup_sup @ assn @ Y @ Z2 ) )
= ( sup_sup @ assn @ Y @ ( sup_sup @ assn @ X @ Z2 ) ) ) ).
% assn_aci(7)
thf(fact_450_assn__aci_I5_J,axiom,
( ( sup_sup @ assn )
= ( ^ [X3: assn,Y3: assn] : ( sup_sup @ assn @ Y3 @ X3 ) ) ) ).
% assn_aci(5)
thf(fact_451_norm__assertion__simps_I32_J,axiom,
! [X: assn] :
( ( sup_sup @ assn @ X @ X )
= X ) ).
% norm_assertion_simps(32)
thf(fact_452_norm__assertion__simps_I15_J,axiom,
! [X: assn,Y: assn,Z2: assn] :
( ( sup_sup @ assn @ ( sup_sup @ assn @ X @ Y ) @ Z2 )
= ( sup_sup @ assn @ X @ ( sup_sup @ assn @ Y @ Z2 ) ) ) ).
% norm_assertion_simps(15)
thf(fact_453_concat__map__maps,axiom,
! [A: $tType,B: $tType,F: B > ( list @ A ),Xs: list @ B] :
( ( concat @ A @ ( map @ B @ ( list @ A ) @ F @ Xs ) )
= ( maps @ B @ A @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_454_maps__def,axiom,
! [B: $tType,A: $tType] :
( ( maps @ A @ B )
= ( ^ [F3: A > ( list @ B ),Xs3: list @ A] : ( concat @ B @ ( map @ A @ ( list @ B ) @ F3 @ Xs3 ) ) ) ) ).
% maps_def
thf(fact_455_map__concat,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ ( list @ B )] :
( ( map @ B @ A @ F @ ( concat @ B @ Xs ) )
= ( concat @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_456_rev__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( rev @ A @ ( concat @ A @ Xs ) )
= ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ ( rev @ ( list @ A ) @ Xs ) ) ) ) ).
% rev_concat
thf(fact_457_List_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind @ A @ B )
= ( ^ [Xs3: list @ A,F3: A > ( list @ B )] : ( concat @ B @ ( map @ A @ ( list @ B ) @ F3 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_458_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_459_rotate1__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( rotate1 @ A @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( rotate1 @ B @ Xs ) ) ) ).
% rotate1_map
thf(fact_460_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_461_concat_Osimps_I2_J,axiom,
! [A: $tType,X: list @ A,Xs: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X @ Xs ) )
= ( append @ A @ X @ ( concat @ A @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_462_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( maps @ B @ A @ F @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_463_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_464_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_465_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_466_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_467_merge__list_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc22: list @ ( list @ A ),L12: list @ A,L23: list @ A,Ls: list @ ( list @ A )] :
( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ Acc22 ) @ Ls ) ) ) ).
% merge_list.simps(5)
thf(fact_468_concat__eq__appendD,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys: list @ A,Zs: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys @ Zs ) )
=> ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs2: list @ A,Xs4: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs4 ) @ Xss23 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append @ A @ Xs4 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_469_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X6: A] :
( ( member2 @ A @ X6 @ S )
& ( ord_less @ B @ ( F @ X6 ) @ ( F @ ( lattic7623131987881927897min_on @ A @ B @ F @ S ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_470_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ A @ ( lattic7623131987881927897min_on @ A @ B @ F @ S ) @ S ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_471_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B4 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_472_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic7752659483105999362nf_fin @ A @ B4 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_473_product_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Uu: list @ B] :
( ( product @ A @ B @ ( nil @ A ) @ Uu )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% product.simps(1)
thf(fact_474_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_475_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% last_snoc
thf(fact_476_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( ord_less @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_477_concat__conv__foldr,axiom,
! [A: $tType] :
( ( concat @ A )
= ( ^ [Xss3: list @ ( list @ A )] : ( foldr @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss3 @ ( nil @ A ) ) ) ) ).
% concat_conv_foldr
thf(fact_478_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( ord_less @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_479_SuccI,axiom,
! [A: $tType,Kl: list @ A,K: A,Kl2: set @ ( list @ A )] :
( ( member2 @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl2 )
=> ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_480_foldr__append,axiom,
! [B: $tType,A: $tType,F: B > A > A,Xs: list @ B,Ys: list @ B,A3: A] :
( ( foldr @ B @ A @ F @ ( append @ B @ Xs @ Ys ) @ A3 )
= ( foldr @ B @ A @ F @ Xs @ ( foldr @ B @ A @ F @ Ys @ A3 ) ) ) ).
% foldr_append
thf(fact_481_last__appendL,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_482_last__appendR,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ).
% last_appendR
thf(fact_483_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ A3 @ A5 )
=> ( ( inf_inf @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= A3 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_484_append__butlast__last__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_485_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_486_map__butlast,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( map @ B @ A @ F @ ( butlast @ B @ Xs ) )
= ( butlast @ A @ ( map @ B @ A @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_487_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= Ys )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_488_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_489_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A5 ) ) ) ) ).
% Min_in
thf(fact_490_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ A5 ) ) ) ) ).
% Max_in
thf(fact_491_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_492_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_493_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_494_last__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ B @ ( map @ A @ B @ F @ Xs ) )
= ( F @ ( last @ A @ Xs ) ) ) ) ).
% last_map
thf(fact_495_last__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ) ).
% last_append
thf(fact_496_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs4: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Xs4 @ Ss ) )
& ( Ys
= ( append @ A @ Ys4 @ Ss ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys4
= ( nil @ A ) )
| ( ( last @ A @ Xs4 )
!= ( last @ A @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_497_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_498_butlast__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_499_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_500_Misc_Ofoldr__Cons,axiom,
! [A: $tType,Xs: list @ A] :
( ( foldr @ A @ ( list @ A ) @ ( cons @ A ) @ Xs @ ( nil @ A ) )
= Xs ) ).
% Misc.foldr_Cons
thf(fact_501_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ A5 )
& ( ord_less @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_502_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ A5 )
& ( ord_less @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_503_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_504_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_505_SuccD,axiom,
! [A: $tType,K: A,Kl2: set @ ( list @ A ),Kl: list @ A] :
( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl ) )
=> ( member2 @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_506_shift__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_shift @ A @ B )
= ( ^ [Lab: ( list @ A ) > B,K2: A,Kl3: list @ A] : ( Lab @ ( cons @ A @ K2 @ Kl3 ) ) ) ) ).
% shift_def
thf(fact_507_empty__Shift,axiom,
! [A: $tType,Kl2: set @ ( list @ A ),K: A] :
( ( member2 @ ( list @ A ) @ ( nil @ A ) @ Kl2 )
=> ( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ ( nil @ A ) ) )
=> ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_508_Succ__Shift,axiom,
! [A: $tType,Kl2: set @ ( list @ A ),K: A,Kl: list @ A] :
( ( bNF_Greatest_Succ @ A @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) @ Kl )
= ( bNF_Greatest_Succ @ A @ Kl2 @ ( cons @ A @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_509_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_510_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_511_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_512_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B4 ) @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_513_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B4 ) @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_514_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member2 @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_515_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member2 @ A @ ( sup_sup @ A @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member2 @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_516_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_517_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_518_subsetI,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ A5 )
=> ( member2 @ A @ X2 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% subsetI
thf(fact_519_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( A5 = B4 ) ) ) ).
% subset_antisym
thf(fact_520_insertCI,axiom,
! [A: $tType,A3: A,B4: set @ A,B2: A] :
( ( ~ ( member2 @ A @ A3 @ B4 )
=> ( A3 = B2 ) )
=> ( member2 @ A @ A3 @ ( insert2 @ A @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_521_insert__iff,axiom,
! [A: $tType,A3: A,B2: A,A5: set @ A] :
( ( member2 @ A @ A3 @ ( insert2 @ A @ B2 @ A5 ) )
= ( ( A3 = B2 )
| ( member2 @ A @ A3 @ A5 ) ) ) ).
% insert_iff
thf(fact_522_insert__absorb2,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ X @ A5 ) )
= ( insert2 @ A @ X @ A5 ) ) ).
% insert_absorb2
thf(fact_523_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C3 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% inf.bounded_iff
thf(fact_524_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_525_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_526_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_527_empty__subsetI,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% empty_subsetI
thf(fact_528_subset__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_529_singletonI,axiom,
! [A: $tType,A3: A] : ( member2 @ A @ A3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_530_insert__subset,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A5 ) @ B4 )
= ( ( member2 @ A @ X @ B4 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% insert_subset
thf(fact_531_Int__subset__iff,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C4 @ A5 )
& ( ord_less_eq @ ( set @ A ) @ C4 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_532_Int__insert__right__if1,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ A3 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ B4 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_533_Int__insert__right__if0,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ~ ( member2 @ A @ A3 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_534_insert__inter__insert,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A5 ) @ ( insert2 @ A @ A3 @ B4 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_535_Int__insert__left__if1,axiom,
! [A: $tType,A3: A,C4: set @ A,B4: set @ A] :
( ( member2 @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B4 ) @ C4 )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_536_Int__insert__left__if0,axiom,
! [A: $tType,A3: A,C4: set @ A,B4: set @ A] :
( ~ ( member2 @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B4 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_537_Un__subset__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_538_Un__insert__left,axiom,
! [A: $tType,A3: A,B4: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ B4 ) @ C4 )
= ( insert2 @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_539_Un__insert__right,axiom,
! [A: $tType,A5: set @ A,A3: A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ B4 ) )
= ( insert2 @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_540_psubsetI,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( A5 != B4 )
=> ( ord_less @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% psubsetI
thf(fact_541_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A3: A,A5: set @ A] :
( ( ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ A3 @ A5 ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_542_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A5: set @ A,B2: A] :
( ( ( insert2 @ A @ A3 @ A5 )
= ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_543_disjoint__insert_I2_J,axiom,
! [A: $tType,A5: set @ A,B2: A,B4: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ ( insert2 @ A @ B2 @ B4 ) ) )
= ( ~ ( member2 @ A @ B2 @ A5 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_544_disjoint__insert_I1_J,axiom,
! [A: $tType,B4: set @ A,A3: A,A5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B4 @ ( insert2 @ A @ A3 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member2 @ A @ A3 @ B4 )
& ( ( inf_inf @ ( set @ A ) @ B4 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_545_insert__disjoint_I2_J,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A5 ) @ B4 ) )
= ( ~ ( member2 @ A @ A3 @ B4 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_546_insert__disjoint_I1_J,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ A5 ) @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member2 @ A @ A3 @ B4 )
& ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_547_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_548_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Min_singleton
thf(fact_549_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_550_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_551_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_552_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_553_in__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member2 @ A @ X @ A5 )
=> ( member2 @ A @ X @ B4 ) ) ) ).
% in_mono
thf(fact_554_insertE,axiom,
! [A: $tType,A3: A,B2: A,A5: set @ A] :
( ( member2 @ A @ A3 @ ( insert2 @ A @ B2 @ A5 ) )
=> ( ( A3 != B2 )
=> ( member2 @ A @ A3 @ A5 ) ) ) ).
% insertE
thf(fact_555_subsetD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member2 @ A @ C3 @ A5 )
=> ( member2 @ A @ C3 @ B4 ) ) ) ).
% subsetD
thf(fact_556_insertI1,axiom,
! [A: $tType,A3: A,B4: set @ A] : ( member2 @ A @ A3 @ ( insert2 @ A @ A3 @ B4 ) ) ).
% insertI1
thf(fact_557_insertI2,axiom,
! [A: $tType,A3: A,B4: set @ A,B2: A] :
( ( member2 @ A @ A3 @ B4 )
=> ( member2 @ A @ A3 @ ( insert2 @ A @ B2 @ B4 ) ) ) ).
% insertI2
thf(fact_558_equalityE,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% equalityE
thf(fact_559_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
! [X3: A] :
( ( member2 @ A @ X3 @ A7 )
=> ( member2 @ A @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_560_equalityD1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% equalityD1
thf(fact_561_equalityD2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ).
% equalityD2
thf(fact_562_Set_Oset__insert,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( member2 @ A @ X @ A5 )
=> ~ ! [B7: set @ A] :
( ( A5
= ( insert2 @ A @ X @ B7 ) )
=> ( member2 @ A @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_563_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
! [T2: A] :
( ( member2 @ A @ T2 @ A7 )
=> ( member2 @ A @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_564_insert__mono,axiom,
! [A: $tType,C4: set @ A,D2: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ C4 ) @ ( insert2 @ A @ A3 @ D2 ) ) ) ).
% insert_mono
thf(fact_565_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_566_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_567_insert__ident,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ~ ( member2 @ A @ X @ A5 )
=> ( ~ ( member2 @ A @ X @ B4 )
=> ( ( ( insert2 @ A @ X @ A5 )
= ( insert2 @ A @ X @ B4 ) )
= ( A5 = B4 ) ) ) ) ).
% insert_ident
thf(fact_568_subset__trans,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_569_insert__absorb,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member2 @ A @ A3 @ A5 )
=> ( ( insert2 @ A @ A3 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_570_insert__eq__iff,axiom,
! [A: $tType,A3: A,A5: set @ A,B2: A,B4: set @ A] :
( ~ ( member2 @ A @ A3 @ A5 )
=> ( ~ ( member2 @ A @ B2 @ B4 )
=> ( ( ( insert2 @ A @ A3 @ A5 )
= ( insert2 @ A @ B2 @ B4 ) )
= ( ( ( A3 = B2 )
=> ( A5 = B4 ) )
& ( ( A3 != B2 )
=> ? [C5: set @ A] :
( ( A5
= ( insert2 @ A @ B2 @ C5 ) )
& ~ ( member2 @ A @ B2 @ C5 )
& ( B4
= ( insert2 @ A @ A3 @ C5 ) )
& ~ ( member2 @ A @ A3 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_571_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z5: set @ A] : Y5 = Z5 )
= ( ^ [A7: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_572_subset__insert,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ~ ( member2 @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% subset_insert
thf(fact_573_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( insert2 @ A @ X @ ( insert2 @ A @ Y @ A5 ) )
= ( insert2 @ A @ Y @ ( insert2 @ A @ X @ A5 ) ) ) ).
% insert_commute
thf(fact_574_subset__insertI,axiom,
! [A: $tType,B4: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert2 @ A @ A3 @ B4 ) ) ).
% subset_insertI
thf(fact_575_subset__insertI2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ B2 @ B4 ) ) ) ).
% subset_insertI2
thf(fact_576_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_577_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member2 @ A @ A3 @ A5 )
=> ? [B7: set @ A] :
( ( A5
= ( insert2 @ A @ A3 @ B7 ) )
& ~ ( member2 @ A @ A3 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_578_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( ord_less_eq @ A @ A3 @ B2 ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( B2 != A3 ) ) ) ) ).
% nle_le
thf(fact_579_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_580_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_581_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_582_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_583_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_584_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_585_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_586_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_587_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
& ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_588_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_589_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_590_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% antisym
thf(fact_591_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_592_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_593_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_594_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_595_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
& ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_596_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_597_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C3 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_598_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ( C3 = D3 )
=> ( ord_less_eq @ A @ A3 @ D3 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_599_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_600_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ S )
=> ( P @ X3 ) ) ) ) ).
% subset_Collect_conv
thf(fact_601_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_602_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B2: B,C3: B] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_603_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_604_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_605_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_606_subset__singletonD,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( A5
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_607_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_608_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B,S2: set @ B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [Y4: B] :
( ( member2 @ B @ Y4 @ S2 )
=> ( ord_less_eq @ A @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert2 @ B @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_609_finite__subset__induct_H,axiom,
! [A: $tType,F2: set @ A,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ F2 @ A5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F4: set @ A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( member2 @ A @ A6 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A5 )
=> ( ~ ( member2 @ A @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert2 @ A @ A6 @ F4 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_610_finite__subset__induct,axiom,
! [A: $tType,F2: set @ A,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ F2 @ A5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F4: set @ A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( member2 @ A @ A6 @ A5 )
=> ( ~ ( member2 @ A @ A6 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert2 @ A @ A6 @ F4 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_611_singleton__inject,axiom,
! [A: $tType,A3: A,B2: A] :
( ( ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B2 ) ) ).
% singleton_inject
thf(fact_612_insert__not__empty,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( insert2 @ A @ A3 @ A5 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_613_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B2: A,C3: A,D3: A] :
( ( ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ C3 @ ( insert2 @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C3 )
& ( B2 = D3 ) )
| ( ( A3 = D3 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_614_singleton__iff,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member2 @ A @ B2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_615_singletonD,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member2 @ A @ B2 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_616_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_617_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_618_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( ord_less @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% nless_le
thf(fact_619_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_620_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_621_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y: A] :
( ! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_622_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z2: A] :
( ! [X2: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_623_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% less_le_not_le
thf(fact_624_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_625_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_626_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_627_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans1
thf(fact_628_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% order.strict_trans2
thf(fact_629_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
& ~ ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_630_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W3: A] :
( ( ord_less @ A @ Z2 @ W3 )
=> ( ( ord_less @ A @ W3 @ X )
=> ( ord_less_eq @ A @ Y @ W3 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_631_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W3: A] :
( ( ord_less @ A @ X @ W3 )
=> ( ( ord_less @ A @ W3 @ Y )
=> ( ord_less_eq @ A @ W3 @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_632_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_633_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_634_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_635_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_636_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
& ~ ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_637_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_638_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_639_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ) ).
% order_le_less
thf(fact_640_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ) ).
% order_less_le
thf(fact_641_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_642_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_643_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_644_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_le_neq_trans
thf(fact_645_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 != B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_neq_le_trans
thf(fact_646_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_647_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_648_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less @ B @ X2 @ Y2 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_649_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F @ B2 ) @ C3 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_650_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C3: B] :
( ( ord_less @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_651_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C3: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C3 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_652_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_653_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_654_Int__insert__right,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ( ( member2 @ A @ A3 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ B4 ) )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) )
& ( ~ ( member2 @ A @ A3 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_655_Int__insert__left,axiom,
! [A: $tType,A3: A,C4: set @ A,B4: set @ A] :
( ( ( member2 @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B4 ) @ C4 )
= ( insert2 @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) )
& ( ~ ( member2 @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert2 @ A @ A3 @ B4 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_656_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_657_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_658_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_659_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_660_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% inf.coboundedI2
thf(fact_661_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% inf.coboundedI1
thf(fact_662_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( ( inf_inf @ A @ A4 @ B3 )
= B3 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_663_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( ( inf_inf @ A @ A4 @ B3 )
= A4 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_664_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 ) ) ).
% inf.cobounded2
thf(fact_665_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ A3 ) ) ).
% inf.cobounded1
thf(fact_666_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( A4
= ( inf_inf @ A @ A4 @ B3 ) ) ) ) ) ).
% inf.order_iff
thf(fact_667_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Z2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ) ).
% inf_greatest
thf(fact_668_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ) ).
% inf.boundedI
thf(fact_669_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% inf.boundedE
thf(fact_670_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( inf_inf @ A @ X @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_671_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( inf_inf @ A @ X @ Y )
= X ) ) ) ).
% inf_absorb1
thf(fact_672_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
thf(fact_673_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb1
thf(fact_674_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y3: A] :
( ( inf_inf @ A @ X3 @ Y3 )
= X3 ) ) ) ) ).
% le_iff_inf
thf(fact_675_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F: A > A > A,X: A,Y: A] :
( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: A,Y2: A,Z4: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ X2 @ Z4 )
=> ( ord_less_eq @ A @ X2 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf @ A @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_676_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( inf_inf @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% inf.orderI
thf(fact_677_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( inf_inf @ A @ A3 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_678_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI2
thf(fact_679_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI1
thf(fact_680_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ ( inf_inf @ A @ C3 @ D3 ) ) ) ) ) ).
% inf_mono
thf(fact_681_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_682_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A3 )
=> ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% le_infE
thf(fact_683_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_684_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_le1
thf(fact_685_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_sup_ord(1)
thf(fact_686_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_687_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_688_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_689_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A3 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_690_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_691_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_692_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_693_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_694_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_695_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A3: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C3 @ D3 ) @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_696_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ ( sup_sup @ A @ C3 @ D3 ) ) ) ) ) ).
% sup_mono
thf(fact_697_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A,Z2: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X ) ) ) ) ).
% sup_least
thf(fact_698_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y3: A] :
( ( sup_sup @ A @ X3 @ Y3 )
= Y3 ) ) ) ) ).
% le_iff_sup
thf(fact_699_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_700_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% sup.orderI
thf(fact_701_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F: A > A > A,X: A,Y: A] :
( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: A,Y2: A,Z4: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ Z4 @ X2 )
=> ( ord_less_eq @ A @ ( F @ Y2 @ Z4 ) @ X2 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_702_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_703_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_704_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_705_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_706_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_707_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_708_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( A4
= ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% sup.order_iff
thf(fact_709_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_710_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_711_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( ( sup_sup @ A @ A4 @ B3 )
= A4 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_712_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( ( sup_sup @ A @ A4 @ B3 )
= B3 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_713_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_714_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_715_Int__Collect__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ A5 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B4 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_716_Int__greatest,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_717_Int__absorb2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= A5 ) ) ).
% Int_absorb2
thf(fact_718_Int__absorb1,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_719_Int__lower2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_720_Int__lower1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ A5 ) ).
% Int_lower1
thf(fact_721_Int__mono,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B4: set @ A,D2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( inf_inf @ ( set @ A ) @ C4 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_722_inter__eq__subsetI,axiom,
! [A: $tType,S: set @ A,S3: set @ A,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ S3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ S3 )
= ( inf_inf @ ( set @ A ) @ B4 @ S3 ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ S )
= ( inf_inf @ ( set @ A ) @ B4 @ S ) ) ) ) ).
% inter_eq_subsetI
thf(fact_723_Un__mono,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B4: set @ A,D2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_724_Un__least,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C4 ) ) ) ).
% Un_least
thf(fact_725_Un__upper1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_upper1
thf(fact_726_Un__upper2,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_upper2
thf(fact_727_Un__absorb1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_728_Un__absorb2,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= A5 ) ) ).
% Un_absorb2
thf(fact_729_subset__UnE,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ~ ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ A5 )
=> ! [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ B4 )
=> ( C4
!= ( sup_sup @ ( set @ A ) @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_730_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A7 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_731_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B5 )
| ( A7 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_732_subset__psubset__trans,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_733_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
& ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_734_psubset__subset__trans,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_735_psubset__imp__subset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_736_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
& ( A7 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_737_psubsetE,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% psubsetE
thf(fact_738_less__eq__assn__def,axiom,
( ( ord_less_eq @ assn )
= ( ^ [A4: assn,B3: assn] :
( A4
= ( inf_inf @ assn @ A4 @ B3 ) ) ) ) ).
% less_eq_assn_def
thf(fact_739_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: set @ A,N4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M @ N4 )
=> ( ( M
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M ) @ ( lattic643756798349783984er_Max @ A @ N4 ) ) ) ) ) ) ).
% Max_mono
thf(fact_740_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ ( lattic643756798349783984er_Max @ A @ B4 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_741_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: set @ A,N4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M @ N4 )
=> ( ( M
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N4 ) @ ( lattic643756798350308766er_Min @ A @ M ) ) ) ) ) ) ).
% Min_antimono
thf(fact_742_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B4 ) @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_743_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B4 ) @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_744_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B4 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_745_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A5: set @ A] :
( ! [A9: set @ A] :
( ~ ( finite_finite2 @ A @ A9 )
=> ( P @ A9 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,F4: set @ A] :
( ( finite_finite2 @ A @ F4 )
=> ( ~ ( member2 @ A @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert2 @ A @ X2 @ F4 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ).
% infinite_finite_induct
thf(fact_746_finite__ne__induct,axiom,
! [A: $tType,F2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F2 )
=> ( ( F2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] : ( P @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X2: A,F4: set @ A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( F4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member2 @ A @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert2 @ A @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_747_finite__induct,axiom,
! [A: $tType,F2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,F4: set @ A] :
( ( finite_finite2 @ A @ F4 )
=> ( ~ ( member2 @ A @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert2 @ A @ X2 @ F4 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_748_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,B3: A] :
( ( A4
= ( insert2 @ A @ B3 @ A7 ) )
& ( finite_finite2 @ A @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_749_finite_Ocases,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A9: set @ A] :
( ? [A6: A] :
( A3
= ( insert2 @ A @ A6 @ A9 ) )
=> ~ ( finite_finite2 @ A @ A9 ) ) ) ) ).
% finite.cases
thf(fact_750_insert__is__Un,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A4: A] : ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_751_Un__singleton__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A5
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A5
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_752_singleton__Un__iff,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ( ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A5
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A5
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_753_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member2 @ A @ X2 @ A5 )
& ! [Xa2: A] :
( ( member2 @ A @ Xa2 @ A5 )
=> ( ( ord_less_eq @ A @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_754_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member2 @ A @ X2 @ A5 )
& ! [Xa2: A] :
( ( member2 @ A @ Xa2 @ A5 )
=> ( ( ord_less_eq @ A @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_755_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% distrib_inf_le
thf(fact_756_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% distrib_sup_le
thf(fact_757_disjoint__mono,axiom,
! [A: $tType,A3: set @ A,A10: set @ A,B2: set @ A,B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ A10 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ B9 )
=> ( ( ( inf_inf @ ( set @ A ) @ A10 @ B9 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% disjoint_mono
thf(fact_758_Un__Int__assoc__eq,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C4 @ A5 ) ) ).
% Un_Int_assoc_eq
thf(fact_759_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B6: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X6: A] :
( ( member2 @ A @ X6 @ A9 )
=> ( ord_less @ A @ X6 @ B6 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert2 @ A @ B6 @ A9 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_760_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B6: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X6: A] :
( ( member2 @ A @ X6 @ A9 )
=> ( ord_less @ A @ B6 @ X6 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert2 @ A @ B6 @ A9 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_761_ShiftD,axiom,
! [A: $tType,Kl: list @ A,Kl2: set @ ( list @ A ),K: A] :
( ( member2 @ ( list @ A ) @ Kl @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) )
=> ( member2 @ ( list @ A ) @ ( cons @ A @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_762_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member2 @ A @ A6 @ A5 )
=> ( ord_less_eq @ A @ A6 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_763_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
=> ! [A11: A] :
( ( member2 @ A @ A11 @ A5 )
=> ( ord_less_eq @ A @ A11 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_764_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ( member2 @ A @ M2 @ A5 )
& ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ X3 @ M2 ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_765_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_766_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A5 )
= M2 )
= ( ( member2 @ A @ M2 @ A5 )
& ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ X3 @ M2 ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_767_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member2 @ A @ A6 @ A5 )
=> ( ord_less_eq @ A @ X @ A6 ) )
=> ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_768_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
=> ! [A11: A] :
( ( member2 @ A @ A11 @ A5 )
=> ( ord_less_eq @ A @ X @ A11 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_769_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ( member2 @ A @ M2 @ A5 )
& ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ M2 @ X3 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_770_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_771_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A5 )
= M2 )
= ( ( member2 @ A @ M2 @ A5 )
& ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ M2 @ X3 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_772_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ X3 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_773_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_774_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member2 @ A @ A6 @ A5 )
=> ( ord_less_eq @ A @ A6 @ X ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_775_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X )
=> ! [A11: A] :
( ( member2 @ A @ A11 @ A5 )
=> ( ord_less_eq @ A @ A11 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_776_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member2 @ A @ A6 @ A5 )
=> ( ord_less_eq @ A @ X @ A6 ) )
=> ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_777_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
=> ! [A11: A] :
( ( member2 @ A @ A11 @ A5 )
=> ( ord_less_eq @ A @ X @ A11 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_778_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S: set @ A,Y: A,F: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member2 @ A @ Y @ S )
=> ( ord_less_eq @ B @ ( F @ ( lattic7623131987881927897min_on @ A @ B @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_779_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member2 @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_780_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member2 @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member2 @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_781_mergesort__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( mergesort @ A )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% mergesort_def
thf(fact_782_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_783_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_784_subset__emptyI,axiom,
! [A: $tType,A5: set @ A] :
( ! [X2: A] :
~ ( member2 @ A @ X2 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_785_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_786_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert2 @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_787_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_788_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_789_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_790_is__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( is_singleton @ A @ A5 )
=> ~ ! [X2: A] :
( A5
!= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_791_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B5: A > B,X3: A] : ( minus_minus @ B @ ( A7 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% minus_apply
thf(fact_792_Diff__idemp,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ).
% Diff_idemp
thf(fact_793_Diff__iff,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( ( member2 @ A @ C3 @ A5 )
& ~ ( member2 @ A @ C3 @ B4 ) ) ) ).
% Diff_iff
thf(fact_794_DiffI,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ A5 )
=> ( ~ ( member2 @ A @ C3 @ B4 )
=> ( member2 @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% DiffI
thf(fact_795_Diff__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Diff_empty
thf(fact_796_empty__Diff,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_797_Diff__cancel,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_798_insert__Diff1,axiom,
! [A: $tType,X: A,B4: set @ A,A5: set @ A] :
( ( member2 @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A5 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_799_Diff__insert0,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ~ ( member2 @ A @ X @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ B4 ) )
= ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_800_Un__Diff__cancel2,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) @ A5 )
= ( sup_sup @ ( set @ A ) @ B4 @ A5 ) ) ).
% Un_Diff_cancel2
thf(fact_801_Un__Diff__cancel,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_802_Diff__eq__empty__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_803_insert__Diff__single,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert2 @ A @ A3 @ A5 ) ) ).
% insert_Diff_single
thf(fact_804_Diff__disjoint,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_805_less__assn__def,axiom,
( ( ord_less @ assn )
= ( ^ [A4: assn,B3: assn] :
( ( ord_less_eq @ assn @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% less_assn_def
thf(fact_806_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_807_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C3 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_808_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( A3 = B2 )
= ( C3 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_809_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B5: A > B,X3: A] : ( minus_minus @ B @ ( A7 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% fun_diff_def
thf(fact_810_DiffD2,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
=> ~ ( member2 @ A @ C3 @ B4 ) ) ).
% DiffD2
thf(fact_811_DiffD1,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
=> ( member2 @ A @ C3 @ A5 ) ) ).
% DiffD1
thf(fact_812_DiffE,axiom,
! [A: $tType,C3: A,A5: set @ A,B4: set @ A] :
( ( member2 @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
=> ~ ( ( member2 @ A @ C3 @ A5 )
=> ( member2 @ A @ C3 @ B4 ) ) ) ).
% DiffE
thf(fact_813_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
= ( ord_less_eq @ A @ C3 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_814_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% diff_right_mono
thf(fact_815_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A3 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_816_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ D3 @ C3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_817_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_818_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C3 @ A3 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_819_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
= ( ord_less @ A @ C3 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_820_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ D3 @ C3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_821_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% left_diff_distrib
thf(fact_822_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% right_diff_distrib
thf(fact_823_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C3 ) @ A3 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C3 @ A3 ) ) ) ) ).
% left_diff_distrib'
thf(fact_824_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% right_diff_distrib'
thf(fact_825_double__diff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B4 @ ( minus_minus @ ( set @ A ) @ C4 @ A5 ) )
= A5 ) ) ) ).
% double_diff
thf(fact_826_Diff__subset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ A5 ) ).
% Diff_subset
thf(fact_827_Diff__mono,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,D2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D2 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_828_insert__Diff__if,axiom,
! [A: $tType,X: A,B4: set @ A,A5: set @ A] :
( ( ( member2 @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A5 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) )
& ( ~ ( member2 @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A5 ) @ B4 )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_829_Int__Diff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Int_Diff
thf(fact_830_Diff__Int2,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_831_Diff__Diff__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_832_Diff__Int__distrib,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ C4 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C4 @ A5 ) @ ( inf_inf @ ( set @ A ) @ C4 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_833_Diff__Int__distrib2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Diff_Int_distrib2
thf(fact_834_set__diff__diff__left,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( minus_minus @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% set_diff_diff_left
thf(fact_835_Un__Diff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ C4 ) @ ( minus_minus @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Un_Diff
thf(fact_836_psubset__imp__ex__mem,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ? [B6: A] : ( member2 @ A @ B6 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_837_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_838_subset__minus__empty,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_839_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ~ ( member2 @ A @ X @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A5 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_840_Diff__insert2,axiom,
! [A: $tType,A5: set @ A,A3: A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_841_insert__Diff,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member2 @ A @ A3 @ A5 )
=> ( ( insert2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_842_Diff__insert,axiom,
! [A: $tType,A5: set @ A,A3: A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_843_insert__minus__eq,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( X != Y )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A5 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% insert_minus_eq
thf(fact_844_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X5: set @ A] :
( ~ ( member2 @ A @ X @ X5 )
=> ( ( minus_minus @ ( set @ A ) @ X5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X5 ) ) ).
% set_minus_singleton_eq
thf(fact_845_subset__Diff__insert,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X: A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ ( insert2 @ A @ X @ C4 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ C4 ) )
& ~ ( member2 @ A @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_846_Int__Diff__disjoint,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_847_Diff__triv,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ B4 )
= A5 ) ) ).
% Diff_triv
thf(fact_848_disjoint__alt__simp1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B4 )
= A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp1
thf(fact_849_disjoint__alt__simp2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B4 )
!= A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp2
thf(fact_850_Diff__subset__conv,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_851_Diff__partition,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_852_Diff__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ ( minus_minus @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Diff_Un
thf(fact_853_Diff__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ ( minus_minus @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Diff_Int
thf(fact_854_Int__Diff__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= A5 ) ).
% Int_Diff_Un
thf(fact_855_Un__Diff__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= A5 ) ).
% Un_Diff_Int
thf(fact_856_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( A7
= ( insert2 @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_857_is__singletonI_H,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] :
( ( member2 @ A @ X2 @ A5 )
=> ( ( member2 @ A @ Y2 @ A5 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton @ A @ A5 ) ) ) ).
% is_singletonI'
thf(fact_858_infinite__remove,axiom,
! [A: $tType,S: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_859_infinite__coinduct,axiom,
! [A: $tType,X5: ( set @ A ) > $o,A5: set @ A] :
( ( X5 @ A5 )
=> ( ! [A9: set @ A] :
( ( X5 @ A9 )
=> ? [X6: A] :
( ( member2 @ A @ X6 @ A9 )
& ( ( X5 @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% infinite_coinduct
thf(fact_860_finite__empty__induct,axiom,
! [A: $tType,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P @ A5 )
=> ( ! [A6: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( member2 @ A @ A6 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_861_Diff__single__insert,axiom,
! [A: $tType,A5: set @ A,X: A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_862_subset__insert__iff,axiom,
! [A: $tType,A5: set @ A,X: A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ B4 ) )
= ( ( ( member2 @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) )
& ( ~ ( member2 @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_863_remove__subset,axiom,
! [A: $tType,X: A,S: set @ A] :
( ( member2 @ A @ X @ S )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ S ) ) ).
% remove_subset
thf(fact_864_disjoint__alt__simp3,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp3
thf(fact_865_finite__remove__induct,axiom,
! [A: $tType,B4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B4 )
=> ( ! [X6: A] :
( ( member2 @ A @ X6 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_866_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B4: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B4 )
=> ( P @ B4 ) )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B4 )
=> ( ! [X6: A] :
( ( member2 @ A @ X6 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert2 @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_867_finite__induct__select,axiom,
! [A: $tType,S: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T3: set @ A] :
( ( ord_less @ ( set @ A ) @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X6: A] :
( ( member2 @ A @ X6 @ ( minus_minus @ ( set @ A ) @ S @ T3 ) )
& ( P @ ( insert2 @ A @ X6 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_868_psubset__insert__iff,axiom,
! [A: $tType,A5: set @ A,X: A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ B4 ) )
= ( ( ( member2 @ A @ X @ B4 )
=> ( ord_less @ ( set @ A ) @ A5 @ B4 ) )
& ( ~ ( member2 @ A @ X @ B4 )
=> ( ( ( member2 @ A @ X @ A5 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) )
& ( ~ ( member2 @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_869_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
? [X3: A] :
( A7
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_870_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D2: A,Q: A > $o] :
( ! [X2: A,K3: A] :
( ( P @ X2 )
= ( P @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K3 @ D2 ) ) ) )
=> ( ! [X2: A,K3: A] :
( ( Q @ X2 )
= ( Q @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K3 @ D2 ) ) ) )
=> ! [X6: A,K4: A] :
( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D2 ) ) )
| ( Q @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D2 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_871_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D2: A,Q: A > $o] :
( ! [X2: A,K3: A] :
( ( P @ X2 )
= ( P @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K3 @ D2 ) ) ) )
=> ( ! [X2: A,K3: A] :
( ( Q @ X2 )
= ( Q @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K3 @ D2 ) ) ) )
=> ! [X6: A,K4: A] :
( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D2 ) ) )
& ( Q @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D2 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_872_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X3: A,A7: set @ A] : ( minus_minus @ ( set @ A ) @ A7 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_873_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_874_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_875_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_876_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_877_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_878_Inf__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic4895041142388067077er_set @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_879_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min_insert
thf(fact_880_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_min @ A @ A3 @ A3 )
= A3 ) ) ).
% min.idem
thf(fact_881_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_min @ A @ A3 @ ( ord_min @ A @ A3 @ B2 ) )
= ( ord_min @ A @ A3 @ B2 ) ) ) ).
% min.left_idem
thf(fact_882_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
= ( ord_min @ A @ A3 @ B2 ) ) ) ).
% min.right_idem
thf(fact_883_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_max @ A @ A3 @ A3 )
= A3 ) ) ).
% max.idem
thf(fact_884_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_max @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) )
= ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.left_idem
thf(fact_885_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ B2 )
= ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.right_idem
thf(fact_886_member__remove,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( member2 @ A @ X @ ( remove @ A @ Y @ A5 ) )
= ( ( member2 @ A @ X @ A5 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_887_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min.absorb1
thf(fact_888_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_889_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C3 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% min.bounded_iff
thf(fact_890_min__arg__le_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [M2: A,N: A] :
( ( ord_less_eq @ A @ M2 @ ( ord_min @ A @ M2 @ N ) )
= ( ( ord_min @ A @ M2 @ N )
= M2 ) ) ) ).
% min_arg_le(2)
thf(fact_891_min__arg__le_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A,M2: A] :
( ( ord_less_eq @ A @ N @ ( ord_min @ A @ M2 @ N ) )
= ( ( ord_min @ A @ M2 @ N )
= N ) ) ) ).
% min_arg_le(1)
thf(fact_892_min__eq__arg_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ( ord_min @ A @ M2 @ N )
= N )
= ( ord_less_eq @ A @ N @ M2 ) ) ) ).
% min_eq_arg(2)
thf(fact_893_min__eq__arg_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ( ord_min @ A @ M2 @ N )
= M2 )
= ( ord_less_eq @ A @ M2 @ N ) ) ) ).
% min_eq_arg(1)
thf(fact_894_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_max @ A @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb1
thf(fact_895_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_max @ A @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_896_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% max.bounded_iff
thf(fact_897_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min.absorb3
thf(fact_898_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min.absorb4
thf(fact_899_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ ( ord_min @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z2 @ X )
& ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_900_min__simps_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min_simps(2)
thf(fact_901_min__simps_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min_simps(1)
thf(fact_902_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% min_less_self_conv(2)
thf(fact_903_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% min_less_self_conv(1)
thf(fact_904_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M2 @ N ) @ N ) )
= ( ( ord_min @ A @ M2 @ N )
= N ) ) ) ).
% min_arg_not_ge(2)
thf(fact_905_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M2 @ N ) @ M2 ) )
= ( ( ord_min @ A @ M2 @ N )
= M2 ) ) ) ).
% min_arg_not_ge(1)
thf(fact_906_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_max @ A @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb3
thf(fact_907_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_max @ A @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb4
thf(fact_908_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y ) @ Z2 )
= ( ( ord_less @ A @ X @ Z2 )
& ( ord_less @ A @ Y @ Z2 ) ) ) ) ).
% max_less_iff_conj
thf(fact_909_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_910_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_911_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_912_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_913_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_914_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_915_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_916_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_917_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ).
% Max_insert
thf(fact_918_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ C3 )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C3 ) ) ) ) ).
% max.assoc
thf(fact_919_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ C3 )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C3 ) ) ) ) ).
% min.assoc
thf(fact_920_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_max @ A )
= ( ^ [A4: A,B3: A] : ( ord_max @ A @ B3 @ A4 ) ) ) ) ).
% max.commute
thf(fact_921_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_min @ A )
= ( ^ [A4: A,B3: A] : ( ord_min @ A @ B3 @ A4 ) ) ) ) ).
% min.commute
thf(fact_922_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_max @ A @ B2 @ ( ord_max @ A @ A3 @ C3 ) )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C3 ) ) ) ) ).
% max.left_commute
thf(fact_923_max__min__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_max @ A @ ( ord_min @ A @ B2 @ C3 ) @ A3 )
= ( ord_min @ A @ ( ord_max @ A @ B2 @ A3 ) @ ( ord_max @ A @ C3 @ A3 ) ) ) ) ).
% max_min_distrib1
thf(fact_924_max__min__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_max @ A @ A3 @ ( ord_min @ A @ B2 @ C3 ) )
= ( ord_min @ A @ ( ord_max @ A @ A3 @ B2 ) @ ( ord_max @ A @ A3 @ C3 ) ) ) ) ).
% max_min_distrib2
thf(fact_925_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_min @ A @ B2 @ ( ord_min @ A @ A3 @ C3 ) )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C3 ) ) ) ) ).
% min.left_commute
thf(fact_926_min__max__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_min @ A @ ( ord_max @ A @ B2 @ C3 ) @ A3 )
= ( ord_max @ A @ ( ord_min @ A @ B2 @ A3 ) @ ( ord_min @ A @ C3 @ A3 ) ) ) ) ).
% min_max_distrib1
thf(fact_927_min__max__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_min @ A @ A3 @ ( ord_max @ A @ B2 @ C3 ) )
= ( ord_max @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ A3 @ C3 ) ) ) ) ).
% min_max_distrib2
thf(fact_928_remove1__commute,axiom,
! [A: $tType,X: A,Y: A,Zs: list @ A] :
( ( remove1 @ A @ X @ ( remove1 @ A @ Y @ Zs ) )
= ( remove1 @ A @ Y @ ( remove1 @ A @ X @ Zs ) ) ) ).
% remove1_commute
thf(fact_929_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A3: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C3 @ D3 ) @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_930_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_931_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_max @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% max.orderI
thf(fact_932_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% max.boundedE
thf(fact_933_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A3 ) ) ) ) ).
% max.boundedI
thf(fact_934_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( A4
= ( ord_max @ A @ A4 @ B3 ) ) ) ) ) ).
% max.order_iff
thf(fact_935_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_936_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_937_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less_eq @ A @ Z2 @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less_eq @ A @ Z2 @ X )
| ( ord_less_eq @ A @ Z2 @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_938_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_max @ A @ A4 @ B3 )
= A4 ) ) ) ) ).
% max.absorb_iff1
thf(fact_939_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_max @ A @ A4 @ B3 )
= B3 ) ) ) ) ).
% max.absorb_iff2
thf(fact_940_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_941_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_942_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A4: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B3 ) @ B3 @ A4 ) ) ) ) ).
% max_def
thf(fact_943_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_944_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_945_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z2 @ X )
| ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_946_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less @ A @ ( ord_max @ A @ B2 @ C3 ) @ A3 )
=> ~ ( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ C3 @ A3 ) ) ) ) ).
% max.strict_boundedE
thf(fact_947_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A4: A] :
( ( A4
= ( ord_max @ A @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_948_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ A3 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_949_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_950_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ C3 @ D3 ) ) ) ) ) ).
% min.mono
thf(fact_951_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_952_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_min @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% min.orderI
thf(fact_953_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% min.boundedE
thf(fact_954_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C3 ) ) ) ) ) ).
% min.boundedI
thf(fact_955_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( A4
= ( ord_min @ A @ A4 @ B3 ) ) ) ) ) ).
% min.order_iff
thf(fact_956_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 ) ) ).
% min.cobounded1
thf(fact_957_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_958_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_min @ A @ A4 @ B3 )
= A4 ) ) ) ) ).
% min.absorb_iff1
thf(fact_959_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_min @ A @ A4 @ B3 )
= B3 ) ) ) ) ).
% min.absorb_iff2
thf(fact_960_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% min.coboundedI1
thf(fact_961_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% min.coboundedI2
thf(fact_962_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X @ Z2 )
| ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% min_le_iff_disj
thf(fact_963_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A4: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A4 @ B3 ) @ A4 @ B3 ) ) ) ) ).
% min_def
thf(fact_964_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_965_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_966_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
= ( ( ord_less @ A @ X @ Z2 )
| ( ord_less @ A @ Y @ Z2 ) ) ) ) ).
% min_less_iff_disj
thf(fact_967_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ ( ord_min @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ A3 @ C3 ) ) ) ) ).
% min.strict_boundedE
thf(fact_968_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B3: A] :
( ( A4
= ( ord_min @ A @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_969_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ A3 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% min.strict_coboundedI1
thf(fact_970_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% min.strict_coboundedI2
thf(fact_971_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_972_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_973_sup__max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% sup_max
thf(fact_974_inf__min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% inf_min
thf(fact_975_remove1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( remove1 @ A @ X @ ( cons @ A @ Y @ Xs ) )
= Xs ) )
& ( ( X != Y )
=> ( ( remove1 @ A @ X @ ( cons @ A @ Y @ Xs ) )
= ( cons @ A @ Y @ ( remove1 @ A @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_976_remove1_Osimps_I1_J,axiom,
! [A: $tType,X: A] :
( ( remove1 @ A @ X @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remove1.simps(1)
thf(fact_977_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B4 ) @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ).
% Max.subset
thf(fact_978_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member2 @ A @ ( ord_max @ A @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member2 @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Max.closed
thf(fact_979_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member2 @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert2 @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_980_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ ( lattic643756798349783984er_Max @ A @ B4 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_981_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B4 ) @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.subset
thf(fact_982_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member2 @ A @ ( ord_min @ A @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member2 @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Min.closed
thf(fact_983_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member2 @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert2 @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_984_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( lattic643756798350308766er_Min @ A @ B4 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_985_complete__linorder__inf__min,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% complete_linorder_inf_min
thf(fact_986_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A5: set @ A,B4: set @ A] :
( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F @ B4 ) @ ( lattic1715443433743089157tice_F @ A @ F @ A5 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_987_remove__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remove @ A @ X @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( insert @ A @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_988_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_989_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_990_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_991_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_992_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_993_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_994_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_995_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_996_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_997_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_998_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_999_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ( times_times @ A @ C3 @ A3 )
= ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_1000_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ( times_times @ A @ A3 @ C3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_1001_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1002_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_1003_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_1004_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_1005_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_1006_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_1007_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% zero_comp_diff_simps(1)
thf(fact_1008_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_1009_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% zero_comp_diff_simps(2)
thf(fact_1010_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A,C3: A] :
( ( ( times_times @ A @ A3 @ C3 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_1011_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B2: A] :
( ( C3
= ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_1012_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,A3: A] :
( ( ( times_times @ A @ C3 @ A3 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_1013_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B2: A] :
( ( C3
= ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_1014_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_1015_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_1016_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_1017_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_1018_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_1019_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_1020_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_1021_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_1022_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ M2 @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_1023_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_1024_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_1025_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_1026_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_1027_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_1028_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C3 @ A3 )
= ( times_times @ A @ C3 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_1029_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_1030_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_1031_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1032_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1033_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1034_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ A3 ) ) ) ).
% zero_le_square
thf(fact_1035_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_1036_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1037_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1038_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1039_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1040_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1041_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_1042_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_1043_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1044_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1045_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1046_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1047_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1048_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1049_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_1050_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_1051_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_1052_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1053_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_1054_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_1055_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_1056_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_1057_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_1058_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_1059_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_1060_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1061_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_1062_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_1063_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1064_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1065_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1066_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1067_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1068_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1069_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1070_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1071_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1072_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_1073_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_1074_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_1075_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1076_Inf__fin__def,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic1715443433743089157tice_F @ A @ ( inf_inf @ A ) ) ) ) ).
% Inf_fin_def
thf(fact_1077_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1078_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1079_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1080_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1081_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1082_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1083_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1084_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1085_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1086_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1087_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1088_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1089_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1090_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1091_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C3: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ A3 ) ) ) ) ).
% mult_left_le
thf(fact_1092_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1093_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_1094_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_1095_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A5: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A5 ) )
=> ! [A11: A] :
( ( member2 @ A @ A11 @ A5 )
=> ( Less_eq @ X @ A11 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1096_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A5: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member2 @ A @ A6 @ A5 )
=> ( Less_eq @ X @ A6 ) )
=> ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A5 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1097_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A5: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A5 ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( Less_eq @ X @ X3 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_1098_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C3: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1099_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1100_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_1101_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_1102_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_1103_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_1104_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_1105_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_1106_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_1107_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_1108_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_1109_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_1110_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_1111_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F: A > A > A,A5: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert2 @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert2 @ A @ X @ A5 ) )
= ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_1112_semilattice__set_Oremove,axiom,
! [A: $tType,F: A > A > A,A5: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ A5 )
= ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1113_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_1114_semilattice__set_Ounion,axiom,
! [A: $tType,F: A > A > A,A5: set @ A,B4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( F @ ( lattic1715443433743089157tice_F @ A @ F @ A5 ) @ ( lattic1715443433743089157tice_F @ A @ F @ B4 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_1115_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F: A > A > A,A5: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member2 @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert2 @ A @ X @ A5 ) )
= ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A5 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_1116_semilattice__set_Oinsert,axiom,
! [A: $tType,F: A > A > A,A5: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert2 @ A @ X @ A5 ) )
= ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A5 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_1117_semilattice__set_Oclosed,axiom,
! [A: $tType,F: A > A > A,A5: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member2 @ A @ ( F @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member2 @ A @ ( lattic1715443433743089157tice_F @ A @ F @ A5 ) @ A5 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_1118_semilattice__set_Osubset,axiom,
! [A: $tType,F: A > A > A,A5: set @ A,B4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( F @ ( lattic1715443433743089157tice_F @ A @ F @ B4 ) @ ( lattic1715443433743089157tice_F @ A @ F @ A5 ) )
= ( lattic1715443433743089157tice_F @ A @ F @ A5 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_1119_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ( ord_less @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1120_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B2 = C3 ) ) ) ).
% add_right_cancel
thf(fact_1121_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add_left_cancel
thf(fact_1122_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_1123_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_1124_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_1125_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_1126_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_1127_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ A3 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_1128_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ B2 @ A3 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_1129_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_1130_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B2: A,A3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_1131_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_1132_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_1133_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_1134_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_1135_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel_right'
thf(fact_1136_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_1137_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_1138_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_1139_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% diff_add_cancel
thf(fact_1140_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel
thf(fact_1141_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_1142_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_1143_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_1144_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_1145_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1146_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1147_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1148_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1149_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_1150_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_1151_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_1152_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_1153_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_1154_exists__leI,axiom,
! [N: nat,P: nat > $o] :
( ( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ N )
=> ~ ( P @ N5 ) )
=> ( P @ N ) )
=> ? [N6: nat] :
( ( ord_less_eq @ nat @ N6 @ N )
& ( P @ N6 ) ) ) ).
% exists_leI
thf(fact_1155_add_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% add.right_commute
thf(fact_1156_add_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add.right_assoc
thf(fact_1157_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
=> ( B2 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_1158_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C3 ) )
=> ( B2 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_1159_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_1160_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_1161_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C3 @ A3 ) )
= ( B2 = C3 ) ) ) ).
% add.right_cancel
thf(fact_1162_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add.left_cancel
thf(fact_1163_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_1164_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: A,K: A,B2: A,A3: A] :
( ( B4
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A3 @ B4 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_1165_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( plus_plus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_1166_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1167_mlex__leI,axiom,
! [A3: nat,A10: nat,B2: nat,B9: nat,N4: nat] :
( ( ord_less_eq @ nat @ A3 @ A10 )
=> ( ( ord_less_eq @ nat @ B2 @ B9 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N4 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A10 @ N4 ) @ B9 ) ) ) ) ).
% mlex_leI
thf(fact_1168_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X3: A] : ( plus_plus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_1169_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_1170_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_1171_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
? [C6: A] :
( B3
= ( plus_plus @ A @ A4 @ C6 ) ) ) ) ) ).
% le_iff_add
thf(fact_1172_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add_right_mono
thf(fact_1173_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ! [C2: A] :
( B2
!= ( plus_plus @ A @ A3 @ C2 ) ) ) ) ).
% less_eqE
thf(fact_1174_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_1175_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_1176_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1177_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1178_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1179_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.group_left_neutral
thf(fact_1180_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.comm_neutral
thf(fact_1181_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_1182_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_1183_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_1184_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add_strict_right_mono
thf(fact_1185_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_1186_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_1187_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1188_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1189_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1190_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1191_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1192_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1193_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ).
% distrib_left
thf(fact_1194_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% distrib_right
thf(fact_1195_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,E3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E3 ) @ C3 ) ) ) ).
% combine_common_factor
thf(fact_1196_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% diff_diff_eq
thf(fact_1197_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ( plus_plus @ A @ C3 @ B2 )
= A3 )
=> ( C3
= ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_1198_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1199_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_1200_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_1201_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% add_diff_eq
thf(fact_1202_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( A3
= ( minus_minus @ A @ C3 @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= C3 ) ) ) ).
% eq_diff_eq
thf(fact_1203_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= C3 )
= ( A3
= ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_1204_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( minus_minus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1205_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_1206_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_1207_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_1208_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_1209_Inf__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic149705377957585745ce_set @ A @ ( inf_inf @ A ) ) ) ).
% Inf_fin.semilattice_set_axioms
thf(fact_1210_add_Osafe__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [X: A,Y: A,A3: A,B2: A] :
( ( syntax7388354845996824322omatch @ A @ A @ ( plus_plus @ A @ X @ Y ) @ A3 )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ B2 @ A3 ) ) ) ) ).
% add.safe_commute
thf(fact_1211_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_1212_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_increasing
thf(fact_1213_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_1214_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_increasing2
thf(fact_1215_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1216_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_1217_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_1218_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_1219_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_1220_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_1221_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1222_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1223_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_add_strict
thf(fact_1224_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ! [C2: A] :
( ( B2
= ( plus_plus @ A @ A3 @ C2 ) )
=> ( C2
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1225_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_1226_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_1227_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_1228_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% le_diff_eq
thf(fact_1229_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1230_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A3 ) ) ) ) ).
% le_add_diff
thf(fact_1231_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1232_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1233_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B2 ) @ A3 )
= ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1234_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1235_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A3 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1236_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ C3 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1237_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1238_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A3 )
= C3 )
= ( B2
= ( plus_plus @ A @ C3 @ A3 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1239_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_1240_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_1241_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( ord_less @ A @ A3 @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_1242_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C3 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% less_diff_eq
thf(fact_1243_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_1244_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( C3
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E3 ) @ D3 ) ) ) ) ).
% eq_add_iff2
thf(fact_1245_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E3 ) @ C3 )
= D3 ) ) ) ).
% eq_add_iff1
thf(fact_1246_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1247_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1248_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_1249_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_1250_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_1251_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_1252_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_1253_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_1254_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_1255_le__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E3 ) @ D3 ) ) ) ) ).
% le_add_iff2
thf(fact_1256_le__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E3 ) @ C3 ) @ D3 ) ) ) ).
% le_add_iff1
thf(fact_1257_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E3 ) @ C3 ) @ D3 ) ) ) ).
% less_add_iff1
thf(fact_1258_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E3 ) @ D3 ) ) ) ) ).
% less_add_iff2
thf(fact_1259_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_1260_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1261_semilattice__set_Osingleton,axiom,
! [A: $tType,F: A > A > A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% semilattice_set.singleton
thf(fact_1262_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_1263_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_1264_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_1265_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_1266_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R3: A,A3: A,B2: A,C3: A,D3: A] :
( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B2 )
& ( C3 != D3 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R3 @ C3 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R3 @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1267_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( A3 != B2 )
& ( C3 != D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A3 @ D3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_1268_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_1269_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X3: A] : ( minus_minus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_1270_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_1271_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] :
( ( slice @ A @ Begin @ End @ ( nil @ A ) )
= ( nil @ A ) ) ).
% slice_Nil
thf(fact_1272_slice__eq__bounds__empty,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( slice @ A @ I @ I @ Xs )
= ( nil @ A ) ) ).
% slice_eq_bounds_empty
thf(fact_1273_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_1274_mlex__snd__decrI,axiom,
! [A3: nat,A10: nat,B2: nat,B9: nat,N4: nat] :
( ( A3 = A10 )
=> ( ( ord_less @ nat @ B2 @ B9 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N4 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A10 @ N4 ) @ B9 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_1275_mlex__fst__decrI,axiom,
! [A3: nat,A10: nat,B2: nat,N4: nat,B9: nat] :
( ( ord_less @ nat @ A3 @ A10 )
=> ( ( ord_less @ nat @ B2 @ N4 )
=> ( ( ord_less @ nat @ B9 @ N4 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N4 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A10 @ N4 ) @ B9 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_1276_mlex__bound,axiom,
! [A3: nat,A5: nat,B2: nat,N4: nat] :
( ( ord_less @ nat @ A3 @ A5 )
=> ( ( ord_less @ nat @ B2 @ N4 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N4 ) @ B2 ) @ ( times_times @ nat @ A5 @ N4 ) ) ) ) ).
% mlex_bound
thf(fact_1277_count__list_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) )
& ( ( X != Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( count_list @ A @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_1278_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ nat @ N @ M2 ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_1279_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_1280_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_1281_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1282_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P6: A,M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1283_card__Diff__singleton,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( member2 @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_1284_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( ( member2 @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member2 @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1285_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_1286_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ( power_power @ A @ A3 @ M2 )
= ( power_power @ A @ A3 @ N ) )
= ( M2 = N ) ) ) ) ).
% power_inject_exp
thf(fact_1287_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_1288_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
= X ) ).
% nth_Cons_0
thf(fact_1289_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_1290_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_1291_card__0__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_1292_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1293_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_1294_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ Y )
= ( times_times @ A @ Y @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_1295_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A3 @ B2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_1296_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_commutes
thf(fact_1297_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F: B > A,L: list @ B,L3: list @ B,I: nat] :
( ( ( map @ B @ A @ F @ L )
= ( map @ B @ A @ F @ L3 ) )
=> ( ( F @ ( nth @ B @ L @ I ) )
= ( F @ ( nth @ B @ L3 @ I ) ) ) ) ).
% map_eq_nth_eq
thf(fact_1298_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% one_le_power
thf(fact_1299_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_1300_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_1301_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_add
thf(fact_1302_card__eq__0__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% card_eq_0_iff
thf(fact_1303_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_1304_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1305_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1306_power__0__left,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% power_0_left
thf(fact_1307_card__1__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( one_one @ nat ) )
=> ~ ! [X2: A] :
( A5
!= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_1308_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_1309_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A3: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N4 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_1310_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N4 ) ) ) ) ) ).
% power_increasing
thf(fact_1311_card__gt__0__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A5 ) )
= ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A5 ) ) ) ).
% card_gt_0_iff
thf(fact_1312_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1313_card__1__singletonI,axiom,
! [A: $tType,S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( finite_card @ A @ S )
= ( one_one @ nat ) )
=> ( ( member2 @ A @ X @ S )
=> ( S
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_1_singletonI
thf(fact_1314_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A3: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N4 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1315_card__Diff1__le,axiom,
! [A: $tType,A5: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ).
% card_Diff1_le
thf(fact_1316_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N4 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1317_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_1318_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_1319_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_1320_card__Un__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_1321_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_1322_card__Diff__subset__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1323_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] :
( ( count_list @ A @ ( nil @ A ) @ Y )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_1324_card__Diff1__less__iff,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) )
= ( ( finite_finite2 @ A @ A5 )
& ( member2 @ A @ X @ A5 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1325_card__Diff2__less,axiom,
! [A: $tType,A5: set @ A,X: A,Y: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( member2 @ A @ Y @ A5 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1326_card__Diff1__less,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ) ) ).
% card_Diff1_less
thf(fact_1327_card__Un__disjoint,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_1328_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A,N: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1329_foldr__max__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ( Xs
= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y )
= Y ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y )
= ( ord_max @ A @ ( nth @ A @ Xs @ ( zero_zero @ nat ) ) @ Y ) ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_1330_slice__nth,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A,I: nat] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ To @ From ) )
=> ( ( nth @ A @ ( slice @ A @ From @ To @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ From @ I ) ) ) ) ) ) ).
% slice_nth
thf(fact_1331_last__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% last_conv_nth
thf(fact_1332_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs: list @ A,V: num] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_1333_card__insert__disjoint_H,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member2 @ A @ X @ A5 )
=> ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ A5 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_insert_disjoint'
thf(fact_1334_card_Oremove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( finite_card @ A @ A5 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1335_card_Oinsert__remove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ A @ ( insert2 @ A @ X @ A5 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1336_card__Suc__Diff1,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1337_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_1338_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_1339_min__Suc__gt_I1_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_min @ nat @ ( suc @ A3 ) @ B2 )
= ( suc @ A3 ) ) ) ).
% min_Suc_gt(1)
thf(fact_1340_min__Suc__gt_I2_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_min @ nat @ B2 @ ( suc @ A3 ) )
= ( suc @ A3 ) ) ) ).
% min_Suc_gt(2)
thf(fact_1341_length__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_1342_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us2: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us2 )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us2 )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_1343_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_1344_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_1345_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V: num,B2: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C3 ) ) ) ) ).
% distrib_left_numeral
thf(fact_1346_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A3: A,B2: A,V: num] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_1347_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V: num,B2: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C3 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1348_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A3: A,B2: A,V: num] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1349_length__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( zero_zero @ nat ) )
= ( Xs
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_1350_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: num,N: num,B2: A] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M2 ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% power_add_numeral2
thf(fact_1351_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M2 ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_1352_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_1353_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_1354_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_1355_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_1356_nth__Cons__Suc,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ N ) )
= ( nth @ A @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1357_length__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_append
thf(fact_1358_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A5 ) )
= ( finite_card @ A @ A5 ) ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_1359_slice__complete,axiom,
! [A: $tType,Xs: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs )
= Xs ) ).
% slice_complete
thf(fact_1360_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_1361_length__greater__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( Xs
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_1362_length__concat__rev,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( concat @ A @ ( rev @ ( list @ A ) @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) ) ) ).
% length_concat_rev
thf(fact_1363_nth__append__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_1364_Suc__diff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( suc @ ( minus_minus @ nat @ N @ M2 ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_diff
thf(fact_1365_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F @ Xs ) @ N )
= ( F @ ( nth @ A @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1366_nth__append__first,axiom,
! [A: $tType,I: nat,L: list @ A,L3: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( append @ A @ L @ L3 ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_append_first
thf(fact_1367_nth__append__length__plus,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,N: nat] :
( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
= ( nth @ A @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1368_length__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_butlast
thf(fact_1369_rotate1__length01,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate1 @ A @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_1370_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_1371_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_1372_length__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_Cons
thf(fact_1373_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y3: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1374_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y3: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1375_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1376_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1377_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ ( suc @ I2 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1378_sorted__wrt__less__idx,axiom,
! [Ns: list @ nat,I: nat] :
( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ nat ) @ Ns ) )
=> ( ord_less_eq @ nat @ I @ ( nth @ nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_1379_sorted__wrt__nth__less,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ P @ Xs )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1380_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P3: A > A > $o,Xs3: list @ A] :
! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ ( nth @ A @ Xs3 @ I2 ) @ ( nth @ A @ Xs3 @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1381_sorted__wrt01,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_1382_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X3: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1383_sorted__wrt_Osimps_I1_J,axiom,
! [A: $tType,P: A > A > $o] : ( sorted_wrt @ A @ P @ ( nil @ A ) ) ).
% sorted_wrt.simps(1)
thf(fact_1384_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys6: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys6 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1385_finite__maxlen,axiom,
! [A: $tType,M: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M )
=> ? [N7: nat] :
! [X6: list @ A] :
( ( member2 @ ( list @ A ) @ X6 @ M )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X6 ) @ N7 ) ) ) ).
% finite_maxlen
thf(fact_1386_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,Xs: list @ B,G: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F @ Xs )
= ( map @ C @ A @ G @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1387_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I2 ) ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_1388_sorted__wrt__mergesort__by__rel,axiom,
! [X7: $tType,R: X7 > X7 > $o,Xs: list @ X7] :
( ! [X2: X7,Y2: X7] :
( ( R @ X2 @ Y2 )
| ( R @ Y2 @ X2 ) )
=> ( ! [X2: X7,Y2: X7,Z4: X7] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z4 )
=> ( R @ X2 @ Z4 ) ) )
=> ( sorted_wrt @ X7 @ R @ ( mergesort_by_rel @ X7 @ R @ Xs ) ) ) ) ).
% sorted_wrt_mergesort_by_rel
thf(fact_1389_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1390_sorted01,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted01
thf(fact_1391_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X22 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_1392_length__Suc__rev__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Ys3: list @ A,Y3: A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_rev_conv
thf(fact_1393_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y3: A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1394_length__append__singleton,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1395_strict__sorted__imp__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% strict_sorted_imp_sorted
thf(fact_1396_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1397_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1398_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Zs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted2
thf(fact_1399_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_1400_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_1401_sorted__wrt1,axiom,
! [A: $tType,P: A > A > $o,X: A] : ( sorted_wrt @ A @ P @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% sorted_wrt1
thf(fact_1402_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A3 @ Xs ) ) ) ) ).
% sorted_remove1
thf(fact_1403_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_1404_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_1405_rev__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_1406_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_1407_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,Ws2: list @ D,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B,Ys2: list @ B,Z4: C,Zs3: list @ C,W3: D,Ws3: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs3 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs3 )
= ( size_size @ ( list @ D ) @ Ws3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) @ ( cons @ C @ Z4 @ Zs3 ) @ ( cons @ D @ W3 @ Ws3 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws2 ) ) ) ) ) ) ).
% list_induct4
thf(fact_1408_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B,Ys2: list @ B,Z4: C,Zs3: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) @ ( cons @ C @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1409_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1410_one__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% one_le_numeral
thf(fact_1411_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_1412_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_1413_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_1414_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_1415_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1416_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_Suc
thf(fact_1417_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 ) ) ) ).
% power_Suc2
thf(fact_1418_list__rest__coinc,axiom,
! [A: $tType,S22: list @ A,S1: list @ A,R1: list @ A,R2: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ S22 ) @ ( size_size @ ( list @ A ) @ S1 ) )
=> ( ( ( append @ A @ S1 @ R1 )
= ( append @ A @ S22 @ R2 ) )
=> ? [R1p: list @ A] :
( R2
= ( append @ A @ R1p @ R1 ) ) ) ) ).
% list_rest_coinc
thf(fact_1419_nat__compl__induct_H,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N7: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N7 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N7 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct'
thf(fact_1420_nat__compl__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N7: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N7 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N7 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct
thf(fact_1421_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1422_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ? [X8: A] : ( P @ I2 @ X8 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( P @ I2 @ ( nth @ A @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1423_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z5: list @ A] : Y5 = Z5 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I2 )
= ( nth @ A @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1424_obtain__list__from__elements,axiom,
! [A: $tType,N: nat,P: A > nat > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ? [Li: A] : ( P @ Li @ I3 ) )
=> ~ ! [L2: list @ A] :
( ( ( size_size @ ( list @ A ) @ L2 )
= N )
=> ~ ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ ( nth @ A @ L2 @ I4 ) @ I4 ) ) ) ) ).
% obtain_list_from_elements
thf(fact_1425_nat__in__between__eq_I1_J,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less @ nat @ A3 @ B2 )
& ( ord_less_eq @ nat @ B2 @ ( suc @ A3 ) ) )
= ( B2
= ( suc @ A3 ) ) ) ).
% nat_in_between_eq(1)
thf(fact_1426_nat__in__between__eq_I2_J,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B2 )
& ( ord_less @ nat @ B2 @ ( suc @ A3 ) ) )
= ( B2 = A3 ) ) ).
% nat_in_between_eq(2)
thf(fact_1427_Suc__to__right,axiom,
! [N: nat,M2: nat] :
( ( ( suc @ N )
= M2 )
=> ( N
= ( minus_minus @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_to_right
thf(fact_1428_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_1429_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J2 ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_1430_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_1431_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_1432_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_1433_gen__length__def,axiom,
! [A: $tType] :
( ( gen_length @ A )
= ( ^ [N3: nat,Xs3: list @ A] : ( plus_plus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_1434_gen__length__code_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list @ B] :
( ( gen_length @ B @ N @ ( cons @ B @ X @ Xs ) )
= ( gen_length @ B @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1435_sorted__butlast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( butlast @ A @ Xs ) ) ) ) ) ).
% sorted_butlast
thf(fact_1436_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_1437_length__compl__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [E: A,L2: list @ A] :
( ! [Ll2: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll2 ) @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( P @ Ll2 ) )
=> ( P @ ( cons @ A @ E @ L2 ) ) )
=> ( P @ L ) ) ) ).
% length_compl_induct
thf(fact_1438_same__length__different,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Pre: list @ A,X2: A,Xs4: list @ A,Y2: A,Ys4: list @ A] :
( ( X2 != Y2 )
& ( Xs
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ Xs4 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1439_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_1440_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_1441_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_1442_nth__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_1443_sorted__in__between,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: nat,J: nat,L: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ X )
=> ( ( ord_less @ A @ X @ ( nth @ A @ L @ J ) )
=> ~ ! [K3: nat] :
( ( ord_less_eq @ nat @ I @ K3 )
=> ( ( ord_less @ nat @ K3 @ J )
=> ( ( 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_1444_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% power_Suc_le_self
thf(fact_1445_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_1446_length__compl__rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [L2: list @ A,E: A] :
( ! [Ll2: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll2 ) @ ( size_size @ ( list @ A ) @ L2 ) )
=> ( P @ Ll2 ) )
=> ( P @ ( append @ A @ L2 @ ( cons @ A @ E @ ( nil @ A ) ) ) ) )
=> ( P @ L ) ) ) ).
% length_compl_rev_induct
thf(fact_1447_card__Suc__eq,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K ) )
= ( ? [B3: A,B5: set @ A] :
( ( A5
= ( insert2 @ A @ B3 @ B5 ) )
& ~ ( member2 @ A @ B3 @ B5 )
& ( ( finite_card @ A @ B5 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1448_card__eq__SucD,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K ) )
=> ? [B6: A,B7: set @ A] :
( ( A5
= ( insert2 @ A @ B6 @ B7 ) )
& ~ ( member2 @ A @ B6 @ B7 )
& ( ( finite_card @ A @ B7 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B7
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_1449_card__1__singleton__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X3: A] :
( A5
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1450_nth__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Xs @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1451_nz__le__conv__less,axiom,
! [K: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ M2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ K @ ( suc @ ( zero_zero @ nat ) ) ) @ M2 ) ) ) ).
% nz_le_conv_less
thf(fact_1452_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_1453_Suc__n__minus__m__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( suc @ ( minus_minus @ nat @ N @ M2 ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_n_minus_m_eq
thf(fact_1454_length__product__lists,axiom,
! [B: $tType,Xss2: list @ ( list @ B )] :
( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss2 ) )
= ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss2 ) @ ( one_one @ nat ) ) ) ).
% length_product_lists
thf(fact_1455_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_1456_last__take__nth__conv,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( take @ A @ N @ L ) )
= ( nth @ A @ L @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% last_take_nth_conv
thf(fact_1457_take__Suc__conv__app__nth,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ ( suc @ I ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1458_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A5 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A5 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A5 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_1459_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert2 @ A @ ( infini527867602293511546merate @ A @ S @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N ) ) ) ).
% enumerate_Suc'
thf(fact_1460_distinct__sorted__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
= ( ord_less_eq @ nat @ I @ J ) ) ) ) ) ) ) ).
% distinct_sorted_mono_iff
thf(fact_1461_take__minus__one__conv__butlast,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( take @ A @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( butlast @ A @ ( take @ A @ N @ L ) ) ) ) ).
% take_minus_one_conv_butlast
thf(fact_1462_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_1463_map__eq__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,G: B > A] :
( ( ( map @ B @ A @ F @ Xs )
= ( map @ B @ A @ G @ Xs ) )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_1464_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X3: list @ A] : X3 ) ) ).
% drop0
thf(fact_1465_set__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rev @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rev
thf(fact_1466_drop__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( drop @ A @ M2 @ Xs ) )
= ( drop @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs ) ) ).
% drop_drop
thf(fact_1467_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( ( nil @ A )
= ( concat @ A @ Xss2 ) )
= ( ! [X3: list @ A] :
( ( member2 @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( X3
= ( nil @ A ) ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1468_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( ( concat @ A @ Xss2 )
= ( nil @ A ) )
= ( ! [X3: list @ A] :
( ( member2 @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( X3
= ( nil @ A ) ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1469_distinct__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rev @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rev
thf(fact_1470_in__set__remove1,axiom,
! [A: $tType,A3: A,B2: A,Xs: list @ A] :
( ( A3 != B2 )
=> ( ( member2 @ A @ A3 @ ( set2 @ A @ ( remove1 @ A @ B2 @ Xs ) ) )
= ( member2 @ A @ A3 @ ( set2 @ A @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1471_set__mergesort__by__rel,axiom,
! [A: $tType,R: A > A > $o,Xs: list @ A] :
( ( set2 @ A @ ( mergesort_by_rel @ A @ R @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_mergesort_by_rel
thf(fact_1472_take__take,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( take @ A @ N @ ( take @ A @ M2 @ Xs ) )
= ( take @ A @ ( ord_min @ nat @ N @ M2 ) @ Xs ) ) ).
% take_take
thf(fact_1473_set__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rotate1 @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate1
thf(fact_1474_distinct1__rotate,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rotate1 @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct1_rotate
thf(fact_1475_in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1476_distinct__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( insert @ A @ X @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_insert
thf(fact_1477_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_1478_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_1479_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X22 ) )
= ( insert2 @ A @ X21 @ ( set2 @ A @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1480_take__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( take @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_1481_take__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( take @ A @ N @ Xs ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil2
thf(fact_1482_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= ( nil @ A ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_1483_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs3: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_1484_drop__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( drop @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1485_take__all,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( take @ A @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_1486_take__all__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= Xs )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_1487_nth__take,axiom,
! [A: $tType,I: nat,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ I ) ) ) ).
% nth_take
thf(fact_1488_length__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( drop @ A @ N @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_drop
thf(fact_1489_List_Olast__in__set,axiom,
! [A: $tType,As2: list @ A] :
( ( As2
!= ( nil @ A ) )
=> ( member2 @ A @ ( last @ A @ As2 ) @ ( set2 @ A @ As2 ) ) ) ).
% List.last_in_set
thf(fact_1490_Misc_Olast__in__set,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( member2 @ A @ ( last @ A @ L ) @ ( set2 @ A @ L ) ) ) ).
% Misc.last_in_set
thf(fact_1491_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A5 ) )
= A5 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_1492_append__take__drop__id,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( append @ A @ ( take @ A @ N @ Xs ) @ ( drop @ A @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_1493_length__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( take @ A @ N @ Xs ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_take
thf(fact_1494_not__in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= ( cons @ A @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1495_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_1496_count__notin,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_1497_drop__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( drop @ A @ N @ Xs ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_1498_drop__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( nil @ A ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1499_drop__all,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( drop @ A @ N @ Xs )
= ( nil @ A ) ) ) ).
% drop_all
thf(fact_1500_take__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( take @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( take @ A @ N @ Xs ) @ ( take @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_1501_drop__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( drop @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( drop @ A @ N @ Xs ) @ ( drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_1502_last__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( last @ A @ ( drop @ A @ N @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_drop
thf(fact_1503_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_1504_take__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_1505_drop__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_1506_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_1507_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_1508_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_1509_take__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( take @ A @ N @ ( drop @ A @ M2 @ Xs ) )
= ( drop @ A @ M2 @ ( take @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs ) ) ) ).
% take_drop
thf(fact_1510_drop__take,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( take @ A @ M2 @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ M2 @ N ) @ ( drop @ A @ N @ Xs ) ) ) ).
% drop_take
thf(fact_1511_subseqs__distinctD,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( member2 @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( ( distinct @ A @ Xs )
=> ( distinct @ A @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_1512_distinct__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ B @ Ys )
=> ( distinct @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_1513_distinct__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A ) @ Xs )
=> ( ! [Ys2: list @ A] :
( ( member2 @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys2 ) )
=> ( ! [Ys2: list @ A,Zs3: list @ A] :
( ( member2 @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member2 @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys2 != Zs3 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys2 ) @ ( set2 @ A @ Zs3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1514_distinct__map__eq,axiom,
! [A: $tType,B: $tType,F: B > A,L: list @ B,X: B,Y: B] :
( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member2 @ B @ X @ ( set2 @ B @ L ) )
=> ( ( member2 @ B @ Y @ ( set2 @ B @ L ) )
=> ( X = Y ) ) ) ) ) ).
% distinct_map_eq
thf(fact_1515_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
= ( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1516_finite__distinct__list,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [Xs2: list @ A] :
( ( ( set2 @ A @ Xs2 )
= A5 )
& ( distinct @ A @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1517_take__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ! [I3: nat] :
( ( take @ A @ I3 @ Xs )
= ( take @ A @ I3 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_1518_distinct__take,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( take @ A @ I @ Xs ) ) ) ).
% distinct_take
thf(fact_1519_distinct__drop,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( drop @ A @ I @ Xs ) ) ) ).
% distinct_drop
thf(fact_1520_in__set__takeD,axiom,
! [A: $tType,X: A,N: nat,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1521_in__set__dropD,axiom,
! [A: $tType,X: A,N: nat,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1522_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ Xs ) ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_1523_distinct__product__lists,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ! [X2: list @ A] :
( ( member2 @ ( list @ A ) @ X2 @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( distinct @ A @ X2 ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_1524_distinct__set__subseqs,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ ( set @ A ) @ ( map @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_1525_set__drop__subset,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_drop_subset
thf(fact_1526_set__take__subset,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_take_subset
thf(fact_1527_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Vs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I @ Vs ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_1528_append__eq__conv__conj,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) )
& ( Ys
= ( drop @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_1529_drop__take__drop__unsplit,axiom,
! [A: $tType,I: nat,J: nat,L: list @ A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( append @ A @ ( drop @ A @ I @ ( take @ A @ J @ L ) ) @ ( drop @ A @ J @ L ) )
= ( drop @ A @ I @ L ) ) ) ).
% drop_take_drop_unsplit
thf(fact_1530_take__add,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A] :
( ( take @ A @ ( plus_plus @ nat @ I @ J ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( take @ A @ J @ ( drop @ A @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_1531_set__take__subset__set__take,axiom,
! [A: $tType,M2: nat,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M2 @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1532_set__drop__subset__set__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M2 @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1533_not__distinct__conv__prefix,axiom,
! [A: $tType,As2: list @ A] :
( ( ~ ( distinct @ A @ As2 ) )
= ( ? [Xs3: list @ A,Y3: A,Ys3: list @ A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Xs3 ) )
& ( distinct @ A @ Xs3 )
& ( As2
= ( append @ A @ Xs3 @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1534_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_1535_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
=> ( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_1536_card__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( finite_card @ A @ ( set2 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Xs ) ) ).
% card_distinct
thf(fact_1537_distinct__card,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( finite_card @ A @ ( set2 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% distinct_card
thf(fact_1538_take__Nil,axiom,
! [A: $tType,N: nat] :
( ( take @ A @ N @ ( nil @ A ) )
= ( nil @ A ) ) ).
% take_Nil
thf(fact_1539_drop__Nil,axiom,
! [A: $tType,N: nat] :
( ( drop @ A @ N @ ( nil @ A ) )
= ( nil @ A ) ) ).
% drop_Nil
thf(fact_1540_drop__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% drop_0
thf(fact_1541_slice__def,axiom,
! [A: $tType] :
( ( slice @ A )
= ( ^ [From2: nat,To2: nat,List2: list @ A] : ( take @ A @ ( minus_minus @ nat @ To2 @ From2 ) @ ( drop @ A @ From2 @ List2 ) ) ) ) ).
% slice_def
thf(fact_1542_distinct__length__2__or__more,axiom,
! [A: $tType,A3: A,B2: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ A3 @ ( cons @ A @ B2 @ Xs ) ) )
= ( ( A3 != B2 )
& ( distinct @ A @ ( cons @ A @ A3 @ Xs ) )
& ( distinct @ A @ ( cons @ A @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_1543_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_1544_take__map,axiom,
! [A: $tType,B: $tType,N: nat,F: B > A,Xs: list @ B] :
( ( take @ A @ N @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( take @ B @ N @ Xs ) ) ) ).
% take_map
thf(fact_1545_drop__map,axiom,
! [A: $tType,B: $tType,N: nat,F: B > A,Xs: list @ B] :
( ( drop @ A @ N @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( drop @ B @ N @ Xs ) ) ) ).
% drop_map
thf(fact_1546_distinct__mapI,axiom,
! [A: $tType,B: $tType,F: B > A,L: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
=> ( distinct @ B @ L ) ) ).
% distinct_mapI
thf(fact_1547_sorted__wrt__take,axiom,
! [A: $tType,F: A > A > $o,Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ F @ Xs )
=> ( sorted_wrt @ A @ F @ ( take @ A @ N @ Xs ) ) ) ).
% sorted_wrt_take
thf(fact_1548_sorted__wrt__drop,axiom,
! [A: $tType,F: A > A > $o,Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ F @ Xs )
=> ( sorted_wrt @ A @ F @ ( drop @ A @ N @ Xs ) ) ) ).
% sorted_wrt_drop
thf(fact_1549_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y: A,X22: list @ A,X21: A] :
( ( member2 @ A @ Y @ ( set2 @ A @ X22 ) )
=> ( member2 @ A @ Y @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1550_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] : ( member2 @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_1551_list_Oset__cases,axiom,
! [A: $tType,E3: A,A3: list @ A] :
( ( member2 @ A @ E3 @ ( set2 @ A @ A3 ) )
=> ( ! [Z22: list @ A] :
( A3
!= ( cons @ A @ E3 @ Z22 ) )
=> ~ ! [Z1: A,Z22: list @ A] :
( ( A3
= ( cons @ A @ Z1 @ Z22 ) )
=> ~ ( member2 @ A @ E3 @ ( set2 @ A @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1552_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list @ A] :
( ( member2 @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member2 @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1553_finite__list,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [Xs2: list @ A] :
( ( set2 @ A @ Xs2 )
= A5 ) ) ).
% finite_list
thf(fact_1554_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B4 )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( member2 @ A @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_1555_ex__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ B,F: A > B] :
( ( ? [Xs3: list @ A] :
( Ys
= ( map @ A @ B @ F @ Xs3 ) ) )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ ( set2 @ B @ Ys ) )
=> ? [Y3: A] :
( X3
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1556_map__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F: A > B,G: A > B] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map @ A @ B @ F @ Xs )
= ( map @ A @ B @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_1557_map__idI,axiom,
! [A: $tType,Xs: list @ A,F: A > A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map @ A @ A @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1558_map__ext,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B,G: A > B] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map @ A @ B @ F @ Xs )
= ( map @ A @ B @ G @ Xs ) ) ) ).
% map_ext
thf(fact_1559_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: list @ A,Xa: list @ A,F: A > B,Fa: A > B] :
( ! [Z4: A,Za: A] :
( ( member2 @ A @ Z4 @ ( set2 @ A @ X ) )
=> ( ( member2 @ A @ Za @ ( set2 @ A @ Xa ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map @ A @ B @ F @ X )
= ( map @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_1560_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: list @ A,F: A > B,G: A > B] :
( ! [Z4: A] :
( ( member2 @ A @ Z4 @ ( set2 @ A @ X ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map @ A @ B @ F @ X )
= ( map @ A @ B @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_1561_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X: list @ A,Ya: list @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z4: A] :
( ( member2 @ A @ Z4 @ ( set2 @ A @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map @ A @ B @ F @ X )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1562_drop__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( butlast @ A @ Xs ) )
= ( butlast @ A @ ( drop @ A @ N @ Xs ) ) ) ).
% drop_butlast
thf(fact_1563_sorted__wrt__mono__rel,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o,Q: A > A > $o] :
( ! [X2: A,Y2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member2 @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) ) ) )
=> ( ( sorted_wrt @ A @ P @ Xs )
=> ( sorted_wrt @ A @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1564_distinct__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( butlast @ A @ Xs ) ) ) ).
% distinct_butlast
thf(fact_1565_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] : ( distinct @ A @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_1566_distinct__remove1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( remove1 @ A @ X @ Xs ) ) ) ).
% distinct_remove1
thf(fact_1567_in__set__butlastD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1568_foldr__cong,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,L: list @ B,K: list @ B,F: B > A > A,G: B > A > A] :
( ( A3 = B2 )
=> ( ( L = K )
=> ( ! [A6: A,X2: B] :
( ( member2 @ B @ X2 @ ( set2 @ B @ L ) )
=> ( ( F @ X2 @ A6 )
= ( G @ X2 @ A6 ) ) )
=> ( ( foldr @ B @ A @ F @ L @ A3 )
= ( foldr @ B @ A @ G @ K @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_1569_notin__set__remove1,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member2 @ A @ X @ ( set2 @ A @ ( remove1 @ A @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1570_remove1__idem,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1571_append__eq__append__conv__if,axiom,
! [A: $tType,Xs_1: list @ A,Xs_2: list @ A,Ys_1: list @ A,Ys_2: list @ A] :
( ( ( append @ A @ Xs_1 @ Xs_2 )
= ( append @ A @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
=> ( ( Xs_1
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
=> ( ( ( take @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_1572_drop__rev,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) @ Xs ) ) ) ).
% drop_rev
thf(fact_1573_rev__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( rev @ A @ ( drop @ A @ I @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ).
% rev_drop
thf(fact_1574_rev__take,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( rev @ A @ ( take @ A @ I @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ).
% rev_take
thf(fact_1575_take__rev,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( take @ A @ N @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) @ Xs ) ) ) ).
% take_rev
thf(fact_1576_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ~ ( distinct @ A @ Xs )
=> ~ ! [Y2: A,Ys2: list @ A] :
( ( distinct @ A @ Ys2 )
=> ( ( member2 @ A @ Y2 @ ( set2 @ A @ Ys2 ) )
=> ! [Zs3: list @ A] :
( Xs
!= ( append @ A @ Ys2 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_1577_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [X2: list @ A] :
( ( ( set2 @ A @ X2 )
= A5 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X2 )
& ( distinct @ A @ X2 )
& ! [Y4: list @ A] :
( ( ( ( set2 @ A @ Y4 )
= A5 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y4 )
& ( distinct @ A @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_1578_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [X2: nat] :
( ( ord_less @ nat @ X2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X2 )
= X )
& ! [Y4: nat] :
( ( ( ord_less @ nat @ Y4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y4 )
= X ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1579_subseqs__refl,axiom,
! [A: $tType,Xs: list @ A] : ( member2 @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs_refl
thf(fact_1580_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
= Xs ) ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_1581_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F: A > ( list @ B ),G: A > ( list @ B )] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( bind @ A @ B @ Xs @ F )
= ( bind @ A @ B @ Ys @ G ) ) ) ) ).
% list_bind_cong
thf(fact_1582_list__ex1__iff,axiom,
! [A: $tType] :
( ( list_ex1 @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs3 ) )
& ( P3 @ X3 )
& ! [Y3: A] :
( ( ( member2 @ A @ Y3 @ ( set2 @ A @ Xs3 ) )
& ( P3 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_1583_merge__list__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Ls: list @ ( list @ A ),As2: list @ ( list @ A )] :
( ! [L2: list @ A] :
( ( member2 @ ( list @ A ) @ L2 @ ( set2 @ ( list @ A ) @ Ls ) )
=> ( ( distinct @ A @ L2 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L2 ) ) )
=> ( ! [L2: list @ A] :
( ( member2 @ ( list @ A ) @ L2 @ ( set2 @ ( list @ A ) @ As2 ) )
=> ( ( distinct @ A @ L2 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L2 ) ) )
=> ( ( distinct @ A @ ( merge_list @ A @ As2 @ Ls ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge_list @ A @ As2 @ Ls ) )
& ( ( set2 @ A @ ( merge_list @ A @ As2 @ Ls ) )
= ( set2 @ A @ ( concat @ A @ ( append @ ( list @ A ) @ As2 @ Ls ) ) ) ) ) ) ) ) ).
% merge_list_correct
thf(fact_1584_in__set__member,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
= ( member @ A @ Xs @ X ) ) ).
% in_set_member
thf(fact_1585_in__set__drop__conv__nth,axiom,
! [A: $tType,X: A,N: nat,L: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ ( drop @ A @ N @ L ) ) )
= ( ? [I2: nat] :
( ( ord_less_eq @ nat @ N @ I2 )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
& ( X
= ( nth @ A @ L @ I2 ) ) ) ) ) ).
% in_set_drop_conv_nth
thf(fact_1586_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_1587_drop__eq__ConsD,axiom,
! [A: $tType,N: nat,Xs: list @ A,X: A,Xs6: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( cons @ A @ X @ Xs6 ) )
=> ( ( drop @ A @ ( suc @ N ) @ Xs )
= Xs6 ) ) ).
% drop_eq_ConsD
thf(fact_1588_distinct__singleton,axiom,
! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_1589_sorted__take,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N @ Xs ) ) ) ) ).
% sorted_take
thf(fact_1590_sorted__drop,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N @ Xs ) ) ) ) ).
% sorted_drop
thf(fact_1591_nth__via__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ( ( nth @ A @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_1592_distinct__match,axiom,
! [A: $tType,Al: list @ A,E3: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( distinct @ A @ ( append @ A @ Al @ ( cons @ A @ E3 @ Bl ) ) )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E3 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E3 @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
thf(fact_1593_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_1594_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_1595_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1596_split__list,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_1597_split__list__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member2 @ A @ X @ ( set2 @ A @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_1598_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_1599_split__list__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member2 @ A @ X @ ( set2 @ A @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_1600_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_1601_append__Cons__eq__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Xs6: list @ A,Ys7: list @ A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( member2 @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ( ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) )
= ( append @ A @ Xs6 @ ( cons @ A @ X @ Ys7 ) ) )
= ( ( Xs = Xs6 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1602_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs2: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1603_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list @ A,X2: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa2: A] :
( ( member2 @ A @ Xa2 @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1604_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa2: A] :
( ( member2 @ A @ Xa2 @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1605_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list @ A,X2: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: A] :
( ( member2 @ A @ Xa2 @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1606_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list @ A,X2: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa2: A] :
( ( member2 @ A @ Xa2 @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1607_in__set__conv__decomp__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member2 @ A @ X @ ( set2 @ A @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1608_in__set__conv__decomp__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member2 @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1609_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list @ A,X3: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1610_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list @ A,X3: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1611_in__set__list__format,axiom,
! [A: $tType,E3: A,L: list @ A] :
( ( member2 @ A @ E3 @ ( set2 @ A @ L ) )
=> ~ ! [L1: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ E3 @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_1612_xy__in__set__cases,axiom,
! [A: $tType,X: A,L: list @ A,Y: A] :
( ( member2 @ A @ X @ ( set2 @ A @ L ) )
=> ( ( member2 @ 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_1613_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs ) )
=> ( Ys = Xs ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_1614_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_1615_sorted__wrt__append,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ P @ Xs )
& ( sorted_wrt @ A @ P @ Ys )
& ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Ys ) )
=> ( P @ X3 @ Y3 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1616_in__set__butlast__appendI,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( ( member2 @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
| ( member2 @ A @ X @ ( set2 @ A @ ( butlast @ A @ Ys ) ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ ( butlast @ A @ ( append @ A @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1617_set__remove1__subset,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_remove1_subset
thf(fact_1618_remove1__append,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remove1 @ A @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( remove1 @ A @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1619_Cons__in__subseqsD,axiom,
! [A: $tType,Y: A,Ys: list @ A,Xs: list @ A] :
( ( member2 @ ( list @ A ) @ ( cons @ A @ Y @ Ys ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( member2 @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_1620_subset__code_I2_J,axiom,
! [B: $tType,A5: set @ B,Ys: list @ B] :
( ( ord_less_eq @ ( set @ B ) @ A5 @ ( coset @ B @ Ys ) )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ ( set2 @ B @ Ys ) )
=> ~ ( member2 @ B @ X3 @ A5 ) ) ) ) ).
% subset_code(2)
thf(fact_1621_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X3: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( member2 @ A @ X3 @ ( set2 @ A @ Xs3 ) ) @ Xs3 @ ( cons @ A @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_1622_arg__min__list__in,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs: list @ A,F: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( member2 @ A @ ( arg_min_list @ A @ B @ F @ Xs ) @ ( set2 @ A @ Xs ) ) ) ) ).
% arg_min_list_in
thf(fact_1623_in__set__product__lists__length,axiom,
! [A: $tType,Xs: list @ A,Xss2: list @ ( list @ A )] :
( ( member2 @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_1624_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N: nat,Xs: list @ A] :
( ( member2 @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1625_id__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( Xs
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1626_strict__sorted__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
& ( distinct @ A @ L ) ) ) ) ).
% strict_sorted_iff
thf(fact_1627_not__distinct__decomp,axiom,
! [A: $tType,Ws2: list @ A] :
( ~ ( distinct @ A @ Ws2 )
=> ? [Xs2: list @ A,Ys2: list @ A,Zs3: list @ A,Y2: A] :
( Ws2
= ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ ( append @ A @ Ys2 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_1628_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_1629_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs3: list @ A] :
! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [J2: nat] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( I2 != J2 )
=> ( ( nth @ A @ Xs3 @ I2 )
!= ( nth @ A @ Xs3 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_1630_take__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ N @ ( butlast @ A @ Xs ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_1631_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1632_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X @ X3 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_1633_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( ord_less @ A @ X @ X3 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_1634_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
& ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ).
% sorted_append
thf(fact_1635_all__set__conv__nth,axiom,
! [A: $tType,L: list @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ L ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( P @ ( nth @ A @ L @ I2 ) ) ) ) ) ).
% all_set_conv_nth
thf(fact_1636_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1637_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1638_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1639_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1640_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member2 @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_1641_card__length,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% card_length
thf(fact_1642_butlast__subset,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( butlast @ A @ Xs ) ) @ A5 ) ) ) ).
% butlast_subset
thf(fact_1643_remove1__split,axiom,
! [A: $tType,A3: A,Xs: list @ A,Ys: list @ A] :
( ( member2 @ A @ A3 @ ( set2 @ A @ Xs ) )
=> ( ( ( remove1 @ A @ A3 @ Xs )
= Ys )
= ( ? [Ls3: list @ A,Rs: list @ A] :
( ( Xs
= ( append @ A @ Ls3 @ ( cons @ A @ A3 @ Rs ) ) )
& ~ ( member2 @ A @ A3 @ ( set2 @ A @ Ls3 ) )
& ( Ys
= ( append @ A @ Ls3 @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1644_subset__code_I3_J,axiom,
! [C: $tType] :
~ ( ord_less_eq @ ( set @ C ) @ ( coset @ C @ ( nil @ C ) ) @ ( set2 @ C @ ( nil @ C ) ) ) ).
% subset_code(3)
thf(fact_1645_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_1646_drop__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ Xs ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_1647_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( ( take @ A @ K @ Xs )
= ( take @ A @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1648_distinct__idx,axiom,
! [B: $tType,A: $tType,F: B > A,L: list @ B,I: nat,J: nat] :
( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ( F @ ( nth @ B @ L @ I ) )
= ( F @ ( nth @ B @ L @ J ) ) )
=> ( I = J ) ) ) ) ) ).
% distinct_idx
thf(fact_1649_butlast__conv__take,axiom,
! [A: $tType] :
( ( butlast @ A )
= ( ^ [Xs3: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( one_one @ nat ) ) @ Xs3 ) ) ) ).
% butlast_conv_take
thf(fact_1650_length__remove1,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_remove1
thf(fact_1651_take__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_1652_Cons__nth__drop__Suc,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) )
= ( drop @ A @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1653_butlast__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_take
thf(fact_1654_sorted__append__bigger,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X2 @ Y ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_append_bigger
thf(fact_1655_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1656_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ~ ! [L2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L2 )
=> ( ( ( set2 @ A @ L2 )
= A5 )
=> ( ( size_size @ ( list @ A ) @ L2 )
!= ( finite_card @ A @ A5 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_1657_distinct__sorted__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( distinct @ A @ L )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) ) ) ) ) ) ) ).
% distinct_sorted_mono
thf(fact_1658_distinct__sorted__strict__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
= ( ord_less @ nat @ I @ J ) ) ) ) ) ) ) ).
% distinct_sorted_strict_mono_iff
thf(fact_1659_set__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_union
thf(fact_1660_can__select__set__list__ex1,axiom,
! [A: $tType,P: A > $o,A5: list @ A] :
( ( can_select @ A @ P @ ( set2 @ A @ A5 ) )
= ( list_ex1 @ A @ P @ A5 ) ) ).
% can_select_set_list_ex1
thf(fact_1661_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_1662_length__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( zero_zero @ nat ) ) )
& ( ( Xs
!= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_1663_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ~ ! [L2: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F @ L2 ) )
=> ( ( ( set2 @ B @ L2 )
= A5 )
=> ( ( size_size @ ( list @ B ) @ L2 )
!= ( finite_card @ B @ A5 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_1664_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_1665_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F: B > A,A3: A,Xs: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( append @ B @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_1666_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] :
( ( member2 @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys3 ) )
& ! [Ys3: list @ A,Zs2: list @ A] :
( ( ( member2 @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( member2 @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( Ys3 != Zs2 ) )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% distinct_concat_iff
thf(fact_1667_removeAll__id,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( removeAll @ A @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1668_removeAll__append,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( removeAll @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( removeAll @ A @ X @ Xs ) @ ( removeAll @ A @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_1669_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F: B > A,A3: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_1670_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F: B > A,A3: A,X: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( cons @ B @ X @ Xs ) )
= ( plus_plus @ A @ ( F @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_1671_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_1672_can__select__def,axiom,
! [A: $tType] :
( ( can_select @ A )
= ( ^ [P3: A > $o,A7: set @ A] :
? [X3: A] :
( ( member2 @ A @ X3 @ A7 )
& ( P3 @ X3 )
& ! [Y3: A] :
( ( ( member2 @ A @ Y3 @ A7 )
& ( P3 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_1673_transpose_Osimps_I1_J,axiom,
! [A: $tType] :
( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ).
% transpose.simps(1)
thf(fact_1674_removeAll_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( removeAll @ A @ X @ ( cons @ A @ Y @ Xs ) )
= ( removeAll @ A @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( removeAll @ A @ X @ ( cons @ A @ Y @ Xs ) )
= ( cons @ A @ Y @ ( removeAll @ A @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_1675_removeAll_Osimps_I1_J,axiom,
! [A: $tType,X: A] :
( ( removeAll @ A @ X @ ( nil @ A ) )
= ( nil @ A ) ) ).
% removeAll.simps(1)
thf(fact_1676_distinct__removeAll,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_1677_transpose_Osimps_I2_J,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
= ( transpose @ A @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_1678_transpose__empty,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( ( transpose @ A @ Xs )
= ( nil @ ( list @ A ) ) )
= ( ! [X3: list @ A] :
( ( member2 @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( X3
= ( nil @ A ) ) ) ) ) ).
% transpose_empty
thf(fact_1679_length__removeAll__less__eq,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_1680_transpose__map__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ ( list @ B )] :
( ( transpose @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F ) @ Xs ) )
= ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F ) @ ( transpose @ B @ Xs ) ) ) ).
% transpose_map_map
thf(fact_1681_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( distinct @ A @ ( map @ B @ A @ F @ Xs ) )
=> ( distinct @ B @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_1682_distinct__remove1__removeAll,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X @ Xs )
= ( removeAll @ A @ X @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_1683_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_1684_remove__code_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remove @ A @ X @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_1685_length__removeAll__less,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1686_sorted__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) ) ) ).
% sorted_transpose
thf(fact_1687_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A,A5: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F @ L ) )
& ( ( set2 @ B @ L )
= A5 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A5 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_1688_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,F: B > A,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_1689_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ ( set2 @ B @ Xs ) @ S )
=> ( ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ ( set2 @ B @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_1690_take__hd__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( append @ A @ ( take @ A @ N @ Xs ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N @ Xs ) ) @ ( nil @ A ) ) )
= ( take @ A @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_1691_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N: nat,X: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ ( nth @ A @ Xs @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1692_is__empty__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( is_empty @ A @ ( set2 @ A @ Xs ) )
= ( null @ A @ Xs ) ) ).
% is_empty_set
thf(fact_1693_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,F: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 )
= ( nil @ B ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1694_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S: list @ A,I5: ( set @ A ) > B > $o,Sigma_0: B,F: B > A > B] :
( ( distinct @ A @ S )
=> ( ( I5 @ ( set2 @ A @ S ) @ Sigma_0 )
=> ( ! [X2: A,It: set @ A,Sigma: B] :
( ( member2 @ A @ X2 @ It )
=> ( ( ord_less_eq @ ( set @ A ) @ It @ ( set2 @ A @ S ) )
=> ( ( I5 @ It @ Sigma )
=> ( I5 @ ( minus_minus @ ( set @ A ) @ It @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( F @ Sigma @ X2 ) ) ) ) )
=> ( I5 @ ( bot_bot @ ( set @ A ) ) @ ( foldl @ B @ A @ F @ Sigma_0 @ S ) ) ) ) ) ).
% distinct_foldl_invar
thf(fact_1695_list__update__overwrite,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A,Y: A] :
( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X ) @ I @ Y )
= ( list_update @ A @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_1696_list__update__nonempty,axiom,
! [A: $tType,Xs: list @ A,K: nat,X: A] :
( ( ( list_update @ A @ Xs @ K @ X )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% list_update_nonempty
thf(fact_1697_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_1698_nth__list__update__neq,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A,X: A] :
( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_1699_list__update__id,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_1700_foldl__append,axiom,
! [A: $tType,B: $tType,F: A > B > A,A3: A,Xs: list @ B,Ys: list @ B] :
( ( foldl @ A @ B @ F @ A3 @ ( append @ B @ Xs @ Ys ) )
= ( foldl @ A @ B @ F @ ( foldl @ A @ B @ F @ A3 @ Xs ) @ Ys ) ) ).
% foldl_append
thf(fact_1701_take__update,axiom,
! [A: $tType,N: nat,L: list @ A,I: nat,X: A] :
( ( take @ A @ N @ ( list_update @ A @ L @ I @ X ) )
= ( list_update @ A @ ( take @ A @ N @ L ) @ I @ X ) ) ).
% take_update
thf(fact_1702_take__update__swap,axiom,
! [A: $tType,M2: nat,Xs: list @ A,N: nat,X: A] :
( ( take @ A @ M2 @ ( list_update @ A @ Xs @ N @ X ) )
= ( list_update @ A @ ( take @ A @ M2 @ Xs ) @ N @ X ) ) ).
% take_update_swap
thf(fact_1703_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1704_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_1705_take__update__cancel,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A,Y: A] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( take @ A @ N @ ( list_update @ A @ Xs @ M2 @ Y ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_1706_drop__upd__irrelevant,axiom,
! [A: $tType,M2: nat,N: nat,L: list @ A,X: A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( drop @ A @ N @ ( list_update @ A @ L @ M2 @ X ) )
= ( drop @ A @ N @ L ) ) ) ).
% drop_upd_irrelevant
thf(fact_1707_drop__update__cancel,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N @ M2 )
=> ( ( drop @ A @ M2 @ ( list_update @ A @ Xs @ N @ X ) )
= ( drop @ A @ M2 @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_1708_list__update__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( list_update @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) @ Y )
= ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1709_nth__update__invalid,axiom,
! [A: $tType,I: nat,L: list @ A,J: nat,X: A] :
( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ J @ X ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_update_invalid
thf(fact_1710_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_1711_hd__take,axiom,
! [A: $tType,J: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ J )
=> ( ( hd @ A @ ( take @ A @ J @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_take
thf(fact_1712_set__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_1713_distinct__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( distinct @ A @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_1714_foldl__foldl__conv__concat,axiom,
! [A: $tType,B: $tType,F: A > B > A,A3: A,Xs: list @ ( list @ B )] :
( ( foldl @ A @ ( list @ B ) @ ( foldl @ A @ B @ F ) @ A3 @ Xs )
= ( foldl @ A @ B @ F @ A3 @ ( concat @ B @ Xs ) ) ) ).
% foldl_foldl_conv_concat
thf(fact_1715_foldl__A1__eq,axiom,
! [A: $tType,F: A > A > A,N: A,I: A,Ww: list @ A] :
( ! [E: A] :
( ( F @ N @ E )
= E )
=> ( ! [E: A] :
( ( F @ E @ N )
= E )
=> ( ! [A6: A,B6: A,C2: A] :
( ( F @ A6 @ ( F @ B6 @ C2 ) )
= ( F @ ( F @ A6 @ B6 ) @ C2 ) )
=> ( ( foldl @ A @ A @ F @ I @ Ww )
= ( F @ I @ ( foldl @ A @ A @ F @ N @ Ww ) ) ) ) ) ) ).
% foldl_A1_eq
thf(fact_1716_list__update__swap,axiom,
! [A: $tType,I: nat,I6: nat,Xs: list @ A,X: A,X9: A] :
( ( I != I6 )
=> ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X ) @ I6 @ X9 )
= ( list_update @ A @ ( list_update @ A @ Xs @ I6 @ X9 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_1717_linorder_Osorted__key__list__of__set_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( sorted8670434370408473282of_set @ A @ B )
= ( sorted8670434370408473282of_set @ A @ B ) ) ).
% linorder.sorted_key_list_of_set.cong
thf(fact_1718_list__update_Osimps_I1_J,axiom,
! [A: $tType,I: nat,V: A] :
( ( list_update @ A @ ( nil @ A ) @ I @ V )
= ( nil @ A ) ) ).
% list_update.simps(1)
thf(fact_1719_list__update__code_I1_J,axiom,
! [A: $tType,I: nat,Y: A] :
( ( list_update @ A @ ( nil @ A ) @ I @ Y )
= ( nil @ A ) ) ).
% list_update_code(1)
thf(fact_1720_map__update,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,K: nat,Y: B] :
( ( map @ B @ A @ F @ ( list_update @ B @ Xs @ K @ Y ) )
= ( list_update @ A @ ( map @ B @ A @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).
% map_update
thf(fact_1721_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_1722_butlast__update_H,axiom,
! [A: $tType,L: list @ A,I: nat,X: A] :
( ( list_update @ A @ ( butlast @ A @ L ) @ I @ X )
= ( butlast @ A @ ( list_update @ A @ L @ I @ X ) ) ) ).
% butlast_update'
thf(fact_1723_foldl__Cons,axiom,
! [B: $tType,A: $tType,F: B > A > B,A3: B,X: A,Xs: list @ A] :
( ( foldl @ B @ A @ F @ A3 @ ( cons @ A @ X @ Xs ) )
= ( foldl @ B @ A @ F @ ( F @ A3 @ X ) @ Xs ) ) ).
% foldl_Cons
thf(fact_1724_foldl__Nil,axiom,
! [A: $tType,B: $tType,F: B > A > B,A3: B] :
( ( foldl @ B @ A @ F @ A3 @ ( nil @ A ) )
= A3 ) ).
% foldl_Nil
thf(fact_1725_foldl__cong,axiom,
! [A: $tType,B: $tType,A3: A,B2: A,L: list @ B,K: list @ B,F: A > B > A,G: A > B > A] :
( ( A3 = B2 )
=> ( ( L = K )
=> ( ! [A6: A,X2: B] :
( ( member2 @ B @ X2 @ ( set2 @ B @ L ) )
=> ( ( F @ A6 @ X2 )
= ( G @ A6 @ X2 ) ) )
=> ( ( foldl @ A @ B @ F @ A3 @ L )
= ( foldl @ A @ B @ G @ B2 @ K ) ) ) ) ) ).
% foldl_cong
thf(fact_1726_hd__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( Xs
!= ( nil @ ( list @ A ) ) )
=> ( ( ( hd @ ( list @ A ) @ Xs )
!= ( nil @ A ) )
=> ( ( hd @ A @ ( concat @ A @ Xs ) )
= ( hd @ A @ ( hd @ ( list @ A ) @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_1727_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) @ Y )
= ( cons @ A @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1728_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I: nat,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons @ A @ X @ ( list_update @ A @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1729_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A,X: A,I: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X ) ) @ A5 ) ) ) ).
% set_update_subsetI
thf(fact_1730_hd__in__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( member2 @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ).
% hd_in_set
thf(fact_1731_list_Oset__sel_I1_J,axiom,
! [A: $tType,A3: list @ A] :
( ( A3
!= ( nil @ A ) )
=> ( member2 @ A @ ( hd @ A @ A3 ) @ ( set2 @ A @ A3 ) ) ) ).
% list.set_sel(1)
thf(fact_1732_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A3: list @ A,F: A > B] :
( ( A3
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F @ A3 ) )
= ( F @ ( hd @ A @ A3 ) ) ) ) ).
% list.map_sel(1)
thf(fact_1733_hd__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F @ Xs ) )
= ( F @ ( hd @ A @ Xs ) ) ) ) ).
% hd_map
thf(fact_1734_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Ys ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_1735_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ps2: list @ A,Xs4: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Ps2 @ Xs4 ) )
& ( Ys
= ( append @ A @ Ps2 @ Ys4 ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys4
= ( nil @ A ) )
| ( ( hd @ A @ Xs4 )
!= ( hd @ A @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_1736_hd__Nil__eq__last,axiom,
! [A: $tType] :
( ( hd @ A @ ( nil @ A ) )
= ( last @ A @ ( nil @ A ) ) ) ).
% hd_Nil_eq_last
thf(fact_1737_last__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( last @ A @ ( rev @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% last_rev
thf(fact_1738_hd__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( rev @ A @ Xs ) )
= ( last @ A @ Xs ) ) ).
% hd_rev
thf(fact_1739_foldl__conc__empty__eq,axiom,
! [A: $tType,I: list @ A,Ww: list @ ( list @ A )] :
( ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ I @ Ww )
= ( append @ A @ I @ ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ ( nil @ A ) @ Ww ) ) ) ).
% foldl_conc_empty_eq
thf(fact_1740_foldl__absorb1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Zs: list @ A] :
( ( times_times @ A @ X @ ( foldl @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) @ Zs ) )
= ( foldl @ A @ A @ ( times_times @ A ) @ X @ Zs ) ) ) ).
% foldl_absorb1
thf(fact_1741_foldl__un__empty__eq,axiom,
! [A: $tType,I: set @ A,Ww: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ I @ Ww )
= ( sup_sup @ ( set @ A ) @ I @ ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ Ww ) ) ) ).
% foldl_un_empty_eq
thf(fact_1742_foldl__list__update,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F: B > A > B,A3: B,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( foldl @ B @ A @ F @ A3 @ ( list_update @ A @ Xs @ N @ X ) )
= ( foldl @ B @ A @ F @ ( F @ ( foldl @ B @ A @ F @ A3 @ ( take @ A @ N @ Xs ) ) @ X ) @ ( drop @ A @ ( suc @ N ) @ Xs ) ) ) ) ).
% foldl_list_update
thf(fact_1743_null__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
~ ( null @ A @ ( cons @ A @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_1744_eq__Nil__null,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
= ( nil @ A ) )
= ( null @ A @ Xs ) ) ).
% eq_Nil_null
thf(fact_1745_null__rec_I2_J,axiom,
! [B: $tType] : ( null @ B @ ( nil @ B ) ) ).
% null_rec(2)
thf(fact_1746_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X ) ) @ ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1747_in__set__upd__eq__aux,axiom,
! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member2 @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X = Y )
| ! [Y3: A] : ( member2 @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y3 ) ) ) ) ) ) ).
% in_set_upd_eq_aux
thf(fact_1748_in__set__upd__cases,axiom,
! [A: $tType,X: A,L: list @ A,I: nat,Y: A] :
( ( member2 @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
=> ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( X != Y ) )
=> ( member2 @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_set_upd_cases
thf(fact_1749_in__set__upd__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member2 @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X = Y )
| ( ( member2 @ A @ X @ ( set2 @ A @ L ) )
& ! [Y3: A] : ( member2 @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y3 ) ) ) ) ) ) ) ).
% in_set_upd_eq
thf(fact_1750_set__update__memI,axiom,
! [A: $tType,N: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member2 @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1751_list__update__append1,axiom,
! [A: $tType,I: nat,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ I @ X )
= ( append @ A @ ( list_update @ A @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1752_nth__list__update_H,axiom,
! [A: $tType,I: nat,J: nat,L: list @ A,X: A] :
( ( ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
= X ) )
& ( ~ ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
= ( nth @ A @ L @ J ) ) ) ) ).
% nth_list_update'
thf(fact_1753_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1754_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1755_drop__update__swap,axiom,
! [A: $tType,M2: nat,N: nat,Xs: list @ A,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( drop @ A @ M2 @ ( list_update @ A @ Xs @ N @ X ) )
= ( list_update @ A @ ( drop @ A @ M2 @ Xs ) @ ( minus_minus @ nat @ N @ M2 ) @ X ) ) ) ).
% drop_update_swap
thf(fact_1756_hd__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ Xs )
= ( nth @ A @ Xs @ ( zero_zero @ nat ) ) ) ) ).
% hd_conv_nth
thf(fact_1757_distinct__butlast__swap,axiom,
! [A: $tType,Pq: list @ A,I: nat] :
( ( distinct @ A @ Pq )
=> ( distinct @ A @ ( butlast @ A @ ( list_update @ A @ Pq @ I @ ( last @ A @ Pq ) ) ) ) ) ).
% distinct_butlast_swap
thf(fact_1758_foldl__rule,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2] :
( ( I5 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X2 @ L22 ) ) )
=> ( ( I5 @ Sigma @ L1 @ ( cons @ A @ X2 @ L22 ) )
=> ( I5 @ ( F @ Sigma @ X2 ) @ ( append @ A @ L1 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ L22 ) ) )
=> ( I5 @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) @ L0 @ ( nil @ A ) ) ) ) ).
% foldl_rule
thf(fact_1759_foldl__rule__P,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I5 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X2 @ L22 ) ) )
=> ( ( I5 @ Sigma @ L1 @ ( cons @ A @ X2 @ L22 ) )
=> ( I5 @ ( F @ Sigma @ X2 ) @ ( append @ A @ L1 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ L22 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I5 @ Sigma @ L0 @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_P
thf(fact_1760_foldl__rule__aux,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2] :
( ( I5 @ Sigma_0 @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X2 @ L22 ) ) )
=> ( ( I5 @ Sigma @ ( cons @ A @ X2 @ L22 ) )
=> ( I5 @ ( F @ Sigma @ X2 ) @ L22 ) ) )
=> ( I5 @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) @ ( nil @ A ) ) ) ) ).
% foldl_rule_aux
thf(fact_1761_foldl__rule__aux__P,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I5 @ Sigma_0 @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X2 @ L22 ) ) )
=> ( ( I5 @ Sigma @ ( cons @ A @ X2 @ L22 ) )
=> ( I5 @ ( F @ Sigma @ X2 ) @ L22 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I5 @ Sigma @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_aux_P
thf(fact_1762_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A,X: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X )
= ( append @ A @ ( list_update @ A @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X )
= ( append @ A @ Xs @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1763_map__upd__eq,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F: A > B,X: A] :
( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F @ ( nth @ A @ L @ I ) )
= ( F @ X ) ) )
=> ( ( map @ A @ B @ F @ ( list_update @ A @ L @ I @ X ) )
= ( map @ A @ B @ F @ L ) ) ) ).
% map_upd_eq
thf(fact_1764_sorted__hd__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ! [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( hd @ A @ Xs ) @ X6 ) ) ) ) ) ).
% sorted_hd_min
thf(fact_1765_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,X: A] :
( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( butlast @ A @ Xs ) ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( list_update @ A @ ( butlast @ A @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_1766_hd__drop__conv__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( drop @ A @ N @ Xs ) )
= ( nth @ A @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_1767_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_1768_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_1769_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_1770_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,F: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ ( bot_bot @ ( set @ B ) ) )
= ( nil @ B ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
thf(fact_1771_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A,A5: set @ B,B4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ S )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 )
= ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ B4 ) )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( A5 = B4 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
thf(fact_1772_insert__swap__set__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( insert2 @ A @ ( nth @ A @ L @ I ) @ ( set2 @ A @ ( list_update @ A @ L @ I @ X ) ) )
= ( insert2 @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% insert_swap_set_eq
thf(fact_1773_rev__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% rev_update
thf(fact_1774_last__list__update,axiom,
! [A: $tType,Xs: list @ A,K: nat,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( last @ A @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1775_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_1776_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A3: A,I: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member2 @ A @ A3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I @ A3 ) ) ) ) ).
% distinct_list_update
thf(fact_1777_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 ) )
= A5 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
thf(fact_1778_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 ) )
= ( finite_card @ B @ A5 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_1779_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( distinct @ A @ ( map @ B @ A @ F @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_1780_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_1781_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_1782_take__update__last,axiom,
! [A: $tType,N: nat,List: list @ A,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ List ) )
=> ( ( list_update @ A @ ( take @ A @ ( suc @ N ) @ List ) @ N @ X )
= ( append @ A @ ( take @ A @ N @ List ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% take_update_last
thf(fact_1783_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A,A3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I @ A3 )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A3 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1784_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,F: B > A,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ ( insert2 @ B @ X @ A5 ) )
= ( insort_key @ A @ B @ Less_eq @ F @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_1785_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,F: B > A,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member2 @ B @ X @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ ( insert2 @ B @ X @ A5 ) )
= ( insort_key @ A @ B @ Less_eq @ F @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_1786_map__distinct__upd__conv,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F: A > B,X: B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( distinct @ A @ L )
=> ( ( list_update @ B @ ( map @ A @ B @ F @ L ) @ I @ X )
= ( map @ A @ B @ ( fun_upd @ A @ B @ F @ ( nth @ A @ L @ I ) @ X ) @ L ) ) ) ) ).
% map_distinct_upd_conv
thf(fact_1787_min__list__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( min_list @ A @ Xs )
= ( lattic643756798350308766er_Min @ A @ ( set2 @ A @ Xs ) ) ) ) ) ).
% min_list_Min
thf(fact_1788_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_1789_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_1790_Un__set__drop__extend,axiom,
! [A: $tType,J: nat,L: list @ ( set @ A )] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ ( set @ A ) ) @ L ) )
=> ( ( sup_sup @ ( set @ A ) @ ( nth @ ( set @ A ) @ L @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ J @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) @ L ) ) ) ) ) ) ).
% Un_set_drop_extend
thf(fact_1791_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( ( complete_Sup_Sup @ A @ A5 )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( X3
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1792_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A5 ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( X3
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1793_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_1794_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_1795_map__fun__upd,axiom,
! [B: $tType,A: $tType,Y: A,Xs: list @ A,F: A > B,V: B] :
( ~ ( member2 @ A @ Y @ ( set2 @ A @ Xs ) )
=> ( ( map @ A @ B @ ( fun_upd @ A @ B @ F @ Y @ V ) @ Xs )
= ( map @ A @ B @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_1796_butlast__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( tl @ A @ Xs ) ) ) ).
% butlast_rev
thf(fact_1797_length__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_tl
thf(fact_1798_hd__Cons__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_1799_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_1800_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list @ A,X: A] :
( ( L
!= ( nil @ A ) )
=> ( ( ( X
= ( hd @ A @ L ) )
| ( member2 @ A @ X @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
= ( member2 @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_1801_linorder_Oinsort__key_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( insort_key @ A @ B )
= ( insort_key @ A @ B ) ) ).
% linorder.insort_key.cong
thf(fact_1802_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_1803_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_1804_in__set__tlD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_tlD
thf(fact_1805_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_1806_Union__empty__conv,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: set @ A] :
( ( member2 @ ( set @ A ) @ X3 @ A5 )
=> ( X3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_1807_empty__Union__conv,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A5 ) )
= ( ! [X3: set @ A] :
( ( member2 @ ( set @ A ) @ X3 @ A5 )
=> ( X3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_1808_map__tl,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( map @ B @ A @ F @ ( tl @ B @ Xs ) )
= ( tl @ A @ ( map @ B @ A @ F @ Xs ) ) ) ).
% map_tl
thf(fact_1809_Union__take__drop__id,axiom,
! [A: $tType,N: nat,L: list @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ N @ L ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( take @ ( set @ A ) @ N @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ L ) ) ) ).
% Union_take_drop_id
thf(fact_1810_distinct__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( tl @ A @ Xs ) ) ) ).
% distinct_tl
thf(fact_1811_tl__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( tl @ A @ ( drop @ A @ N @ Xs ) )
= ( drop @ A @ N @ ( tl @ A @ Xs ) ) ) ).
% tl_drop
thf(fact_1812_butlast__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% butlast_tl
thf(fact_1813_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,U: A] :
( ! [V2: A] :
( ( member2 @ A @ V2 @ A5 )
=> ( ord_less_eq @ A @ U @ V2 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_1814_tl__obtain__elem,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( tl @ A @ Xs )
= ( nil @ A ) )
=> ~ ! [E: A] :
( Xs
!= ( cons @ A @ E @ ( nil @ A ) ) ) ) ) ).
% tl_obtain_elem
thf(fact_1815_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X3: A] :
( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_1816_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X3: A] :
( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_1817_list_Oset__sel_I2_J,axiom,
! [A: $tType,A3: list @ A,X: A] :
( ( A3
!= ( nil @ A ) )
=> ( ( member2 @ A @ X @ ( set2 @ A @ ( tl @ A @ A3 ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ A3 ) ) ) ) ).
% list.set_sel(2)
thf(fact_1818_list_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A3: list @ A,F: A > B] :
( ( A3
!= ( nil @ A ) )
=> ( ( tl @ B @ ( map @ A @ B @ F @ A3 ) )
= ( map @ A @ B @ F @ ( tl @ A @ A3 ) ) ) ) ).
% list.map_sel(2)
thf(fact_1819_sorted__tl,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( tl @ A @ Xs ) ) ) ) ).
% sorted_tl
thf(fact_1820_insert__partition,axiom,
! [A: $tType,X: set @ A,F2: set @ ( set @ A )] :
( ~ ( member2 @ ( set @ A ) @ X @ F2 )
=> ( ! [X2: set @ A] :
( ( member2 @ ( set @ A ) @ X2 @ ( insert2 @ ( set @ A ) @ X @ F2 ) )
=> ! [Xa3: set @ A] :
( ( member2 @ ( set @ A ) @ Xa3 @ ( insert2 @ ( set @ A ) @ X @ F2 ) )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X2 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ X @ ( complete_Sup_Sup @ ( set @ A ) @ F2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_partition
thf(fact_1821_Union__disjoint,axiom,
! [A: $tType,C4: set @ ( set @ A ),A5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: set @ A] :
( ( member2 @ ( set @ A ) @ X3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_1822_Union__Int__subset,axiom,
! [A: $tType,A5: set @ ( set @ A ),B4: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A5 @ B4 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) ) ) ).
% Union_Int_subset
thf(fact_1823_take__tl,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( take @ A @ N @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( take @ A @ ( suc @ N ) @ Xs ) ) ) ).
% take_tl
thf(fact_1824_drop__Suc,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( drop @ A @ ( suc @ N ) @ Xs )
= ( drop @ A @ N @ ( tl @ A @ Xs ) ) ) ).
% drop_Suc
thf(fact_1825_list_Oexpand,axiom,
! [A: $tType,List: list @ A,List3: list @ A] :
( ( ( List
= ( nil @ A ) )
= ( List3
= ( nil @ A ) ) )
=> ( ( ( List
!= ( nil @ A ) )
=> ( ( List3
!= ( nil @ A ) )
=> ( ( ( hd @ A @ List )
= ( hd @ A @ List3 ) )
& ( ( tl @ A @ List )
= ( tl @ A @ List3 ) ) ) ) )
=> ( List = List3 ) ) ) ).
% list.expand
thf(fact_1826_not__hd__in__tl,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( X
!= ( hd @ A @ Xs ) )
=> ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member2 @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% not_hd_in_tl
thf(fact_1827_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_1828_last__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( Xs
= ( nil @ A ) )
| ( ( tl @ A @ Xs )
!= ( nil @ A ) ) )
=> ( ( last @ A @ ( tl @ A @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_tl
thf(fact_1829_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_1830_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_1831_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] :
( ( member2 @ A @ X2 @ A5 )
=> ( ( member2 @ A @ Y2 @ A5 )
=> ( member2 @ A @ ( sup_sup @ A @ X2 @ Y2 ) @ A5 ) ) )
=> ( member2 @ A @ ( complete_Sup_Sup @ A @ A5 ) @ A5 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_1832_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B4 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_1833_tl__subset,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( tl @ A @ Xs ) ) @ A5 ) ) ) ).
% tl_subset
thf(fact_1834_Misc_Onth__tl,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( suc @ N ) ) ) ) ).
% Misc.nth_tl
thf(fact_1835_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 ) )
=> ( ! [N7: nat] :
( ( ord_less @ nat @ N7 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ B @ Ys @ N7 ) @ ( nth @ A @ Xs @ N7 ) ) )
=> ! [N8: nat] :
( ( ord_less @ nat @ N8 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( P @ ( nth @ B @ ( tl @ B @ Ys ) @ N8 ) @ ( nth @ A @ ( tl @ A @ Xs ) @ N8 ) ) ) ) ) ).
% list_take_induct_tl2
thf(fact_1836_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% Max_Sup
thf(fact_1837_list_Oexhaust__sel,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_1838_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_1839_tl__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( tl @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ).
% tl_take
thf(fact_1840_distinct__hd__tl,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( X
= ( hd @ A @ Xs ) )
=> ~ ( member2 @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% distinct_hd_tl
thf(fact_1841_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_1842_remove1__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( remove1 @ A @ ( hd @ A @ Xs ) @ Xs )
= ( tl @ A @ Xs ) ) ) ).
% remove1_tl
thf(fact_1843_list__collect__set__as__map,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F3: B > ( set @ A ),L4: list @ B] : ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( map @ B @ ( set @ A ) @ F3 @ L4 ) ) ) ) ) ).
% list_collect_set_as_map
thf(fact_1844_List_Onth__tl,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( suc @ N ) ) ) ) ).
% List.nth_tl
thf(fact_1845_card__partition,axiom,
! [A: $tType,C4: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ ( set @ A ) @ C4 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
=> ( ! [C2: set @ A] :
( ( member2 @ ( set @ A ) @ C2 @ C4 )
=> ( ( finite_card @ A @ C2 )
= K ) )
=> ( ! [C13: set @ A,C24: set @ A] :
( ( member2 @ ( set @ A ) @ C13 @ C4 )
=> ( ( member2 @ ( set @ A ) @ C24 @ C4 )
=> ( ( C13 != C24 )
=> ( ( inf_inf @ ( set @ A ) @ C13 @ C24 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C4 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_1846_take__Suc,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( take @ A @ ( suc @ N ) @ Xs )
= ( cons @ A @ ( hd @ A @ Xs ) @ ( take @ A @ N @ ( tl @ A @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_1847_rotate1__hd__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( rotate1 @ A @ Xs )
= ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ).
% rotate1_hd_tl
thf(fact_1848_Int__greaterThanLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C3 @ D3 ) )
= ( set_or5935395276787703475ssThan @ A @ ( ord_max @ A @ A3 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanLessThan
thf(fact_1849_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_1850_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_1851_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_1852_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_1853_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_1854_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_1855_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,Z2: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ X5 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X5 ) @ Z2 ) ) ) ) ).
% cSup_least
thf(fact_1856_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A3: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ X5 )
=> ( ord_less_eq @ A @ X2 @ A3 ) )
=> ( ! [Y2: A] :
( ! [X6: A] :
( ( member2 @ A @ X6 @ X5 )
=> ( ord_less_eq @ A @ X6 @ Y2 ) )
=> ( ord_less_eq @ A @ A3 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X5 )
= A3 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1857_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X5: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X5 ) )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X2: A] :
( ( member2 @ A @ X2 @ X5 )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ) ) ).
% less_cSupE
thf(fact_1858_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,Z2: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X5 ) )
=> ? [X2: A] :
( ( member2 @ A @ X2 @ X5 )
& ( ord_less @ A @ Z2 @ X2 ) ) ) ) ) ).
% less_cSupD
thf(fact_1859_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,A3: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X5 ) @ A3 )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ X5 )
=> ( ord_less @ A @ X3 @ A3 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_1860_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X5 )
= ( lattic643756798349783984er_Max @ A @ X5 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_1861_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X5 )
= ( lattic5882676163264333800up_fin @ A @ X5 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_1862_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_1863_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B4: set @ A,A3: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B4 ) @ A3 )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ B4 )
=> ( ( inf_inf @ A @ X3 @ A3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_1864_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F3: A > nat,Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F3 @ ( hd @ A @ Xs3 ) ) @ ( size_list @ A @ F3 @ ( tl @ A @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_1865_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ I ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_1866_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( ~ ( ( ( List
= ( nil @ A ) )
& ~ ( P @ F1 ) )
| ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
& ~ ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% list.split_sel_asm
thf(fact_1867_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( ( ( List
= ( nil @ A ) )
=> ( P @ F1 ) )
& ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
=> ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% list.split_sel
thf(fact_1868_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat,S4: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ S @ ( set_ord_lessThan @ A @ S4 ) ) ) )
=> ( ( infini527867602293511546merate @ A @ ( inf_inf @ ( set @ A ) @ S @ ( set_ord_lessThan @ A @ S4 ) ) @ N )
= ( infini527867602293511546merate @ A @ S @ N ) ) ) ) ) ).
% finite_enumerate_initial_segment
thf(fact_1869_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_1870_PowI,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( member2 @ ( set @ A ) @ A5 @ ( pow @ A @ B4 ) ) ) ).
% PowI
thf(fact_1871_Pow__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( member2 @ ( set @ A ) @ A5 @ ( pow @ A @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% Pow_iff
thf(fact_1872_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_1873_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_1874_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_1875_size__list__append,axiom,
! [A: $tType,F: A > nat,Xs: list @ A,Ys: list @ A] :
( ( size_list @ A @ F @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_list @ A @ F @ Xs ) @ ( size_list @ A @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_1876_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K ) ) @ ( cons @ nat @ K @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_1877_Pow__Int__eq,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( pow @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( pow @ A @ B4 ) ) ) ).
% Pow_Int_eq
thf(fact_1878_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_1879_Int__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( ord_max @ A @ A3 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanAtMost
thf(fact_1880_Pow__empty,axiom,
! [A: $tType] :
( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_1881_Pow__singleton__iff,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A] :
( ( ( pow @ A @ X5 )
= ( insert2 @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
& ( Y6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_1882_Pow__top,axiom,
! [A: $tType,A5: set @ A] : ( member2 @ ( set @ A ) @ A5 @ ( pow @ A @ A5 ) ) ).
% Pow_top
thf(fact_1883_Pow__bottom,axiom,
! [A: $tType,B4: set @ A] : ( member2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B4 ) ) ).
% Pow_bottom
thf(fact_1884_PowD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( member2 @ ( set @ A ) @ A5 @ ( pow @ A @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% PowD
thf(fact_1885_Pow__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( pow @ A @ B4 ) ) ) ).
% Pow_mono
thf(fact_1886_Un__Pow__subset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( pow @ A @ B4 ) ) @ ( pow @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% Un_Pow_subset
thf(fact_1887_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_1888_Pow__not__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( pow @ A @ A5 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_1889_list_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > ( list @ A ) > B,X21: A,X22: list @ A] :
( ( case_list @ B @ A @ F1 @ F22 @ ( cons @ A @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% list.simps(5)
thf(fact_1890_list_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > ( list @ A ) > B] :
( ( case_list @ B @ A @ F1 @ F22 @ ( nil @ A ) )
= F1 ) ).
% list.simps(4)
thf(fact_1891_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(6)
thf(fact_1892_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N: A] :
( ( ( set_ord_lessThan @ A @ N )
= ( bot_bot @ ( set @ A ) ) )
= ( N
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_1893_greaterThan__Int__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( set_ord_lessThan @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% greaterThan_Int_greaterThan
thf(fact_1894_list_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_list @ A @ X @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_1895_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F: A > nat] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y @ ( F @ X ) )
=> ( ord_less @ nat @ Y @ ( size_list @ A @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_1896_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F: A > nat] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F @ X ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_1897_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F: A > nat,G: A > nat] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_1898_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ord_less_eq @ A @ D3 @ C3 )
| ( ord_less_eq @ A @ B2 @ C3 )
| ( ord_less_eq @ A @ D3 @ A3 ) ) ) ) ).
% Ioc_disjoint
thf(fact_1899_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(2)
thf(fact_1900_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_1901_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_1902_list_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_list @ B @ A )
= ( ^ [F12: B,F23: A > ( list @ A ) > B,List2: list @ A] :
( if @ B
@ ( List2
= ( nil @ A ) )
@ F12
@ ( F23 @ ( hd @ A @ List2 ) @ ( tl @ A @ List2 ) ) ) ) ) ).
% list.case_eq_if
thf(fact_1903_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_1904_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_1905_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,Ys: list @ B] :
( ( inj_on @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Ys ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F @ Ys ) )
=> ( ( ( set2 @ B @ Xs )
= ( set2 @ B @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_1906_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linord329482645794927042rt_key @ B @ A @ F @ X @ Xs ) ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_1907_distinct__adj__append__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys ) )
= ( ( distinct_adj @ A @ Xs )
& ( distinct_adj @ A @ Ys )
& ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ A ) )
| ( ( last @ A @ Xs )
!= ( hd @ A @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_1908_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_1909_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% one_plus_numeral
thf(fact_1910_nth__rotate1,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_1911_inj__on__rev,axiom,
! [A: $tType,A5: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ A5 ) ).
% inj_on_rev
thf(fact_1912_distinct__adj__Cons__Cons,axiom,
! [B: $tType,X: B,Y: B,Xs: list @ B] :
( ( distinct_adj @ B @ ( cons @ B @ X @ ( cons @ B @ Y @ Xs ) ) )
= ( ( X != Y )
& ( distinct_adj @ B @ ( cons @ B @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_1913_distinct__adj__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct_adj @ A @ ( rev @ A @ Xs ) )
= ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_rev
thf(fact_1914_numeral__eq__one__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( numeral_numeral @ A @ N )
= ( one_one @ A ) )
= ( N = one2 ) ) ) ).
% numeral_eq_one_iff
thf(fact_1915_one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ N ) )
= ( one2 = N ) ) ) ).
% one_eq_numeral_iff
thf(fact_1916_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_1917_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_1918_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ one2 @ N ) ) ) ).
% one_less_numeral_iff
thf(fact_1919_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_1920_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A3: A,B2: A] :
( ( ( finite_card @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( A3 != B2 ) ) ).
% card_doubleton_eq_2_iff
thf(fact_1921_inj__on__Cons1,axiom,
! [A: $tType,X: A,A5: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ A5 ) ).
% inj_on_Cons1
thf(fact_1922_distinct__adj__map__iff,axiom,
! [B: $tType,A: $tType,F: A > B,Xs: list @ A] :
( ( inj_on @ A @ B @ F @ ( set2 @ A @ Xs ) )
=> ( ( distinct_adj @ B @ ( map @ A @ B @ F @ Xs ) )
= ( distinct_adj @ A @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_1923_distinct__adj__mapI,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
( ( distinct_adj @ A @ Xs )
=> ( ( inj_on @ A @ B @ F @ ( set2 @ A @ Xs ) )
=> ( distinct_adj @ B @ ( map @ A @ B @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_1924_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_1925_folding__insort__key_Oinj__on,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( inj_on @ B @ A @ F @ S ) ) ).
% folding_insort_key.inj_on
thf(fact_1926_distinct__adj__ConsD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct_adj @ A @ ( cons @ A @ X @ Xs ) )
=> ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_1927_distinct__adj__Nil,axiom,
! [A: $tType] : ( distinct_adj @ A @ ( nil @ A ) ) ).
% distinct_adj_Nil
thf(fact_1928_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_1929_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_1930_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_1931_distinct__adj__mapD,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( distinct_adj @ A @ ( map @ B @ A @ F @ Xs ) )
=> ( distinct_adj @ B @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_1932_distinct__adj__appendD2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys ) )
=> ( distinct_adj @ A @ Ys ) ) ).
% distinct_adj_appendD2
thf(fact_1933_distinct__adj__appendD1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys ) )
=> ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_1934_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_1935_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% power2_eq_square
thf(fact_1936_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_1937_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( modulo_modulo @ A @ X @ M2 ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M2 ) @ M2 ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1938_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_1939_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,A5: set @ A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A5 ) ) ) ).
% inj_on_mult
thf(fact_1940_list__decomp__2,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ? [A6: A,B6: A] :
( L
= ( cons @ A @ A6 @ ( cons @ A @ B6 @ ( nil @ A ) ) ) ) ) ).
% list_decomp_2
thf(fact_1941_card__2__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X3: A,Y3: A] :
( ( S
= ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X3 != Y3 ) ) ) ) ).
% card_2_iff
thf(fact_1942_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A3 )
= A3 ) ) ).
% mult_numeral_1
thf(fact_1943_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% mult_numeral_1_right
thf(fact_1944_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_1945_distinct__insort__insert,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F: B > A,X: B] :
( ( distinct @ B @ Xs )
=> ( distinct @ B @ ( linord329482645794927042rt_key @ B @ A @ F @ X @ Xs ) ) ) ) ).
% distinct_insort_insert
thf(fact_1946_length__subseqs,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_subseqs
thf(fact_1947_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_1948_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_1949_distinct__adj__singleton,axiom,
! [B: $tType,X: B] : ( distinct_adj @ B @ ( cons @ B @ X @ ( nil @ B ) ) ) ).
% distinct_adj_singleton
thf(fact_1950_map__inj__on,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F @ Xs )
= ( map @ B @ A @ F @ Ys ) )
=> ( ( inj_on @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_1951_inj__on__map__eq__map,axiom,
! [B: $tType,A: $tType,F: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) )
=> ( ( ( map @ A @ B @ F @ Xs )
= ( map @ A @ B @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_1952_distinct__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F @ Xs ) )
= ( ( distinct @ B @ Xs )
& ( inj_on @ B @ A @ F @ ( set2 @ B @ Xs ) ) ) ) ).
% distinct_map
thf(fact_1953_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_1954_inj__on__nth,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ! [X2: nat] :
( ( member2 @ nat @ X2 @ I5 )
=> ( ord_less @ nat @ X2 @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_1955_map__removeAll__inj__on,axiom,
! [B: $tType,A: $tType,F: A > B,X: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F @ ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) )
=> ( ( map @ A @ B @ F @ ( removeAll @ A @ X @ Xs ) )
= ( removeAll @ B @ ( F @ X ) @ ( map @ A @ B @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_1956_distinct__adj__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ( ( X
!= ( hd @ A @ Xs ) )
& ( distinct_adj @ A @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_1957_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1958_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1959_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_1960_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_1961_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat,M2: nat] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M2 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_1962_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_1963_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_1964_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_1965_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A3 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_1966_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_1967_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C3 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_1968_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B2 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_1969_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_1970_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% mod_mult_eq
thf(fact_1971_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,A10: A,B2: A,B9: A] :
( ( ( modulo_modulo @ A @ A3 @ C3 )
= ( modulo_modulo @ A @ A10 @ C3 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C3 )
= ( modulo_modulo @ A @ B9 @ C3 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A10 @ B9 ) @ C3 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_1972_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% mod_mult_mult2
thf(fact_1973_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( times_times @ A @ C3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ).
% mult_mod_right
thf(fact_1974_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C3 ) @ B2 ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% mod_mult_left_eq
thf(fact_1975_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% mod_mult_right_eq
thf(fact_1976_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ C3 )
= ( modulo_modulo @ A @ B2 @ C3 ) )
=> ~ ! [D4: A] :
( B2
!= ( plus_plus @ A @ A3 @ ( times_times @ A @ C3 @ D4 ) ) ) ) ) ).
% mod_eqE
thf(fact_1977_inj__on__empty,axiom,
! [B: $tType,A: $tType,F: A > B] : ( inj_on @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_1978_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L: list @ A,A5: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L ) @ A5 )
=> ( ( inj_on @ A @ B @ F @ A5 )
=> ( ( map @ B @ A @ ( inv_on @ A @ B @ F @ A5 ) @ ( map @ A @ B @ F @ L ) )
= L ) ) ) ).
% inj_on_map_inv_f
thf(fact_1979_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_1980_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_1981_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_1982_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% one_mod_2_pow_eq
thf(fact_1983_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_1984_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C3 ) ) ) ).
% times_divide_eq_left
thf(fact_1985_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_1986_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_1987_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ).
% times_divide_eq_right
thf(fact_1988_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% div_by_1
thf(fact_1989_bits__div__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% bits_div_by_1
thf(fact_1990_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_1991_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_1992_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,X: A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( inv_on @ A @ B @ F @ A5 @ ( F @ X ) )
= X ) ) ) ).
% inv_on_f_f
thf(fact_1993_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1994_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1995_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1996_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C3 @ A3 ) )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1997_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1998_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1999_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2000_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2001_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2002_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2003_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2004_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_2005_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_2006_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2007_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_2008_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_2009_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A3 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_2010_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_2011_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_2012_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ A3 )
= ( one_one @ A ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_2013_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A3 ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_2014_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_2015_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_2016_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C3 @ A3 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_2017_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C3 @ A3 ) @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_2018_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_2019_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_2020_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% dvd_div_mult_self
thf(fact_2021_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_2022_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_2023_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= A3 ) ) ) ).
% unit_div_1_div_1
thf(fact_2024_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_2025_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_2026_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_2027_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_2028_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_2029_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_2030_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_2031_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_2032_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_2033_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_2034_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_2035_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_2036_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2037_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2038_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_2039_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_2040_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_2041_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C3 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_2042_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_2043_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_2044_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_2045_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_2046_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_2047_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( divide_divide @ A @ B2 @ A3 ) ) ) ) ).
% unit_mult_div_div
thf(fact_2048_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_2049_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_2050_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_2051_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_2052_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_mult_iff
thf(fact_2053_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_2054_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_2055_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_plus_one_iff
thf(fact_2056_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_2057_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_2058_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_2059_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A3 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_2060_one__div__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% one_div_2_pow_eq
thf(fact_2061_bits__1__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% bits_1_div_exp
thf(fact_2062_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_2063_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= ( divide_divide @ A @ C3 @ A3 ) )
= ( B2 = C3 ) ) ) ) ).
% unit_div_cancel
thf(fact_2064_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_2065_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ C3 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_2066_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,D3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( dvd_dvd @ A @ D3 @ C3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( divide_divide @ A @ C3 @ D3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_2067_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_2068_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C3 ) @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_2069_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_2070_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ) ).
% div_mult_swap
thf(fact_2071_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C3 ) ) ) ) ).
% dvd_div_mult
thf(fact_2072_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C3 @ D3 )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= ( divide_divide @ A @ D3 @ C3 ) )
= ( ( times_times @ A @ B2 @ C3 )
= ( times_times @ A @ A3 @ D3 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_2073_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_2074_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_2075_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= C3 )
= ( B2
= ( times_times @ A @ C3 @ A3 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_2076_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_2077_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_2078_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_2079_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_2080_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_2081_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A3
= ( divide_divide @ A @ C3 @ B2 ) )
= ( ( times_times @ A @ A3 @ B2 )
= C3 ) ) ) ) ).
% unit_eq_div2
thf(fact_2082_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= C3 )
= ( A3
= ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_2083_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
& Q ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_conj
thf(fact_2084_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ~ ! [K3: A] :
( A3
!= ( times_times @ A @ B2 @ K3 ) ) ) ) ).
% dvdE
thf(fact_2085_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A3: A,B2: A,K: A] :
( ( A3
= ( times_times @ A @ B2 @ K ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% dvdI
thf(fact_2086_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B3: A,A4: A] :
? [K2: A] :
( A4
= ( times_times @ A @ B3 @ K2 ) ) ) ) ) ).
% dvd_def
thf(fact_2087_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C3 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% dvd_mult
thf(fact_2088_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% dvd_mult2
thf(fact_2089_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
=> ( dvd_dvd @ A @ A3 @ C3 ) ) ) ).
% dvd_mult_left
thf(fact_2090_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) ) ) ).
% dvd_triv_left
thf(fact_2091_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C3 @ D3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_2092_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
=> ( dvd_dvd @ A @ B2 @ C3 ) ) ) ).
% dvd_mult_right
thf(fact_2093_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) ) ) ).
% dvd_triv_right
thf(fact_2094_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_2095_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_2096_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_2097_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% one_dvd
thf(fact_2098_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% unit_imp_dvd
thf(fact_2099_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_2100_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B6: A] :
( ( B6
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B6 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B6 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B6 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B6 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C3 @ A3 )
!= ( times_times @ A @ C3 @ B6 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_2101_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_2102_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_2103_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A3 ) ) ) ).
% even_two_times_div_two
thf(fact_2104_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_2105_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P4
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_2106_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ( P4
=> ( P @ ( one_one @ A ) ) )
& ( ~ P4
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_2107_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_2108_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_2109_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_2110_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ C3 )
= A3 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_2111_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_2112_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A3 @ C3 ) )
=> ( ( divide_divide @ A @ B2 @ C3 )
= A3 ) ) ) ) ).
% divide_eq_imp
thf(fact_2113_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C3 )
= B2 )
=> ( A3
= ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_2114_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C3 )
= A3 )
= ( B2
= ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2115_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( A3
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( times_times @ A @ A3 @ C3 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2116_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( A3 = B2 ) ) ) ) ).
% right_inverse_eq
thf(fact_2117_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A3 )
= ( times_times @ A @ C3 @ A3 ) )
= ( B2 = C3 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_2118_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A3 @ B2 )
= ( times_times @ A @ A3 @ C3 ) )
= ( B2 = C3 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_2119_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_2120_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_2121_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_2122_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C3 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_2123_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_2124_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_one_over
thf(fact_2125_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A3: A] :
( ! [A6: A] :
( ( ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: A,B6: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B6 ) @ ( 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 @ B6 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% bits_induct
thf(fact_2126_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2127_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X3: A] : ( P @ ( times_times @ A @ L @ X3 ) ) )
= ( ? [X3: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X3 @ ( zero_zero @ A ) ) )
& ( P @ X3 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_2128_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_2129_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_2130_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_2131_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_2132_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_2133_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_2134_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_2135_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_2136_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_2137_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_2138_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [C2: A] :
( B2
!= ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% unit_dvdE
thf(fact_2139_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_2140_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_2141_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ( numeral_numeral @ A @ W )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2142_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C3 )
= ( numeral_numeral @ A @ W ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2143_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2144_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2145_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_2146_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_2147_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_2148_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_2149_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_2150_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_2151_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_2152_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2153_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_2154_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_2155_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_2156_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_2157_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_2158_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D2: A,T4: A] :
( ( dvd_dvd @ A @ D3 @ D2 )
=> ! [X6: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X6 @ T4 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D2 ) ) @ T4 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_2159_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D2: A,T4: A] :
( ( dvd_dvd @ A @ D3 @ D2 )
=> ! [X6: A,K4: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X6 @ T4 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X6 @ ( times_times @ A @ K4 @ D2 ) ) @ T4 ) ) ) ) ) ).
% inf_period(3)
thf(fact_2160_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_2161_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A3: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% mult_div_mod_eq
thf(fact_2162_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= A3 ) ) ).
% mod_mult_div_eq
thf(fact_2163_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= A3 ) ) ).
% mod_div_mult_eq
thf(fact_2164_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% div_mult_mod_eq
thf(fact_2165_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( A3
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_2166_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C3 )
= ( plus_plus @ A @ A3 @ C3 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_2167_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C3 )
= ( plus_plus @ A @ A3 @ C3 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_2168_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 ) ) ) ) ).
% div_mult1_eq
thf(fact_2169_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_2170_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_2171_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_2172_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_2173_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_2174_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% even_mask_div_iff'
thf(fact_2175_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N @ M2 ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_2176_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% even_mask_div_iff
thf(fact_2177_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_2178_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_2179_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_2180_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_2181_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_2182_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_2183_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_2184_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_2185_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_2186_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A3 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_2187_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_2188_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_2189_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_2190_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_2191_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2192_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_2193_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B6: A] :
( A3
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B6 ) ) ) ) ).
% evenE
thf(fact_2194_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_2195_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_2196_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% dvd_power
thf(fact_2197_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M2 )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M2 @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2198_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_2199_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_2200_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V: A,R3: A,S4: A] :
( ( ord_less_eq @ A @ U @ V )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R3 )
=> ( ( ord_less_eq @ A @ R3 @ S4 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R3 @ ( minus_minus @ A @ V @ U ) ) @ S4 ) ) @ V ) ) ) ) ) ).
% scaling_mono
thf(fact_2201_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C3 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2202_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_2203_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,N: nat,M2: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ A3 @ ( minus_minus @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_2204_odd__card__imp__not__empty,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A5 ) )
=> ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_2205_dvd__partition,axiom,
! [A: $tType,C4: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
=> ( ! [X2: set @ A] :
( ( member2 @ ( set @ A ) @ X2 @ C4 )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X2 ) ) )
=> ( ! [X2: set @ A] :
( ( member2 @ ( set @ A ) @ X2 @ C4 )
=> ! [Xa3: set @ A] :
( ( member2 @ ( set @ A ) @ Xa3 @ C4 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X2 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_2206_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B6: A] :
( A3
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B6 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_2207_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_2208_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_2209_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M2 @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% exp_mod_exp
thf(fact_2210_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2211_inj__on__Int,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( ( inj_on @ A @ B @ F @ A5 )
| ( inj_on @ A @ B @ F @ B4 ) )
=> ( inj_on @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% inj_on_Int
thf(fact_2212_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_2213_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2214_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_2215_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_2216_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_2217_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_2218_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_2219_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_Suc_1
thf(fact_2220_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_2221_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A4: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2222_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A4: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( 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_rec
thf(fact_2223_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_2224_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_2225_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% xor_one_eq
thf(fact_2226_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_xor_eq
thf(fact_2227_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B2 ) )
= ( A3 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_2228_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_2229_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X3: A] : ( uminus_uminus @ B @ ( A7 @ X3 ) ) ) ) ) ).
% uminus_apply
thf(fact_2230_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y ) )
= ( X = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_2231_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_2232_Compl__subset__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ).
% Compl_subset_Compl_iff
thf(fact_2233_Compl__anti__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% Compl_anti_mono
thf(fact_2234_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_2235_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_2236_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_2237_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( A3
= ( uminus_uminus @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_2238_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_2239_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A3 ) )
= ( ( zero_zero @ A )
= A3 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_2240_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_2241_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_2242_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_2243_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_2244_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_2245_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_2246_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_2247_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A3 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_2248_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_2249_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% minus_diff_eq
thf(fact_2250_Compl__disjoint2,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_2251_Compl__disjoint,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_2252_inter__compl__diff__conv,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ).
% inter_compl_diff_conv
thf(fact_2253_Diff__Compl,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ).
% Diff_Compl
thf(fact_2254_Compl__Diff__eq,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ B4 ) ) ).
% Compl_Diff_eq
thf(fact_2255_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_2256_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_le_0_iff_le
thf(fact_2257_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_2258_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_2259_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_2260_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_pos
thf(fact_2261_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_2262_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_0_iff_less
thf(fact_2263_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_2264_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_2265_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% diff_0
thf(fact_2266_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_2267_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_2268_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% uminus_add_conv_diff
thf(fact_2269_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A3 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_2270_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% div_minus1_right
thf(fact_2271_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_2272_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_2273_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_2274_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_2275_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_2276_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_2277_subset__Compl__singleton,axiom,
! [A: $tType,A5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member2 @ A @ B2 @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_2278_boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( sup_sup @ A @ X @ Y ) )
= ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_2279_boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( inf_inf @ A @ X @ Y ) )
= ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_2280_take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_Suc_1
thf(fact_2281_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_2282_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% signed_take_bit_of_minus_1
thf(fact_2283_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_2284_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_2285_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_2286_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_2287_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( N = one2 ) ) ) ).
% neg_one_eq_numeral_iff
thf(fact_2288_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( N = one2 ) ) ) ).
% numeral_eq_neg_one_iff
thf(fact_2289_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_2290_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A3 ) )
= A3 ) ) ).
% left_minus_one_mult_self
thf(fact_2291_minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% minus_one_mult_self
thf(fact_2292_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_zero @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_2293_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_2294_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_2295_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_2296_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_2297_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) )
= ( M2 != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_2298_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_2299_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_2300_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_2301_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_2302_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( M2 != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_2303_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_2304_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_2305_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_2306_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% take_bit_of_1
thf(fact_2307_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_2308_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_2309_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_2310_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_2311_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_2312_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_2313_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ one2 ) ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_2314_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% diff_numeral_special(3)
thf(fact_2315_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_2316_power__minus1__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% power_minus1_even
thf(fact_2317_neg__one__odd__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% neg_one_odd_power
thf(fact_2318_neg__one__even__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) ) ) ).
% neg_one_even_power
thf(fact_2319_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_of_exp
thf(fact_2320_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_2321_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_2322_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_2323_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X3: A] : ( uminus_uminus @ B @ ( A7 @ X3 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_2324_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_2325_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_2326_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_2327_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_2328_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_2329_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_2330_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_less_iff
thf(fact_2331_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% less_minus_iff
thf(fact_2332_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_2333_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_2334_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_2335_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A,K: A,A3: A] :
( ( A5
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( uminus_uminus @ A @ A5 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_2336_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ A3 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_2337_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_2338_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_2339_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A3 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_2340_minus__assn__def,axiom,
( ( minus_minus @ assn )
= ( ^ [A4: assn,B3: assn] : ( inf_inf @ assn @ A4 @ ( uminus_uminus @ assn @ B3 ) ) ) ) ).
% minus_assn_def
thf(fact_2341_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( divide_divide @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_2342_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_2343_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_2344_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A3 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_2345_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_2346_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add_eq_0_iff
thf(fact_2347_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_2348_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_2349_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_2350_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_2351_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_2352_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ N )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% numeral_neq_neg_one
thf(fact_2353_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( one_one @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% one_neq_neg_numeral
thf(fact_2354_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_2355_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2356_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_2357_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B4: A,K: A,B2: A,A3: A] :
( ( B4
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A3 @ B4 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_2358_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ A3 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_2359_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( inf_inf @ A @ X @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_2360_diff__eq,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( minus_minus @ A )
= ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ) ).
% diff_eq
thf(fact_2361_subset__Compl__self__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_2362_Compl__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) ) ) ).
% Compl_Un
thf(fact_2363_Compl__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) ) ) ).
% Compl_Int
thf(fact_2364_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_2365_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_2366_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K2: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A4 ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_2367_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% gbinomial_index_swap
thf(fact_2368_coset__def,axiom,
! [A: $tType] :
( ( coset @ A )
= ( ^ [Xs3: list @ A] : ( uminus_uminus @ ( set @ A ) @ ( set2 @ A @ Xs3 ) ) ) ) ).
% coset_def
thf(fact_2369_compl__coset,axiom,
! [A: $tType,Xs: list @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( coset @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% compl_coset
thf(fact_2370_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_2371_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_2372_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_2373_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_2374_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_2375_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_le_neg_numeral
thf(fact_2376_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_le_neg_one
thf(fact_2377_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% neg_numeral_le_neg_one
thf(fact_2378_neg__one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_le_numeral
thf(fact_2379_neg__numeral__le__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_le_one
thf(fact_2380_neg__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_less_one
thf(fact_2381_neg__one__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_less_numeral
thf(fact_2382_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_less_neg_one
thf(fact_2383_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_less_neg_numeral
thf(fact_2384_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_2385_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( A3
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C3 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_2386_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) )
= A3 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A3 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_2387_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= C3 )
= ( ( uminus_uminus @ A @ A3 )
= ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_2388_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C3
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( ( times_times @ A @ C3 @ B2 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_2389_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(2)
thf(fact_2390_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(1)
thf(fact_2391_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_2392_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_2393_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_2394_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z2 ) ) ) ) ).
% shunt1
thf(fact_2395_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y ) ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% shunt2
thf(fact_2396_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P4: A,Q3: A,R3: A] :
( ( ord_less_eq @ A @ P4 @ ( sup_sup @ A @ Q3 @ R3 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P4 @ ( uminus_uminus @ A @ Q3 ) ) @ R3 ) ) ) ).
% sup_neg_inf
thf(fact_2397_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus
thf(fact_2398_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_2399_Compl__insert,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ A5 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_2400_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_2401_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_2402_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2403_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_2404_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_2405_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_2406_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_2407_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_2408_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_2409_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C3 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_2410_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2411_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2412_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_2413_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2414_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2415_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_2416_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_2417_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_2418_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_2419_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_2420_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_2421_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_2422_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_2423_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_2424_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_2425_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_2426_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2427_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2428_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_2429_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_2430_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M2: nat,A3: A] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M2 ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M2 ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M2 @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_2431_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A3
= ( one_one @ A ) )
| ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_2432_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2433_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_2434_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_2435_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_2436_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_2437_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_2438_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_2439_minus__one__power__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% minus_one_power_iff
thf(fact_2440_power__minus1__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% power_minus1_odd
thf(fact_2441_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_2442_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_2443_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_2444_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_2445_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ).
% of_nat_Suc
thf(fact_2446_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_2447_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_2448_Compl__eq__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A5 )
= ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( A5 = B4 ) ) ).
% Compl_eq_Compl_iff
thf(fact_2449_Compl__iff,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ( member2 @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( ~ ( member2 @ A @ C3 @ A5 ) ) ) ).
% Compl_iff
thf(fact_2450_ComplI,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ~ ( member2 @ A @ C3 @ A5 )
=> ( member2 @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% ComplI
thf(fact_2451_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M2 @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mult
thf(fact_2452_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( one_one @ A ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_2453_double__complement,axiom,
! [A: $tType,A5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= A5 ) ).
% double_complement
thf(fact_2454_ComplD,axiom,
! [A: $tType,C3: A,A5: set @ A] :
( ( member2 @ A @ C3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
=> ~ ( member2 @ A @ C3 @ A5 ) ) ).
% ComplD
thf(fact_2455_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_2456_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_2457_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_2458_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
=> ~ ! [N7: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N7 ) ) ) @ E3 ) ) ) ).
% nat_approx_posE
thf(fact_2459_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N7: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N7 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_2460_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_2461_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_2462_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_2463_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_2464_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_2465_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_2466_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_2467_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_2468_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_2469_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_2470_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_2471_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_2472_push__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_of_1
thf(fact_2473_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_2474_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_2475_pochhammer__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_0_left
thf(fact_2476_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N3: nat,A4: A] : ( bit_se5824344971392196577ns_xor @ A @ A4 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_2477_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A3 @ A3 ) @ A3 ) ) ) ).
% power3_eq_cube
thf(fact_2478_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_2479_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_2480_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_2481_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_2482_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat,M2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ M2 ) ) ) ) ).
% pochhammer_product'
thf(fact_2483_card__3__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X3: A,Y3: A,Z3: A] :
( ( S
= ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( insert2 @ A @ Z3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X3 != Y3 )
& ( Y3 != Z3 )
& ( X3 != Z3 ) ) ) ) ).
% card_3_iff
thf(fact_2484_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N3: nat,A4: A] : ( times_times @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_2485_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M2: nat,N: nat,Z2: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_2486_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R3 ) @ K ) )
= ( times_times @ A @ R3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R3 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_2487_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_minus
thf(fact_2488_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% pochhammer_minus'
thf(fact_2489_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_2490_fact__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_double
thf(fact_2491_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_2492_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_2493_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_2494_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_2495_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_2496_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_2497_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A3 ) ) ).
% and.right_neutral
thf(fact_2498_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A3 )
= A3 ) ) ).
% and.left_neutral
thf(fact_2499_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_2500_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_2501_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_2502_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_2503_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_2504_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_2505_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_2506_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_2507_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_2508_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_Suc
thf(fact_2509_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_2510_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_2511_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_2512_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_2513_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_2514_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_2515_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_2516_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_2517_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_2518_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_2519_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_2520_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_ge_1
thf(fact_2521_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_2522_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_2523_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N3: nat,A4: A] : ( bit_se1065995026697491101ons_or @ A @ A4 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_2524_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_2525_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_2526_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_2527_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M3 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_2528_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_2529_pochhammer__same,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% pochhammer_same
thf(fact_2530_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_2531_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% or_one_eq
thf(fact_2532_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_or_eq
thf(fact_2533_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K2: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 ) @ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_2534_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A4 ) @ K2 ) ) @ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_2535_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N ) ) ) ) ) ) ).
% mask_numeral
thf(fact_2536_mask__Suc__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ).
% mask_Suc_double
thf(fact_2537_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N3: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_2538_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A,N: nat] :
( ! [J3: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J3 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_2539_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% diff_numeral_special(8)
thf(fact_2540_diff__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% diff_numeral_special(7)
thf(fact_2541_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_2542_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_2543_diff__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ one2 @ N ) ) ) ).
% diff_numeral_special(1)
thf(fact_2544_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% diff_numeral_special(2)
thf(fact_2545_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(1)
thf(fact_2546_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(2)
thf(fact_2547_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% add_neg_numeral_special(3)
thf(fact_2548_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% add_neg_numeral_special(4)
thf(fact_2549_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M2 @ one2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% minus_sub_one_diff_one
thf(fact_2550_bit__1__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% bit_1_iff
thf(fact_2551_not__bit__1__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N ) ) ) ).
% not_bit_1_Suc
thf(fact_2552_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: num] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% bit_numeral_simps(1)
thf(fact_2553_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A4: A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A4 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_2554_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_2555_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_2556_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A4: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A4 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A4 @ N3 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_2557_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K ) ) @ ( cons @ nat @ ( suc @ K ) @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_2558_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_minus'
thf(fact_2559_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_2560_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_2561_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
=> ? [B10: A,C7: A] :
( ( A3
= ( times_times @ A @ B10 @ C7 ) )
& ( dvd_dvd @ A @ B10 @ B2 )
& ( dvd_dvd @ A @ C7 @ C3 ) ) ) ) ).
% division_decomp
thf(fact_2562_Int__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atMost @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% Int_atMost
thf(fact_2563_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_2564_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_2565_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_2566_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_2567_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_2568_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_2569_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_2570_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_2571_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_2572_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_2573_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [H: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H ) ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_2574_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_2575_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_2576_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A4: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A4 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_2577_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A4: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A4 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_2578_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A4: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_2579_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_2580_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_2581_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N3: nat,A4: A] : ( bit_se5824344872417868541ns_and @ A @ A4 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_2582_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_2583_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_2584_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,A3: A,B2: A,C3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_2585_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,C3: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C3 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_2586_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,C3: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C3 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2587_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,A3: A,B2: A,C3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2588_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_2589_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_2590_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P4: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ P4 @ ( times_times @ A @ A3 @ B2 ) )
=> ~ ! [X2: A,Y2: A] :
( ( P4
= ( times_times @ A @ X2 @ Y2 ) )
=> ( ( dvd_dvd @ A @ X2 @ A3 )
=> ~ ( dvd_dvd @ A @ Y2 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_2591_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_2592_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_2593_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_2594_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_2595_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_2596_rotate__drop__take,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N3: nat,Xs3: list @ A] : ( append @ A @ ( drop @ A @ ( modulo_modulo @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) @ ( take @ A @ ( modulo_modulo @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ) ).
% rotate_drop_take
thf(fact_2597_rotate__is__Nil__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( rotate @ A @ N @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate_is_Nil_conv
thf(fact_2598_set__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( set2 @ A @ ( rotate @ A @ N @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate
thf(fact_2599_length__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate @ A @ N @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate
thf(fact_2600_distinct__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( distinct @ A @ ( rotate @ A @ N @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rotate
thf(fact_2601_rotate__Suc,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( rotate @ A @ ( suc @ N ) @ Xs )
= ( rotate1 @ A @ ( rotate @ A @ N @ Xs ) ) ) ).
% rotate_Suc
thf(fact_2602_rotate__length01,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_2603_rotate__id,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
= ( zero_zero @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_2604_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F5: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F5: A > B,A6: A,As: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_2605_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F5: A > B > C] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [F5: A > B > C,A6: A,As: list @ A,B6: B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ ( cons @ B @ B6 @ Bs2 ) ) ) )
=> ( ! [A6: A > B > C,V2: A,Va2: list @ A] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ B ) ) ) )
=> ~ ! [A6: A > B > C,V2: B,Va2: 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 @ V2 @ Va2 ) ) ) ) ) ) ) ).
% zipf.cases
thf(fact_2606_list__assn_Ocases,axiom,
! [A: $tType,C: $tType,X: product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) )] :
( ! [P2: A > C > assn] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ P2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( nil @ C ) ) ) )
=> ( ! [P2: A > C > assn,A6: A,As: list @ A,C2: C,Cs: list @ C] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ P2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ A6 @ As ) @ ( cons @ C @ C2 @ Cs ) ) ) )
=> ( ! [Uu2: A > C > assn,V2: A,Va2: list @ A] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ Uu2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ C ) ) ) )
=> ~ ! [Uu2: A > C > assn,V2: C,Va2: list @ C] :
( X
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ Uu2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( cons @ C @ V2 @ Va2 ) ) ) ) ) ) ) ).
% list_assn.cases
thf(fact_2607_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R5: A > A > $o,X2: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) )
=> ( ! [R5: A > A > $o,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) ) )
=> ~ ! [R5: A > A > $o,V2: A,Va2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V2 @ Va2 ) ) ) ) ) ) ).
% mergesort_by_rel_merge.cases
thf(fact_2608_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R5: A > A > $o,Sl: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ~ ! [R5: A > A > $o,Sl: list @ A,X2: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ).
% quicksort_by_rel.cases
thf(fact_2609_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [P2: A > B > $o] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [P2: A > B > $o,A6: A,As: list @ A,B6: B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ ( cons @ B @ B6 @ Bs2 ) ) ) )
=> ( ! [P2: A > B > $o,V2: A,Va2: list @ A] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ B ) ) ) )
=> ~ ! [P2: A > B > $o,V2: B,Va2: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V2 @ Va2 ) ) ) ) ) ) ) ).
% list_all_zip.cases
thf(fact_2610_bex2I,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( P @ A3 @ B2 ) )
=> ? [A6: A,B6: B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B6 ) @ S )
& ( P @ A6 @ B6 ) ) ) ) ).
% bex2I
thf(fact_2611_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F5: A > B,A6: A,B6: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F5 @ ( product_Pair @ A @ A @ A6 @ B6 ) ) ) ).
% pairself.cases
thf(fact_2612_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,Y: A,Z2: A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Conj @ ( Xor @ Y @ Z2 ) @ X )
= ( Xor @ ( Conj @ Y @ X ) @ ( Conj @ Z2 @ X ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
thf(fact_2613_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ ( Compl @ X ) )
= One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_right
thf(fact_2614_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A,Z2: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Conj @ X @ ( Xor @ Y @ Z2 ) )
= ( Xor @ ( Conj @ X @ Y ) @ ( Conj @ X @ Z2 ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib
thf(fact_2615_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ ( Compl @ Y ) )
= ( Compl @ ( Xor @ X @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_right
thf(fact_2616_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ ( Compl @ X ) @ X )
= One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_left
thf(fact_2617_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ ( Compl @ X ) @ Y )
= ( Compl @ ( Xor @ X @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_left
thf(fact_2618_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ One )
= ( Compl @ X ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_right
thf(fact_2619_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ ( Xor @ X @ Y ) )
= Y ) ) ).
% abstract_boolean_algebra_sym_diff.xor_left_self
thf(fact_2620_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ One @ X )
= ( Compl @ X ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_left
thf(fact_2621_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Y: A,Z2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ ( Conj @ Y @ Z2 ) @ X )
= ( Conj @ ( Disj @ Y @ X ) @ ( Disj @ Z2 @ X ) ) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib2
thf(fact_2622_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Y: A,Z2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ ( Disj @ Y @ Z2 ) @ X )
= ( Disj @ ( Conj @ Y @ X ) @ ( Conj @ Z2 @ X ) ) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib2
thf(fact_2623_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( ( Compl @ X )
= ( Compl @ Y ) )
= ( X = Y ) ) ) ).
% abstract_boolean_algebra.compl_eq_compl_iff
thf(fact_2624_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ X )
= Zero ) ) ).
% abstract_boolean_algebra_sym_diff.xor_self
thf(fact_2625_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ Y )
= ( Conj @ ( Disj @ X @ Y ) @ ( Disj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def2
thf(fact_2626_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A,Z2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ ( Conj @ Y @ Z2 ) )
= ( Conj @ ( Disj @ X @ Y ) @ ( Disj @ X @ Z2 ) ) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib
thf(fact_2627_abstract__boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ ( Compl @ X ) )
= One ) ) ).
% abstract_boolean_algebra.disj_cancel_right
thf(fact_2628_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A,Z2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ ( Disj @ Y @ Z2 ) )
= ( Disj @ ( Conj @ X @ Y ) @ ( Conj @ X @ Z2 ) ) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib
thf(fact_2629_abstract__boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ ( Compl @ X ) )
= Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_right
thf(fact_2630_abstract__boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,A3: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( ( Conj @ A3 @ X )
= Zero )
=> ( ( ( Disj @ A3 @ X )
= One )
=> ( ( ( Conj @ A3 @ Y )
= Zero )
=> ( ( ( Disj @ A3 @ Y )
= One )
=> ( X = Y ) ) ) ) ) ) ).
% abstract_boolean_algebra.complement_unique
thf(fact_2631_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A,X: A,Y: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( ( Xor @ X @ Y )
= ( Disj @ ( Conj @ X @ ( Compl @ Y ) ) @ ( Conj @ ( Compl @ X ) @ Y ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def
thf(fact_2632_abstract__boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ ( Compl @ X ) @ X )
= One ) ) ).
% abstract_boolean_algebra.disj_cancel_left
thf(fact_2633_abstract__boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ ( Compl @ X ) @ X )
= Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_left
thf(fact_2634_abstract__boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ Zero )
= X ) ) ).
% abstract_boolean_algebra.disj_zero_right
thf(fact_2635_abstract__boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ Zero )
= Zero ) ) ).
% abstract_boolean_algebra.conj_zero_right
thf(fact_2636_abstract__boolean__algebra_Odisj__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ One )
= One ) ) ).
% abstract_boolean_algebra.disj_one_right
thf(fact_2637_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ ( Disj @ X @ Y ) )
= ( Conj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ).
% abstract_boolean_algebra.de_Morgan_disj
thf(fact_2638_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ ( Conj @ X @ Y ) )
= ( Disj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ).
% abstract_boolean_algebra.de_Morgan_conj
thf(fact_2639_abstract__boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ Zero @ X )
= Zero ) ) ).
% abstract_boolean_algebra.conj_zero_left
thf(fact_2640_abstract__boolean__algebra_Oconj__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ One )
= X ) ) ).
% abstract_boolean_algebra.conj_one_right
thf(fact_2641_abstract__boolean__algebra_Odisj__one__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ One @ X )
= One ) ) ).
% abstract_boolean_algebra.disj_one_left
thf(fact_2642_abstract__boolean__algebra_Odouble__compl,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ ( Compl @ X ) )
= X ) ) ).
% abstract_boolean_algebra.double_compl
thf(fact_2643_abstract__boolean__algebra_Ocompl__unique,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( ( Conj @ X @ Y )
= Zero )
=> ( ( ( Disj @ X @ Y )
= One )
=> ( ( Compl @ X )
= Y ) ) ) ) ).
% abstract_boolean_algebra.compl_unique
thf(fact_2644_abstract__boolean__algebra_Ocompl__zero,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ Zero )
= One ) ) ).
% abstract_boolean_algebra.compl_zero
thf(fact_2645_abstract__boolean__algebra_Ocompl__one,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ One )
= Zero ) ) ).
% abstract_boolean_algebra.compl_one
thf(fact_2646_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One ) ) ).
% abstract_boolean_algebra_sym_diff.axioms(1)
thf(fact_2647_partition__rev_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
( ! [P2: A > $o,Yes: list @ A,No: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P2 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) )
=> ~ ! [P2: A > $o,Yes: list @ A,No: list @ A,X2: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P2 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ).
% partition_rev.cases
thf(fact_2648_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
( ! [Xs12: list @ A,Xs22: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs22 ) @ ( nil @ A ) ) )
=> ( ! [Xs12: list @ A,Xs22: list @ A,X2: A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs22 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ~ ! [Xs12: list @ A,Xs22: list @ A,X1: A,X23: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs22 ) @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) ) ) ) ) ).
% mergesort_by_rel_split.cases
thf(fact_2649_rotate__rotate,axiom,
! [A: $tType,M2: nat,N: nat,Xs: list @ A] :
( ( rotate @ A @ M2 @ ( rotate @ A @ N @ Xs ) )
= ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) @ Xs ) ) ).
% rotate_rotate
thf(fact_2650_rotate__map,axiom,
! [A: $tType,B: $tType,N: nat,F: B > A,Xs: list @ B] :
( ( rotate @ A @ N @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( rotate @ B @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_2651_rotate1__rotate__swap,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( rotate1 @ A @ ( rotate @ A @ N @ Xs ) )
= ( rotate @ A @ N @ ( rotate1 @ A @ Xs ) ) ) ).
% rotate1_rotate_swap
thf(fact_2652_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F5: A > B,X2: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F5 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ( ! [F5: A > B,X2: A,Y2: A,Zs3: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F5 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs3 ) ) ) )
=> ~ ! [A6: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A6 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_2653_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P2: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( nil @ A ) ) )
=> ~ ! [P2: A > A > $o,X2: A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( cons @ A @ X2 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_2654_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P2: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( nil @ A ) ) )
=> ( ! [P2: A > A > $o,X2: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ~ ! [P2: A > A > $o,X2: A,Y2: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_2655_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ( ! [Xs2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_2656_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 ) )
=> ( ! [V2: A,Va2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ A ) ) )
=> ~ ! [X1: A,L1: list @ A,X23: A,L22: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ).
% merge.cases
thf(fact_2657_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 ) ) ) )
=> ( ! [L2: list @ A] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L2 @ ( 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 ),L2: list @ A] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L2 @ ( nil @ ( list @ A ) ) ) ) )
=> ~ ! [Acc2: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc2 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) ) ) ) ) ) ).
% merge_list.cases
thf(fact_2658_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_2659_rotate__append,axiom,
! [A: $tType,L: list @ A,Q3: list @ A] :
( ( rotate @ A @ ( size_size @ ( list @ A ) @ L ) @ ( append @ A @ L @ Q3 ) )
= ( append @ A @ Q3 @ L ) ) ).
% rotate_append
thf(fact_2660_rotate__conv__mod,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N3: nat,Xs3: list @ A] : ( rotate @ A @ ( modulo_modulo @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_2661_rotate__rev,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( rotate @ A @ N @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( rotate @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_2662_nth__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A,M2: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate @ A @ M2 @ Xs ) @ N )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_2663_hd__rotate__conv__nth,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( rotate @ A @ N @ Xs ) )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_2664_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R3: A,Q3: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R3 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R3 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R3 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R3 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R3 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R3 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_2665_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
= X ) )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I2 @ X )
= ( take @ A @ I2 @ Y ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I2 ) @ ( nth @ A @ Y @ I2 ) ) @ R3 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_2666_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R3 ) )
=> ~ ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I3 @ Xs )
= ( take @ A @ I3 @ Ys ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Ys @ I3 ) ) @ R3 ) ) ) ) ) ).
% lex_take_index
thf(fact_2667_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ~ ! [X2: A,Xs2: list @ A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Ys2 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_2668_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_2669_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ).
% diff_numeral_special(6)
thf(fact_2670_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list @ A,B2: A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R3 ) )
= ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
| ( ( A3 = B2 )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_2671_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R3 ) )
= ( ? [A4: A,X3: list @ A] :
( Y
= ( cons @ A @ A4 @ X3 ) ) ) ) ).
% lexord_Nil_left
thf(fact_2672_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_2673_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_2674_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R3 ) )
= ( ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X = Y )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R3 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_2675_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lex @ A @ R3 ) )
= ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_2676_mergesort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
~ ! [R5: A > A > $o,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R5 @ Xs2 ) ) ).
% mergesort_by_rel.cases
thf(fact_2677_lexord__linear,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A6: A,B6: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B6 ) @ R3 )
| ( A6 = B6 )
| ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ A6 ) @ R3 ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
| ( X = Y )
| ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_linear
thf(fact_2678_lexord__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X2: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R3 )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_irreflexive
thf(fact_2679_lexord__Nil__right,axiom,
! [A: $tType,X: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R3 ) ) ).
% lexord_Nil_right
thf(fact_2680_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V: list @ A,R3: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R3 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_append_leftI
thf(fact_2681_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R3 ) ) ).
% Nil2_notin_lex
thf(fact_2682_Nil__notin__lex,axiom,
! [A: $tType,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) @ ( lex @ A @ R3 ) ) ).
% Nil_notin_lex
thf(fact_2683_lex__append__leftI,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R3 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R3 ) ) ) ).
% lex_append_leftI
thf(fact_2684_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs: list @ A] :
( ! [X2: A,Y2: A,Z4: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z4 ) @ R3 )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z4 ) @ R3 ) ) ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R3 ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R3 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_2685_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R3 ) )
=> ( ! [A6: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R3 )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_append_leftD
thf(fact_2686_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ? [B11: A,Z6: list @ A] :
( Y
= ( cons @ A @ B11 @ Z6 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y ) ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_append_rightI
thf(fact_2687_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs: list @ A,Ys: list @ A,Qs: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R3 ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R3 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_2688_lex__append__left__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X2: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R3 )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R3 ) )
= ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_left_iff
thf(fact_2689_lex__append__leftD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X2: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R3 )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R3 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_leftD
thf(fact_2690_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),Vs: list @ A,Us2: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R3 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us2 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Ys @ Vs ) ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_rightI
thf(fact_2691_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A3 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_append_left_rightI
thf(fact_2692_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R3 ) )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 ) )
| ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_2693_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W: list @ A,R3: set @ ( product_prod @ A @ A ),V: list @ A,Z2: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W ) @ ( lexord @ A @ R3 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V ) @ ( append @ A @ W @ Z2 ) ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_sufI
thf(fact_2694_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_2695_Cons__lenlex__iff,axiom,
! [A: $tType,M2: A,Ms: list @ A,N: A,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M2 @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R3 ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ N ) @ R3 ) )
| ( ( M2 = N )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R3 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_2696_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
= ( ? [Y3: A,N3: nat] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N3 ) @ Y3 ) @ R3 )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N3 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_2697_nth__step__trancl,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A ),N: nat,M2: nat] :
( ! [N7: nat] :
( ( ord_less @ nat @ N7 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ ( suc @ N7 ) ) @ ( nth @ A @ Xs @ N7 ) ) @ R ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ ( nth @ A @ Xs @ M2 ) ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% nth_step_trancl
thf(fact_2698_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_2699_list__assn_Opelims,axiom,
! [A: $tType,C: $tType,X: A > C > assn,Xa: list @ A,Xb: list @ C,Y: assn] :
( ( ( vEBT_List_list_assn @ A @ C @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ C ) )
=> ( ( Y
= ( one_one @ assn ) )
=> ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( nil @ C ) ) ) ) ) ) )
=> ( ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [C2: C,Cs: list @ C] :
( ( Xb
= ( cons @ C @ C2 @ Cs ) )
=> ( ( Y
= ( times_times @ assn @ ( X @ A6 @ C2 ) @ ( vEBT_List_list_assn @ A @ C @ X @ As @ Cs ) ) )
=> ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ A6 @ As ) @ ( cons @ C @ C2 @ Cs ) ) ) ) ) ) )
=> ( ! [V2: A,Va2: list @ A] :
( ( Xa
= ( cons @ A @ V2 @ Va2 ) )
=> ( ( Xb
= ( nil @ C ) )
=> ( ( Y
= ( bot_bot @ assn ) )
=> ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ C ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V2: C,Va2: list @ C] :
( ( Xb
= ( cons @ C @ V2 @ Va2 ) )
=> ( ( Y
= ( bot_bot @ assn ) )
=> ~ ( accp @ ( product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) ) @ ( vEBT_L4249061453398456502sn_rel @ A @ C ) @ ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( cons @ C @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% list_assn.pelims
thf(fact_2700_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( gcd_Gcd @ A @ A5 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_2701_trancl__single,axiom,
! [A: $tType,A3: A,B2: A] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) )
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trancl_single
thf(fact_2702_zip__eq__zip__same__len,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,A10: list @ A,B9: list @ B] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ( ( size_size @ ( list @ A ) @ A10 )
= ( size_size @ ( list @ B ) @ B9 ) )
=> ( ( ( zip @ A @ B @ A3 @ B2 )
= ( zip @ A @ B @ A10 @ B9 ) )
= ( ( A3 = A10 )
& ( B2 = B9 ) ) ) ) ) ).
% zip_eq_zip_same_len
thf(fact_2703_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_2704_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_2705_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R3 ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_2706_zip__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( nil @ ( product_prod @ A @ B ) ) )
= ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ B ) ) ) ) ).
% zip_eq_Nil_iff
thf(fact_2707_Nil__eq__zip__iff,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( nil @ ( product_prod @ A @ B ) )
= ( zip @ A @ B @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ B ) ) ) ) ).
% Nil_eq_zip_iff
thf(fact_2708_zip__Nil,axiom,
! [B: $tType,A: $tType,Ys: list @ B] :
( ( zip @ A @ B @ ( nil @ A ) @ Ys )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip_Nil
thf(fact_2709_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R3 ) )
= ( ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_2710_zip__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Us2: list @ B,Ys: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Us2 ) )
=> ( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ ( append @ B @ Us2 @ Vs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Us2 ) @ ( zip @ A @ B @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_2711_length__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_zip
thf(fact_2712_listrel1__mono,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R3 ) @ ( listrel1 @ A @ S4 ) ) ) ).
% listrel1_mono
thf(fact_2713_trancl__sub,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( transitive_trancl @ A @ R ) ) ).
% trancl_sub
thf(fact_2714_trancl__mono__mp,axiom,
! [A: $tType,U2: set @ ( product_prod @ A @ A ),V3: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V3 )
=> ( ( member2 @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ U2 ) )
=> ( member2 @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ V3 ) ) ) ) ).
% trancl_mono_mp
thf(fact_2715_trancl__sub__insert__trancl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ R ) ) ) ).
% trancl_sub_insert_trancl
thf(fact_2716_zip__inj,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,A10: list @ A,B9: list @ B] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ( ( size_size @ ( list @ A ) @ A10 )
= ( size_size @ ( list @ B ) @ B9 ) )
=> ( ( ( zip @ A @ B @ A3 @ B2 )
= ( zip @ A @ B @ A10 @ B9 ) )
=> ( ( A3 = A10 )
& ( B2 = B9 ) ) ) ) ) ).
% zip_inj
thf(fact_2717_take__zip,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( take @ ( product_prod @ A @ B ) @ N @ ( zip @ A @ B @ Xs @ Ys ) )
= ( zip @ A @ B @ ( take @ A @ N @ Xs ) @ ( take @ B @ N @ Ys ) ) ) ).
% take_zip
thf(fact_2718_drop__zip,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( drop @ ( product_prod @ A @ B ) @ N @ ( zip @ A @ B @ Xs @ Ys ) )
= ( zip @ A @ B @ ( drop @ A @ N @ Xs ) @ ( drop @ B @ N @ Ys ) ) ) ).
% drop_zip
thf(fact_2719_Gcd__1,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( member2 @ A @ ( one_one @ A ) @ A5 )
=> ( ( gcd_Gcd @ A @ A5 )
= ( one_one @ A ) ) ) ) ).
% Gcd_1
thf(fact_2720_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xy: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( cons @ ( product_prod @ A @ B ) @ Xy @ Xys ) )
=> ~ ! [X2: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Xs4 ) )
=> ! [Y2: B,Ys4: list @ B] :
( ( Ys
= ( cons @ B @ Y2 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair @ A @ B @ X2 @ Y2 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_2721_zip__same,axiom,
! [A: $tType,A3: A,B2: A,Xs: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
= ( ( member2 @ A @ A3 @ ( set2 @ A @ Xs ) )
& ( A3 = B2 ) ) ) ).
% zip_same
thf(fact_2722_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ~ ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member2 @ B @ Y @ ( set2 @ B @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_2723_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_2724_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member2 @ B @ Y @ ( set2 @ B @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_2725_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,X: A,Ys: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X ) @ ( list_update @ B @ Ys @ I @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_update
thf(fact_2726_zip__rev,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( zip @ A @ B @ ( rev @ A @ Xs ) @ ( rev @ B @ Ys ) )
= ( rev @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_2727_zip_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Xs: list @ A] :
( ( zip @ A @ B @ Xs @ ( nil @ B ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip.simps(1)
thf(fact_2728_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys ) ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel1I2
thf(fact_2729_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R3 ) ) ).
% not_Nil_listrel1
thf(fact_2730_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R3 ) ) ).
% not_listrel1_Nil
thf(fact_2731_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_2732_append__listrel1I,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),Us2: list @ A,Vs: list @ A] :
( ( ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
& ( Us2 = Vs ) )
| ( ( Xs = Ys )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us2 @ Vs ) @ ( listrel1 @ A @ R3 ) ) ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Ys @ Vs ) ) @ ( listrel1 @ A @ R3 ) ) ) ).
% append_listrel1I
thf(fact_2733_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_2734_Gcd__eq__1__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( member2 @ A @ A3 @ A5 )
=> ( ( gcd_Gcd @ A @ A5 )
= ( one_one @ A ) ) ) ) ) ).
% Gcd_eq_1_I
thf(fact_2735_distinct__zipI2,axiom,
! [B: $tType,A: $tType,Ys: list @ A,Xs: list @ B] :
( ( distinct @ A @ Ys )
=> ( distinct @ ( product_prod @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys ) ) ) ).
% distinct_zipI2
thf(fact_2736_distinct__zipI1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( distinct @ A @ Xs )
=> ( distinct @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% distinct_zipI1
thf(fact_2737_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 ) )
=> ( ( member2 @ B @ Y @ ( set2 @ B @ Ys ) )
=> ~ ! [X2: A] :
~ ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_2738_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 ) )
=> ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ! [Y2: B] :
~ ( member2 @ ( 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_2739_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_2740_lenlex__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X2: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R3 )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R3 ) ) ) ).
% lenlex_irreflexive
thf(fact_2741_zip__obtain__same__length,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ ( product_prod @ A @ B ) ) > $o] :
( ! [Zs3: list @ A,Ws3: list @ B,N7: nat] :
( ( ( size_size @ ( list @ A ) @ Zs3 )
= ( size_size @ ( list @ B ) @ Ws3 ) )
=> ( ( N7
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( Zs3
= ( take @ A @ N7 @ Xs ) )
=> ( ( Ws3
= ( take @ B @ N7 @ Ys ) )
=> ( P @ ( zip @ A @ B @ Zs3 @ Ws3 ) ) ) ) ) )
=> ( P @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_obtain_same_length
thf(fact_2742_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R3 ) ) ).
% Nil_lenlex_iff2
thf(fact_2743_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R3 ) )
=> ( ! [X2: A] :
( ( Xs
= ( cons @ A @ X2 @ Ys ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R3 ) )
=> ~ ! [Zs3: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs3 ) )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs3 @ Ys ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_2744_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ( ! [Y2: A] :
( ( Ys
= ( cons @ A @ Y2 @ Xs ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R3 ) )
=> ~ ! [Zs3: list @ A] :
( ( Ys
= ( cons @ A @ X @ Zs3 ) )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs3 ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_2745_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel1I1
thf(fact_2746_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_2747_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Us2: list @ A,Vs: list @ A,Ys: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( Xs
= ( append @ A @ Us2 @ ( cons @ A @ X @ Vs ) ) )
=> ( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ Y @ Vs ) ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% listrel1I
thf(fact_2748_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ~ ! [X2: A,Y2: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 )
=> ! [Us3: list @ A,Vs3: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X2 @ Vs3 ) ) )
=> ( Ys
!= ( append @ A @ Us3 @ ( cons @ A @ Y2 @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_2749_lenlex__length,axiom,
! [A: $tType,Ms: list @ A,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R3 ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% lenlex_length
thf(fact_2750_lenlex__append1,axiom,
! [A: $tType,Us2: list @ A,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us2 @ Xs ) @ ( lenlex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us2 @ Vs ) @ ( append @ A @ Xs @ Ys ) ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append1
thf(fact_2751_zip__append1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ B] :
( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ ( take @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) @ ( zip @ A @ B @ Ys @ ( drop @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) ) ) ).
% zip_append1
thf(fact_2752_zip__append2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ B] :
( ( zip @ A @ B @ Xs @ ( append @ B @ Ys @ Zs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) @ Ys ) @ ( zip @ A @ B @ ( drop @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) @ Zs ) ) ) ).
% zip_append2
thf(fact_2753_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R3 ) )
= ( ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_2754_map__tailrec__rev_Opelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y = Xb )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xb ) ) ) ) )
=> ~ ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ( ( Y
= ( map_tailrec_rev @ A @ B @ X @ As @ ( cons @ B @ ( X @ A6 ) @ Xb ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_2755_trancl__set__ntrancl,axiom,
! [A: $tType,Xs: list @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) @ ( one_one @ nat ) ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) ) ).
% trancl_set_ntrancl
thf(fact_2756_abstract__boolean__algebra__sym__diff__def,axiom,
! [A: $tType] :
( ( boolea3799213064322606851m_diff @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One2: A,Xor2: A > A > A] :
( ( boolea2506097494486148201lgebra @ A @ Conj2 @ Disj2 @ Compl2 @ Zero2 @ One2 )
& ( boolea5476839437570043046axioms @ A @ Conj2 @ Disj2 @ Compl2 @ Xor2 ) ) ) ) ).
% abstract_boolean_algebra_sym_diff_def
thf(fact_2757_abstract__boolean__algebra__sym__diff_Ointro,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor )
=> ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor ) ) ) ).
% abstract_boolean_algebra_sym_diff.intro
thf(fact_2758_lenlex__append2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Us2: list @ A,Xs: list @ A,Ys: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us2 @ Xs ) @ ( append @ A @ Us2 @ Ys ) ) @ ( lenlex @ A @ R ) )
= ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append2
thf(fact_2759_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_2760_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B,A5: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( member2 @ B @ X @ A5 )
=> ( member2 @ A @ B2 @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_2761_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( ( image2 @ B @ A @ F @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_2762_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image2 @ B @ A @ F @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_2763_image__empty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( image2 @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_2764_image__insert,axiom,
! [A: $tType,B: $tType,F: B > A,A3: B,B4: set @ B] :
( ( image2 @ B @ A @ F @ ( insert2 @ B @ A3 @ B4 ) )
= ( insert2 @ A @ ( F @ A3 ) @ ( image2 @ B @ A @ F @ B4 ) ) ) ).
% image_insert
thf(fact_2765_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F: A > B] :
( ( member2 @ A @ X @ A5 )
=> ( ( insert2 @ B @ ( F @ X ) @ ( image2 @ A @ B @ F @ A5 ) )
= ( image2 @ A @ B @ F @ A5 ) ) ) ).
% insert_image
thf(fact_2766_op__conc__empty__img__id,axiom,
! [A: $tType,L7: set @ ( list @ A )] :
( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( nil @ A ) ) @ L7 )
= L7 ) ).
% op_conc_empty_img_id
thf(fact_2767_image__update,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F: A > B,N: B] :
( ~ ( member2 @ A @ X @ A5 )
=> ( ( image2 @ A @ B @ ( fun_upd @ A @ B @ F @ X @ N ) @ A5 )
= ( image2 @ A @ B @ F @ A5 ) ) ) ).
% image_update
thf(fact_2768_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_2769_list_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: list @ A] :
( ( set2 @ B @ ( map @ A @ B @ F @ V ) )
= ( image2 @ A @ B @ F @ ( set2 @ A @ V ) ) ) ).
% list.set_map
thf(fact_2770_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( irrefl @ A @ R3 )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R3 ) )
= ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_2771_inj__on__insert,axiom,
! [B: $tType,A: $tType,F: A > B,A3: A,A5: set @ A] :
( ( inj_on @ A @ B @ F @ ( insert2 @ A @ A3 @ A5 ) )
= ( ( inj_on @ A @ B @ F @ A5 )
& ~ ( member2 @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_2772_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_2773_abstract__boolean__algebra__sym__diff__axioms_Ointro,axiom,
! [A: $tType,Xor: A > A > A,Disj: A > A > A,Conj: A > A > A,Compl: A > A] :
( ! [X2: A,Y2: A] :
( ( Xor @ X2 @ Y2 )
= ( Disj @ ( Conj @ X2 @ ( Compl @ Y2 ) ) @ ( Conj @ ( Compl @ X2 ) @ Y2 ) ) )
=> ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% abstract_boolean_algebra_sym_diff_axioms.intro
thf(fact_2774_abstract__boolean__algebra__sym__diff__axioms__def,axiom,
! [A: $tType] :
( ( boolea5476839437570043046axioms @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Xor2: A > A > A] :
! [X3: A,Y3: A] :
( ( Xor2 @ X3 @ Y3 )
= ( Disj2 @ ( Conj2 @ X3 @ ( Compl2 @ Y3 ) ) @ ( Conj2 @ ( Compl2 @ X3 ) @ Y3 ) ) ) ) ) ).
% abstract_boolean_algebra_sym_diff_axioms_def
thf(fact_2775_imageI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F: A > B] :
( ( member2 @ A @ X @ A5 )
=> ( member2 @ B @ ( F @ X ) @ ( image2 @ A @ B @ F @ A5 ) ) ) ).
% imageI
thf(fact_2776_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F: B > A,A5: set @ B] :
( ( member2 @ A @ Z2 @ ( image2 @ B @ A @ F @ A5 ) )
= ( ? [X3: B] :
( ( member2 @ B @ X3 @ A5 )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_2777_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
( ? [X6: A] :
( ( member2 @ A @ X6 @ ( image2 @ B @ A @ F @ A5 ) )
& ( P @ X6 ) )
=> ? [X2: B] :
( ( member2 @ B @ X2 @ A5 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_2778_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N4: set @ A,F: A > B,G: A > B] :
( ( M = N4 )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ N4 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image2 @ A @ B @ F @ M )
= ( image2 @ A @ B @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_2779_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( image2 @ B @ A @ F @ A5 ) )
=> ( P @ X2 ) )
=> ! [X6: B] :
( ( member2 @ B @ X6 @ A5 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_2780_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,B2: B,F: A > B] :
( ( member2 @ A @ X @ A5 )
=> ( ( B2
= ( F @ X ) )
=> ( member2 @ B @ B2 @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_2781_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ B,B4: set @ A] :
( ( ( image2 @ B @ A @ F @ A5 )
= B4 )
=> ( ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F ) @ ( pow @ B @ A5 ) )
= ( pow @ A @ B4 ) ) ) ).
% image_Pow_surj
thf(fact_2782_lexord__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( irrefl @ ( list @ A ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irrefl
thf(fact_2783_irrefl__lex,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R3 )
=> ( irrefl @ ( list @ A ) @ ( lex @ A @ R3 ) ) ) ).
% irrefl_lex
thf(fact_2784_image__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ F @ B4 ) ) ) ).
% image_mono
thf(fact_2785_image__subsetI,axiom,
! [A: $tType,B: $tType,A5: set @ A,F: A > B,B4: set @ B] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ A5 )
=> ( member2 @ B @ ( F @ X2 ) @ B4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ B4 ) ) ).
% image_subsetI
thf(fact_2786_subset__imageE,axiom,
! [A: $tType,B: $tType,B4: set @ A,F: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image2 @ B @ A @ F @ A5 ) )
=> ~ ! [C8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C8 @ A5 )
=> ( B4
!= ( image2 @ B @ A @ F @ C8 ) ) ) ) ).
% subset_imageE
thf(fact_2787_image__subset__iff,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ A5 ) @ B4 )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ A5 )
=> ( member2 @ A @ ( F @ X3 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_2788_subset__image__iff,axiom,
! [A: $tType,B: $tType,B4: set @ A,F: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image2 @ B @ A @ F @ A5 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
& ( B4
= ( image2 @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_2789_image__Un,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,B4: set @ B] :
( ( image2 @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F @ A5 ) @ ( image2 @ B @ A @ F @ B4 ) ) ) ).
% image_Un
thf(fact_2790_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ A5 ) @ B4 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F ) @ ( pow @ B @ A5 ) ) @ ( pow @ A @ B4 ) ) ) ).
% image_Pow_mono
thf(fact_2791_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y: A,F: B > A,A5: set @ B] :
( ( member2 @ A @ Y @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( F @ ( inv_on @ B @ A @ F @ A5 @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_2792_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y: A,F: B > A,A5: set @ B] :
( ( member2 @ A @ Y @ ( image2 @ B @ A @ F @ A5 ) )
=> ( member2 @ B @ ( inv_on @ B @ A @ F @ A5 @ Y ) @ A5 ) ) ).
% inv_on_f_range
thf(fact_2793_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F: B > A,X: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ I5 ) )
= X ) ) ) ) ).
% SUP_eq_const
thf(fact_2794_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S4: set @ A,T4: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( inf_inf @ ( set @ A ) @ S4 @ T4 ) )
= ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S4 ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T4 ) ) ) ) ).
% translation_Int
thf(fact_2795_inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B4: set @ A] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ B4 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ A @ A @ ( inf_inf @ A @ A3 ) @ B4 ) ) ) ) ).
% inf_Sup
thf(fact_2796_image__Int__subset,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,B4: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) @ ( inf_inf @ ( set @ A ) @ ( image2 @ B @ A @ F @ A5 ) @ ( image2 @ B @ A @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_2797_image__diff__subset,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,B4: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image2 @ B @ A @ F @ A5 ) @ ( image2 @ B @ A @ F @ B4 ) ) @ ( image2 @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) ) ).
% image_diff_subset
thf(fact_2798_image__set,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( image2 @ B @ A @ F @ ( set2 @ B @ Xs ) )
= ( set2 @ A @ ( map @ B @ A @ F @ Xs ) ) ) ).
% image_set
thf(fact_2799_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B,A3: B] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
=> ( ( member2 @ B @ A3 @ A5 )
=> ( finite_finite2 @ A @ ( B4 @ A3 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_2800_insort__insert__key__triv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Xs: list @ B] :
( ( member2 @ A @ ( F @ X ) @ ( image2 @ B @ A @ F @ ( set2 @ B @ Xs ) ) )
=> ( ( linord329482645794927042rt_key @ B @ A @ F @ X @ Xs )
= Xs ) ) ) ).
% insort_insert_key_triv
thf(fact_2801_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B,X: A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y2: A] :
( ( member2 @ A @ Y2 @ A5 )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_elem @ B @ ( image2 @ A @ B @ F @ A5 ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_2802_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,C3: A,F: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ C3 @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ I5 ) )
= C3 )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ I5 )
=> ( ( F @ X3 )
= C3 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_2803_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,M: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ M ) ) ) ) ).
% cSUP_least
thf(fact_2804_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A5: set @ A,A13: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F3: A > B] :
( ( inj_on @ A @ B @ F3 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ A13 ) ) )
= ( ? [G2: B > A] :
( ( image2 @ B @ A @ G2 @ A13 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_2805_in__image__insert__iff,axiom,
! [A: $tType,B4: set @ ( set @ A ),X: A,A5: set @ A] :
( ! [C8: set @ A] :
( ( member2 @ ( set @ A ) @ C8 @ B4 )
=> ~ ( member2 @ A @ X @ C8 ) )
=> ( ( member2 @ ( set @ A ) @ A5 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X ) @ B4 ) )
= ( ( member2 @ A @ X @ A5 )
& ( member2 @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_2806_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F: A > B,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ F @ B4 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_2807_lexl__not__refl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( irrefl @ A @ R3 )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ X ) @ ( lex @ A @ R3 ) ) ) ).
% lexl_not_refl
thf(fact_2808_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Xor: A > A > A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One @ Xor )
=> ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% abstract_boolean_algebra_sym_diff.axioms(2)
thf(fact_2809_Pow__insert,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( pow @ A @ ( insert2 @ A @ A3 @ A5 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ A3 ) @ ( pow @ A @ A5 ) ) ) ) ).
% Pow_insert
thf(fact_2810_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list @ A,L3: list @ A,F: A > B] :
( ( ( size_size @ ( list @ A ) @ L )
= ( size_size @ ( list @ A ) @ L3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F @ ( nth @ A @ L @ I3 ) )
= ( F @ ( nth @ A @ L3 @ I3 ) ) ) )
=> ( ( image2 @ A @ B @ F @ ( set2 @ A @ L ) )
= ( image2 @ A @ B @ F @ ( set2 @ A @ L3 ) ) ) ) ) ).
% set_image_eq_pointwiseI
thf(fact_2811_in__set__image__conv__nth,axiom,
! [B: $tType,A: $tType,F: B > A,X: B,L: list @ B] :
( ( member2 @ A @ ( F @ X ) @ ( image2 @ B @ A @ F @ ( set2 @ B @ L ) ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ L ) )
& ( ( F @ ( nth @ B @ L @ I2 ) )
= ( F @ X ) ) ) ) ) ).
% in_set_image_conv_nth
thf(fact_2812_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A5: set @ B,F: B > A,Y: A] :
( ( ( member2 @ B @ X @ A5 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X @ Y ) @ A5 )
= ( insert2 @ A @ Y @ ( image2 @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member2 @ B @ X @ A5 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X @ Y ) @ A5 )
= ( image2 @ B @ A @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_2813_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H: A > A,N4: set @ A] :
( ! [X2: A,Y2: A] :
( ( H @ ( inf_inf @ A @ X2 @ Y2 ) )
= ( inf_inf @ A @ ( H @ X2 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic7752659483105999362nf_fin @ A @ N4 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H @ N4 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_2814_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H: A > A,N4: set @ A] :
( ! [X2: A,Y2: A] :
( ( H @ ( sup_sup @ A @ X2 @ Y2 ) )
= ( sup_sup @ A @ ( H @ X2 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic5882676163264333800up_fin @ A @ N4 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H @ N4 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_2815_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N4: set @ A] :
( ! [X2: A,Y2: A] :
( ( H @ ( ord_max @ A @ X2 @ Y2 ) )
= ( ord_max @ A @ ( H @ X2 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798349783984er_Max @ A @ N4 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H @ N4 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_2816_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N4: set @ A] :
( ! [X2: A,Y2: A] :
( ( H @ ( ord_min @ A @ X2 @ Y2 ) )
= ( ord_min @ A @ ( H @ X2 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798350308766er_Min @ A @ N4 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H @ N4 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_2817_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F: A > A > A,H: A > A,N4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ! [X2: A,Y2: A] :
( ( H @ ( F @ X2 @ Y2 ) )
= ( F @ ( H @ X2 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N4 )
=> ( ( N4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic1715443433743089157tice_F @ A @ F @ N4 ) )
= ( lattic1715443433743089157tice_F @ A @ F @ ( image2 @ A @ A @ H @ N4 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_2818_f__arg__min__list__f,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs: list @ A,F: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( F @ ( arg_min_list @ A @ B @ F @ Xs ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F @ ( set2 @ A @ Xs ) ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_2819_subset__subseqs,axiom,
! [A: $tType,X5: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( set2 @ A @ Xs ) )
=> ( member2 @ ( set @ A ) @ X5 @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_2820_finite__set__image,axiom,
! [A: $tType,A5: set @ ( list @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A5 ) )
=> ( ! [Xs2: list @ A] :
( ( member2 @ ( list @ A ) @ Xs2 @ A5 )
=> ( distinct @ A @ Xs2 ) )
=> ( finite_finite2 @ ( list @ A ) @ A5 ) ) ) ).
% finite_set_image
thf(fact_2821_subseqs__powset,axiom,
! [A: $tType,Xs: list @ A] :
( ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
= ( pow @ A @ ( set2 @ A @ Xs ) ) ) ).
% subseqs_powset
thf(fact_2822_inj__on__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ( inj_on @ A @ B @ F @ A5 )
& ( inj_on @ A @ B @ F @ B4 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_2823_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A5: B > ( set @ A ),I: B,B4: set @ A,J4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A5 @ I @ B4 ) @ J4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ B ) @ J4 @ ( insert2 @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member2 @ B @ I @ J4 ) @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_2824_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ ( list @ A )] :
( ( inj_on @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A5 ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F ) @ A5 ) ) ).
% inj_on_mapI
thf(fact_2825_Union__image__empty,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F @ ( bot_bot @ ( set @ B ) ) ) ) )
= A5 ) ).
% Union_image_empty
thf(fact_2826_bijective__Empty,axiom,
! [B: $tType,A: $tType] : ( bijective @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% bijective_Empty
thf(fact_2827_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_2828_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) )
= ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) ) ) ).
% Field_insert
thf(fact_2829_upt__rec__numeral,axiom,
! [M2: num,N: num] :
( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
= ( cons @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) )
& ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
= ( nil @ nat ) ) ) ) ).
% upt_rec_numeral
thf(fact_2830_nth__enumerate__eq,axiom,
! [A: $tType,M2: nat,Xs: list @ A,N: nat] :
( ( ord_less @ nat @ M2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M2 )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M2 ) @ ( nth @ A @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_2831_finite__Field__eq__finite,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ ( field2 @ A @ R ) )
= ( finite_finite2 @ ( product_prod @ A @ A ) @ R ) ) ).
% finite_Field_eq_finite
thf(fact_2832_tl__upt,axiom,
! [M2: nat,N: nat] :
( ( tl @ nat @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ N ) ) ).
% tl_upt
thf(fact_2833_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( hd @ nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_2834_drop__upt,axiom,
! [M2: nat,I: nat,J: nat] :
( ( drop @ nat @ M2 @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus @ nat @ I @ M2 ) @ J ) ) ).
% drop_upt
thf(fact_2835_enumerate__simps_I1_J,axiom,
! [A: $tType,N: nat] :
( ( enumerate @ A @ N @ ( nil @ A ) )
= ( nil @ ( product_prod @ nat @ A ) ) ) ).
% enumerate_simps(1)
thf(fact_2836_length__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( product_prod @ nat @ A ) ) @ ( enumerate @ A @ N @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_enumerate
thf(fact_2837_Field__empty,axiom,
! [A: $tType] :
( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_2838_take__upt,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M2 ) @ N )
=> ( ( take @ nat @ M2 @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus @ nat @ I @ M2 ) ) ) ) ).
% take_upt
thf(fact_2839_upt__0__eq__Nil__conv,axiom,
! [J: nat] :
( ( ( upt @ ( zero_zero @ nat ) @ J )
= ( nil @ nat ) )
= ( J
= ( zero_zero @ nat ) ) ) ).
% upt_0_eq_Nil_conv
thf(fact_2840_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_2841_upt__merge,axiom,
! [I: nat,J: nat,K: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
& ( ord_less_eq @ nat @ J @ K ) )
=> ( ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
= ( upt @ I @ K ) ) ) ).
% upt_merge
thf(fact_2842_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ I ) ) ).
% length_upt
thf(fact_2843_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( last @ nat @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% last_upt
thf(fact_2844_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= ( nil @ nat ) )
= ( ( J
= ( zero_zero @ nat ) )
| ( ord_less_eq @ nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_2845_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list @ B] :
( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_2846_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
=> ( ( nth @ nat @ ( upt @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_2847_map__Suc__upt,axiom,
! [M2: nat,N: nat] :
( ( map @ nat @ nat @ suc @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_2848_distinct__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( distinct @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ).
% distinct_enumerate
thf(fact_2849_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct @ nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_2850_enumerate__eq__zip,axiom,
! [A: $tType] :
( ( enumerate @ A )
= ( ^ [N3: nat,Xs3: list @ A] : ( zip @ nat @ A @ ( upt @ N3 @ ( plus_plus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) @ Xs3 ) ) ) ).
% enumerate_eq_zip
thf(fact_2851_total__on__empty,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R3 ) ).
% total_on_empty
thf(fact_2852_butlast__upt,axiom,
! [M2: nat,N: nat] :
( ( butlast @ nat @ ( upt @ M2 @ N ) )
= ( upt @ M2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% butlast_upt
thf(fact_2853_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% atLeast_upt
thf(fact_2854_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list @ nat,Q3: nat] :
( ( ( cons @ nat @ M2 @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M2 @ Q3 ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M2 ) @ Q3 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_2855_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_2856_sorted__wrt__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M2 @ N ) ) ).
% sorted_wrt_upt
thf(fact_2857_sorted__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M2 @ N ) ) ).
% sorted_upt
thf(fact_2858_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N3: nat,M3: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ ( suc @ M3 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_2859_upt__eq__append__conv,axiom,
! [I: nat,J: nat,Xs: list @ nat,Ys: list @ nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( upt @ I @ J )
= ( append @ nat @ Xs @ Ys ) )
= ( ? [K2: nat] :
( ( ord_less_eq @ nat @ I @ K2 )
& ( ord_less_eq @ nat @ K2 @ J )
& ( ( upt @ I @ K2 )
= Xs )
& ( ( upt @ K2 @ J )
= Ys ) ) ) ) ) ).
% upt_eq_append_conv
thf(fact_2860_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N3: nat,M3: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ M3 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_2861_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) ) ) ) ).
% atMost_upto
thf(fact_2862_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_2863_upt__append,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( append @ nat @ ( upt @ ( zero_zero @ nat ) @ I ) @ ( upt @ I @ J ) )
= ( upt @ ( zero_zero @ nat ) @ J ) ) ) ).
% upt_append
thf(fact_2864_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_2865_bijective__def,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B,Z3: B] :
( ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R6 )
& ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z3 ) @ R6 ) )
=> ( Y3 = Z3 ) )
& ! [X3: A,Y3: A,Z3: B] :
( ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z3 ) @ R6 )
& ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z3 ) @ R6 ) )
=> ( X3 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_2866_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M2: nat,F: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M2 ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F @ ( upt @ M2 @ N ) ) @ I )
= ( F @ ( plus_plus @ nat @ M2 @ I ) ) ) ) ).
% nth_map_upt
thf(fact_2867_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_2868_upt__rec,axiom,
( upt
= ( ^ [I2: nat,J2: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I2 @ J2 ) @ ( cons @ nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_2869_enumerate__append__eq,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( enumerate @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ ( enumerate @ A @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_2870_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,M2: nat,F: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( F @ ( plus_plus @ nat @ M2 @ I3 ) ) ) )
=> ( ( map @ nat @ A @ F @ ( upt @ M2 @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_2871_map__nth__upt__drop__take__conv,axiom,
! [A: $tType,N4: nat,L: list @ A,M: nat] :
( ( ord_less_eq @ nat @ N4 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( map @ nat @ A @ ( nth @ A @ L ) @ ( upt @ M @ N4 ) )
= ( drop @ A @ M @ ( take @ A @ N4 @ L ) ) ) ) ).
% map_nth_upt_drop_take_conv
thf(fact_2872_upt__eq__lel__conv,axiom,
! [L: nat,H: nat,Is1: list @ nat,I: nat,Is2: list @ nat] :
( ( ( upt @ L @ H )
= ( append @ nat @ Is1 @ ( cons @ nat @ I @ Is2 ) ) )
= ( ( Is1
= ( upt @ L @ I ) )
& ( Is2
= ( upt @ ( suc @ I ) @ H ) )
& ( ord_less_eq @ nat @ L @ I )
& ( ord_less @ nat @ I @ H ) ) ) ).
% upt_eq_lel_conv
thf(fact_2873_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_2874_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_2875_nth__image,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( image2 @ nat @ A @ ( nth @ A @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
= ( set2 @ A @ ( take @ A @ L @ Xs ) ) ) ) ).
% nth_image
thf(fact_2876_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_2877_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite2 @ A @ A5 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_2878_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_2879_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_2880_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_2881_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_2882_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_2883_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_2884_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_2885_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_2886_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_2887_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( one_one @ A ) ) ) ) ).
% Gcd_fin.infinite
thf(fact_2888_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A5 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Gcd_fin_iff
thf(fact_2889_sorted__list__of__set__range,axiom,
! [M2: nat,N: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% sorted_list_of_set_range
thf(fact_2890_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_2891_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_2892_Int__atLeastLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( ord_max @ A @ A3 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastLessThan
thf(fact_2893_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_2894_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_2895_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_2896_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_2897_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_2898_atLeastLessThan__upt,axiom,
( ( set_or7035219750837199246ssThan @ nat )
= ( ^ [I2: nat,J2: nat] : ( set2 @ nat @ ( upt @ I2 @ J2 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_2899_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X3: nat] :
( ( member2 @ nat @ X3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_2900_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X3: nat] :
( ( member2 @ nat @ X3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_2901_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(3)
thf(fact_2902_zipf_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,A3: A,As2: list @ A,B2: B,Bs: list @ B] :
( ( zipf @ A @ B @ C @ F @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs ) )
= ( cons @ C @ ( F @ A3 @ B2 ) @ ( zipf @ A @ B @ C @ F @ As2 @ Bs ) ) ) ).
% zipf.simps(2)
thf(fact_2903_zipf_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B > C] :
( ( zipf @ A @ B @ C @ F @ ( nil @ A ) @ ( nil @ B ) )
= ( nil @ C ) ) ).
% zipf.simps(1)
thf(fact_2904_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A3 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A3 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_2905_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_2906_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(1)
thf(fact_2907_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_2908_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_2909_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_2910_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_2911_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_2912_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) @ P4 ) ) ) ).
% ceiling_divide_lower
thf(fact_2913_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_2914_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_2915_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_2916_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_2917_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_2918_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_2919_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T4: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T4 ) )
= ( ! [I2: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I2 ) @ T4 )
& ( ord_less @ A @ T4 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% floor_split
thf(fact_2920_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A3 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_2921_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_2922_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_2923_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_2924_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_2925_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ Q3 ) @ P4 ) ) ) ).
% floor_divide_lower
thf(fact_2926_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% floor_divide_upper
thf(fact_2927_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A3 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_2928_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T4: A] :
( ( P @ ( archimedean_ceiling @ A @ T4 ) )
= ( ! [I2: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) @ T4 )
& ( ord_less_eq @ A @ T4 @ ( ring_1_of_int @ A @ I2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% ceiling_split
thf(fact_2929_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archimedean_ceiling @ A @ X )
= A3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_2930_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_2931_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_2932_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_2933_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_2934_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q3 ) ) ) @ Q3 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2935_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_2936_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_2937_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_2938_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_2939_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_2940_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_2941_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_2942_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_2943_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_2944_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_2945_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_2946_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_2947_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X3: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_2948_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_2949_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X3: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X3 ) ) @ ( archimedean_ceiling @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) ) ) ) ).
% round_altdef
thf(fact_2950_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I2: int,J2: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J2 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) @ ( cons @ int @ J2 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_2951_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N ) ) ) ).
% round_unique'
thf(fact_2952_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_2953_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_2954_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_idempotent
thf(fact_2955_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_2956_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_2957_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_2958_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_2959_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% abs_mult_self_eq
thf(fact_2960_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_2961_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus_cancel
thf(fact_2962_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_2963_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% abs_le_self_iff
thf(fact_2964_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_nonneg
thf(fact_2965_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_2966_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_2967_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_nonpos
thf(fact_2968_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_minus_commute
thf(fact_2969_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_2970_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_mult
thf(fact_2971_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_2972_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_self
thf(fact_2973_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_zero
thf(fact_2974_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_pos
thf(fact_2975_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_2976_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_2977_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_2978_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_2979_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_2980_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C3: A,B2: A,D3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C3 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D3 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C3 @ D3 ) ) ) ) ) ).
% abs_mult_less
thf(fact_2981_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_minus_self
thf(fact_2982_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_2983_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_2984_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_2985_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_2986_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_2987_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_2988_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_2989_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
| ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_2990_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_2991_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C3 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_2992_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_2993_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_neg
thf(fact_2994_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S ) ) @ A3 ) ) ) ) ).
% cSup_abs_le
thf(fact_2995_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_2996_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S ) @ L ) ) @ E3 ) ) ) ) ).
% cSup_asclose
thf(fact_2997_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_2998_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_2999_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_3000_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_3001_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_3002_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_3003_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_3004_upto__rec__numeral_I2_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_3005_upto__rec__numeral_I3_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_3006_upto__rec__numeral_I4_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_3007_upto__rec__numeral_I1_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_3008_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member2 @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A3 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A3 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_3009_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
=> ( ( nth @ int @ ( upto @ I @ J ) @ K )
= ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% nth_upto
thf(fact_3010_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil
thf(fact_3011_upto__Nil2,axiom,
! [I: int,J: int] :
( ( ( nil @ int )
= ( upto @ I @ J ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil2
thf(fact_3012_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less @ int @ J @ I )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_3013_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons @ int @ I @ ( nil @ int ) ) ) ).
% upto_single
thf(fact_3014_distinct__upto,axiom,
! [I: int,J: int] : ( distinct @ int @ ( upto @ I @ J ) ) ).
% distinct_upto
thf(fact_3015_sorted__upto,axiom,
! [M2: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M2 @ N ) ) ).
% sorted_upto
thf(fact_3016_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_3017_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member2 @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member2 @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member2 @ A @ ( times_times @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_3018_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member2 @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_3019_upto__code,axiom,
( upto
= ( ^ [I2: int,J2: int] : ( upto_aux @ I2 @ J2 @ ( nil @ int ) ) ) ) ).
% upto_code
thf(fact_3020_upto__aux__def,axiom,
( upto_aux
= ( ^ [I2: int,J2: int] : ( append @ int @ ( upto @ I2 @ J2 ) ) ) ) ).
% upto_aux_def
thf(fact_3021_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I2: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J2 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_3022_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I2: int,J2: int] : ( set2 @ int @ ( upto @ I2 @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_3023_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member2 @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_3024_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_3025_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_3026_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I2: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_3027_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member2 @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_3028_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member2 @ 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_3029_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member2 @ 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_3030_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( member2 @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member2 @ 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_3031_upto_Osimps,axiom,
( upto
= ( ^ [I2: int,J2: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I2 @ J2 ) @ ( cons @ int @ I2 @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J2 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_3032_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_3033_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_3034_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_3035_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( member2 @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member2 @ 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_3036_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_3037_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: A] :
( ( ( archimedean_frac @ A @ X )
= A3 )
= ( ( member2 @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_3038_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member2 @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A3 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A3 @ B2 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_3039_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_3040_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N3 ) @ ( nth @ B @ Ys @ N3 ) ) @ R3 ) ) ) ) ).
% listrel_iff_nth
thf(fact_3041_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_3042_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_3043_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F: A > nat,Fs: list @ ( A > nat )] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) )
= ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_3044_in__measures_I1_J,axiom,
! [A: $tType,X: A,Y: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_3045_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_3046_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_3047_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_3048_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R3 ) )
=> ( Xs
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_3049_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R3 ) )
=> ( Xs
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_3050_listrel_ONil,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R3 ) ) ).
% listrel.Nil
thf(fact_3051_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R3 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_3052_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,P4: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P4 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P4 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P4 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_3053_listrel__mono,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( listrel @ A @ B @ R3 ) @ ( listrel @ A @ B @ S4 ) ) ) ).
% listrel_mono
thf(fact_3054_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R3: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R3 )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R3 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R3 ) ) ) ) ).
% listrel.Cons
thf(fact_3055_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys ) @ Xs ) @ ( listrel @ A @ B @ R3 ) )
=> ~ ! [Y2: B,Ys2: list @ B] :
( ( Xs
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y2 ) @ R3 )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys2 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_3056_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R3 ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R3 )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_3057_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_3058_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_3059_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_3060_measures__less,axiom,
! [A: $tType,F: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_3061_measures__lesseq,axiom,
! [A: $tType,F: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_3062_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R3 ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X2: A,Y2: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ B] :
( ( A22
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R3 )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_3063_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X3: A,Y3: B,Xs3: list @ A,Ys3: list @ B] :
( ( A1
= ( cons @ A @ X3 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y3 @ Ys3 ) )
& ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys3 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel.simps
thf(fact_3064_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member2 @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A5 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ) ) ).
% prod.insert
thf(fact_3065_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_3066_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_3067_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A5: set @ B,B4: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
=> ( ( F @ X2 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_3068_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A5: set @ B,F: B > A,A3: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( F @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( member2 @ B @ A3 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( F @ A3 ) ) ) )
& ( ~ ( member2 @ B @ A3 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_3069_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A5: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
!= ( one_one @ A ) )
=> ~ ! [A6: B] :
( ( member2 @ B @ A6 @ A5 )
=> ( ( G @ A6 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_3070_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A] :
( ! [X2: B] :
( ( member2 @ B @ X2 @ A5 )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_3071_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X2: B] :
( ( member2 @ B @ X2 @ A5 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F @ X2 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ).
% prod_ge_1
thf(fact_3072_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X2: B] :
( ( member2 @ B @ X2 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X2 ) )
& ( ord_less_eq @ A @ ( F @ X2 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_3073_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S: set @ B,H: B > A,G: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X1: A,Y1: A,X23: A,Y23: A] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( times_times @ A @ X1 @ Y1 ) @ ( times_times @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H @ S ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S ) ) ) ) ) ) ) ).
% prod.related
thf(fact_3074_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member2 @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) )
& ( ~ ( member2 @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A5 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_3075_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T5: set @ C,S: set @ B,I: C > B,J: B > C,T: set @ C,G: B > A,H: C > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A6: B] :
( ( member2 @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S3 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member2 @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S3 ) )
=> ( member2 @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T @ T5 ) ) )
=> ( ! [B6: C] :
( ( member2 @ C @ B6 @ ( minus_minus @ ( set @ C ) @ T @ T5 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: C] :
( ( member2 @ C @ B6 @ ( minus_minus @ ( set @ C ) @ T @ T5 ) )
=> ( member2 @ B @ ( I @ B6 ) @ ( minus_minus @ ( set @ B ) @ S @ S3 ) ) )
=> ( ! [A6: B] :
( ( member2 @ B @ A6 @ S3 )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B6: C] :
( ( member2 @ C @ B6 @ T5 )
=> ( ( H @ B6 )
= ( one_one @ A ) ) )
=> ( ! [A6: B] :
( ( member2 @ B @ A6 @ S )
=> ( ( H @ ( J @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ C @ A @ H @ T ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_3076_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,I: A,F: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( member2 @ A @ I @ I5 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F @ I ) )
=> ( ! [I3: A] :
( ( member2 @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I5 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_3077_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member2 @ A @ I3 @ I5 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_3078_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B4: set @ B,A5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B4 @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B4 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_3079_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T: set @ B,S: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ T )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T @ S ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_3080_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T: set @ B,S: set @ B,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ T )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T @ S ) )
=> ( ( H @ X2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_3081_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T @ S ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_3082_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T @ S ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_3083_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ B,A5: set @ B,B4: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C4 )
=> ( ! [A6: B] :
( ( member2 @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C4 @ A5 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B6: B] :
( ( member2 @ B @ B6 @ ( minus_minus @ ( set @ B ) @ C4 @ B4 ) )
=> ( ( H @ B6 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C4 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B4 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_3084_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ B,A5: set @ B,B4: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C4 )
=> ( ! [A6: B] :
( ( member2 @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C4 @ A5 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B6: B] :
( ( member2 @ B @ B6 @ ( minus_minus @ ( set @ B ) @ C4 @ B4 ) )
=> ( ( H @ B6 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B4 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C4 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_3085_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B4 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_3086_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A,B4: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_3087_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T: set @ B,S: set @ B,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ T )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T @ S ) )
=> ( ( H @ I3 )
= ( one_one @ A ) ) )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S @ T ) )
=> ( ( G @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( inf_inf @ ( set @ B ) @ S @ T ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_3088_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F: B > A,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) )
& ( ord_less @ A @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_3089_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert2 @ B @ X @ A5 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_3090_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member2 @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_3091_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B4 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_3092_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( ( inf_inf @ ( set @ B ) @ A5 @ B4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B4 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_3093_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B4 @ A5 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_3094_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B4: set @ A,A5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ! [B6: A] :
( ( member2 @ A @ B6 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ B6 ) ) )
=> ( ! [A6: A] :
( ( member2 @ A @ A6 @ A5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ A6 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A5 ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ B4 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_3095_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F: B > A,N: A,K: nat] :
( ! [I3: B] :
( ( member2 @ B @ I3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) )
& ( ord_less_eq @ A @ ( F @ I3 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_3096_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_3097_prod_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Xs: list @ B,G: B > A] :
( ( distinct @ B @ Xs )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups5270119922927024881d_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% prod.distinct_set_conv_list
thf(fact_3098_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_3099_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_3100_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_3101_Nil__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_in_shuffles
thf(fact_3102_finite__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( finite_finite2 @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% finite_shuffles
thf(fact_3103_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3104_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3105_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_3106_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert2 @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( B2 = C3 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_3107_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
( ( set_or1337092689740270186AtMost @ A @ A3 @ A3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_3108_euclidean__size__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
= ( one_one @ nat ) ) ) ).
% euclidean_size_1
thf(fact_3109_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_3110_prod__list_ONil,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
= ( one_one @ A ) ) ) ).
% prod_list.Nil
thf(fact_3111_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_3112_Int__atLeastAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMost
thf(fact_3113_Int__atLeastAtMostR1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ C3 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostR1
thf(fact_3114_Int__atLeastAtMostL1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atMost @ A @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostL1
thf(fact_3115_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A3 ) @ ( times_times @ A @ D3 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_3116_shuffles__commutes,axiom,
! [A: $tType] :
( ( shuffles @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] : ( shuffles @ A @ Ys3 @ Xs3 ) ) ) ).
% shuffles_commutes
thf(fact_3117_Cons__in__shuffles__rightI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z2: A] :
( ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member2 @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Z2 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_3118_Cons__in__shuffles__leftI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z2: A] :
( ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member2 @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs ) @ ( shuffles @ A @ ( cons @ A @ Z2 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_3119_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_3120_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_3121_Nil__in__shufflesI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( nil @ A ) )
=> ( ( Ys
= ( nil @ A ) )
=> ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_3122_refl__on__Int,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),B4: set @ A,S4: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A5 @ R3 )
=> ( ( refl_on @ A @ B4 @ S4 )
=> ( refl_on @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S4 ) ) ) ) ).
% refl_on_Int
thf(fact_3123_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_3124_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X3: nat] :
( ( member2 @ nat @ X3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_3125_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X3: nat] :
( ( member2 @ nat @ X3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_3126_shufflesE,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys
!= ( nil @ A ) ) )
=> ( ( ( Zs = Ys )
=> ( Xs
!= ( nil @ A ) ) )
=> ( ! [X2: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Xs4 ) )
=> ! [Z4: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z4 @ Zs4 ) )
=> ( ( X2 = Z4 )
=> ~ ( member2 @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y2: A,Ys4: list @ A] :
( ( Ys
= ( cons @ A @ Y2 @ Ys4 ) )
=> ! [Z4: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z4 @ Zs4 ) )
=> ( ( Y2 = Z4 )
=> ~ ( member2 @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_3127_length__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Zs )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_3128_set__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( set2 @ A @ Zs )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_3129_euclidean__size__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) ) ) ) ).
% euclidean_size_unit
thf(fact_3130_euclidean__size__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_mult
thf(fact_3131_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_3132_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_3133_Cons__shuffles__subset1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_3134_Cons__shuffles__subset2,axiom,
! [A: $tType,Y: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_3135_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(7)
thf(fact_3136_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_3137_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(8)
thf(fact_3138_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(5)
thf(fact_3139_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(4)
thf(fact_3140_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
& ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_iff_euclidean_size
thf(fact_3141_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B2 @ A3 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_3142_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% size_mult_mono
thf(fact_3143_euclidean__size__times__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_times_unit
thf(fact_3144_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N3: nat,M3: nat] : ( set2 @ nat @ ( upt @ N3 @ ( suc @ M3 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_3145_shuffles_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( insert2 @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( insert2 @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( Y
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y2 @ Ys2 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y2 ) @ ( shuffles @ A @ ( cons @ A @ X2 @ Xs2 ) @ Ys2 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_3146_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A4: A,B3: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_3147_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A4: A,B3: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_3148_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_3149_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_3150_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_3151_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_3152_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_3153_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_3154_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ A @ Ys )
=> ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( distinct @ A @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_3155_Cons__in__shuffles__iff,axiom,
! [A: $tType,Z2: A,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member2 @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( ( Xs
!= ( nil @ A ) )
& ( ( hd @ A @ Xs )
= Z2 )
& ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ ( tl @ A @ Xs ) @ Ys ) ) )
| ( ( Ys
!= ( nil @ A ) )
& ( ( hd @ A @ Ys )
= Z2 )
& ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ ( tl @ A @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_3156_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_3157_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_3158_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_3159_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_3160_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P4 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P4 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3161_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ X ) @ ( times_times @ A @ C3 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ Y ) @ ( times_times @ A @ C3 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_3162_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A3: A] :
( ( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( A3
!= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) )
=> ( ( ( B2
!= ( zero_zero @ A ) )
=> ! [Q4: A,R7: A] :
( ( ( euclid7384307370059645450egment @ A @ R7 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R7 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( R7
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= Q4 )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= R7 )
=> ( A3
!= ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R7 ) ) ) ) ) ) ) )
=> ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_3163_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q3: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R3 )
= A3 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= R3 ) ) ) ) ) ) ).
% mod_eqI
thf(fact_3164_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q3: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R3 )
= A3 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= Q3 ) ) ) ) ) ) ).
% div_eqI
thf(fact_3165_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R3 ) @ B2 )
= Q3 ) ) ) ) ) ).
% div_bounded
thf(fact_3166_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_3167_image__ident,axiom,
! [A: $tType,Y6: set @ A] :
( ( image2 @ A @ A
@ ^ [X3: A] : X3
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_3168_subset__mset_OcSUP__const,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X3: B] : C3
@ A5 ) )
= C3 ) ) ).
% subset_mset.cSUP_const
thf(fact_3169_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X3: A] : X3 )
= ( ^ [Xs3: list @ A] : Xs3 ) ) ).
% map_ident
thf(fact_3170_division__segment__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% division_segment_1
thf(fact_3171_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z5: A] : Y5 = Z5
@ A3 ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_3172_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X3: A] : X3 = A3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_3173_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu3: B] : ( one_one @ A )
@ A5 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_3174_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_3175_division__segment__of__nat,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( euclid7384307370059645450egment @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( one_one @ A ) ) ) ).
% division_segment_of_nat
thf(fact_3176_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( bot_bot @ A )
@ A5 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_3177_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B4 @ A5 ) )
= ( bot_bot @ A ) )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ A5 )
=> ( ( B4 @ X3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_3178_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B4 @ A5 ) ) )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ A5 )
=> ( ( B4 @ X3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_3179_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% cSUP_const
thf(fact_3180_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : F
@ A5 ) )
= F ) ) ) ).
% SUP_const
thf(fact_3181_foldl__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I2: nat,X3: A] : ( suc @ I2 )
@ ( zero_zero @ nat )
@ L )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldl_length
thf(fact_3182_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member2 @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A3 = K2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S )
= ( B2 @ A3 ) ) )
& ( ~ ( member2 @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A3 = K2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_3183_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member2 @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S )
= ( B2 @ A3 ) ) )
& ( ~ ( member2 @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_3184_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F: B > A,G: B > A,S: set @ B] :
( ( image2 @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ S )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ S @ ( collect @ B @ P ) ) )
@ ( image2 @ B @ A @ G
@ ( inf_inf @ ( set @ B ) @ S
@ ( collect @ B
@ ^ [X3: B] :
~ ( P @ X3 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_3185_UN__constant,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_3186_sorted__wrt__rev__linord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ L ) ) ) ) ).
% sorted_wrt_rev_linord
thf(fact_3187_division__segment__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) ) )
= A3 ) ) ).
% division_segment_euclidean_size
thf(fact_3188_UN__singleton,axiom,
! [A: $tType,A5: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X3: A] : ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= A5 ) ).
% UN_singleton
thf(fact_3189_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ B,A3: A,B4: B > ( set @ A )] :
( ( ( C4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( insert2 @ A @ A3 @ ( B4 @ X3 ) )
@ C4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( insert2 @ A @ A3 @ ( B4 @ X3 ) )
@ C4 ) )
= ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_3190_UN__simps_I3_J,axiom,
! [E4: $tType,F6: $tType,C4: set @ F6,A5: set @ E4,B4: F6 > ( set @ E4 )] :
( ( ( C4
= ( bot_bot @ ( set @ F6 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E4 )
@ ( image2 @ F6 @ ( set @ E4 )
@ ^ [X3: F6] : ( sup_sup @ ( set @ E4 ) @ A5 @ ( B4 @ X3 ) )
@ C4 ) )
= ( bot_bot @ ( set @ E4 ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ F6 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E4 )
@ ( image2 @ F6 @ ( set @ E4 )
@ ^ [X3: F6] : ( sup_sup @ ( set @ E4 ) @ A5 @ ( B4 @ X3 ) )
@ C4 ) )
= ( sup_sup @ ( set @ E4 ) @ A5 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image2 @ F6 @ ( set @ E4 ) @ B4 @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_3191_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ C,A5: C > ( set @ D ),B4: set @ D] :
( ( ( C4
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X3: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X3 ) @ B4 )
@ C4 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X3: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X3 ) @ B4 )
@ C4 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C4 ) ) @ B4 ) ) ) ) ).
% UN_simps(2)
thf(fact_3192_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ B,C3: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% Max_const
thf(fact_3193_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ B,C3: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% Min_const
thf(fact_3194_foldr__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldr @ A @ nat
@ ^ [X3: A] : suc
@ L
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldr_length
thf(fact_3195_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( concat @ A
@ ( map @ B @ ( list @ A )
@ ^ [X3: B] : ( cons @ A @ ( F @ X3 ) @ ( nil @ A ) )
@ Xs ) )
= ( map @ B @ A @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_3196_set__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( set2 @ A @ ( concat @ A @ Xs ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs ) ) ) ) ).
% set_concat
thf(fact_3197_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_3198_sorted__wrt__true,axiom,
! [A: $tType,Xs: list @ A] :
( sorted_wrt @ A
@ ^ [Uu3: A,Uv: A] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_3199_sorted__wrt__rev,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( sorted_wrt @ A @ P @ ( rev @ A @ Xs ) )
= ( sorted_wrt @ A
@ ^ [X3: A,Y3: A] : ( P @ Y3 @ X3 )
@ Xs ) ) ).
% sorted_wrt_rev
thf(fact_3200_insort__insert__triv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs )
= Xs ) ) ) ).
% insort_insert_triv
thf(fact_3201_tl__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys )
@ ^ [Z3: A,Zs2: list @ A] : ( append @ A @ Zs2 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_3202_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R: A > A > $o,F: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ R @ ( map @ B @ A @ F @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X3: B,Y3: B] : ( R @ ( F @ X3 ) @ ( F @ Y3 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_3203_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A5: set @ B] :
( ( member2 @ A @ B2 @ ( image2 @ B @ A @ F @ A5 ) )
=> ~ ! [X2: B] :
( ( B2
= ( F @ X2 ) )
=> ~ ( member2 @ B @ X2 @ A5 ) ) ) ).
% imageE
thf(fact_3204_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A5: set @ C] :
( ( image2 @ B @ A @ F @ ( image2 @ C @ B @ G @ A5 ) )
= ( image2 @ C @ A
@ ^ [X3: C] : ( F @ ( G @ X3 ) )
@ A5 ) ) ).
% image_image
thf(fact_3205_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ ( image2 @ B @ A @ F @ A5 ) )
& ( P @ X3 ) ) )
= ( image2 @ B @ A @ F
@ ( collect @ B
@ ^ [X3: B] :
( ( member2 @ B @ X3 @ A5 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_3206_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ~ ! [F5: A > nat] :
( ? [N7: nat] :
( ( image2 @ A @ nat @ F5 @ A5 )
= ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N7 ) ) )
=> ~ ( inj_on @ A @ nat @ F5 @ A5 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_3207_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X3: A] : ( member2 @ A @ X3 @ A7 )
@ ^ [X3: A] : ( member2 @ A @ X3 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_3208_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_3209_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ^ [X3: A] :
~ ( member2 @ A @ X3 @ A7 ) ) ) ) ).
% Compl_eq
thf(fact_3210_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_3211_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X3: A] : ( member2 @ A @ X3 @ A7 ) ) ) ) ) ).
% uminus_set_def
thf(fact_3212_list_Omap__ident,axiom,
! [A: $tType,T4: list @ A] :
( ( map @ A @ A
@ ^ [X3: A] : X3
@ T4 )
= T4 ) ).
% list.map_ident
thf(fact_3213_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G: A > B > A,A3: A,F: C > B,Xs: list @ C] :
( ( foldl @ A @ B @ G @ A3 @ ( map @ C @ B @ F @ Xs ) )
= ( foldl @ A @ C
@ ^ [A4: A,X3: C] : ( G @ A4 @ ( F @ X3 ) )
@ A3
@ Xs ) ) ).
% foldl_map
thf(fact_3214_Pow__def,axiom,
! [A: $tType] :
( ( pow @ A )
= ( ^ [A7: set @ A] :
( collect @ ( set @ A )
@ ^ [B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ A7 ) ) ) ) ).
% Pow_def
thf(fact_3215_Collect__subset,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ A5 )
& ( P @ X3 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_3216_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member2 @ A @ X3 @ A7 )
@ ^ [X3: A] : ( member2 @ A @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_3217_insert__compr,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A4: A,B5: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( X3 = A4 )
| ( member2 @ A @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_3218_insert__Collect,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( insert2 @ A @ A3 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U3: A] :
( ( U3 != A3 )
=> ( P @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_3219_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $false ) ) ).
% Set.empty_def
thf(fact_3220_insert__def,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A4: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] : X3 = A4 ) ) ) ) ).
% insert_def
thf(fact_3221_inj__singleton,axiom,
! [A: $tType,A5: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X3: A] : ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) ).
% inj_singleton
thf(fact_3222_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( A3 = X3 )
& ( P @ X3 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( A3 = X3 )
& ( P @ X3 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_3223_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( X3 = A3 )
& ( P @ X3 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X3: A] :
( ( X3 = A3 )
& ( P @ X3 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_3224_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: B > C,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( H @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( case_list @ C @ A @ ( H @ F1 )
@ ^ [X13: A,X25: list @ A] : ( H @ ( F22 @ X13 @ X25 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_3225_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ A7 )
| ( member2 @ A @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_3226_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_3227_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X3: A] : ( member2 @ A @ X3 @ A7 )
@ ^ [X3: A] : ( member2 @ A @ X3 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_3228_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_3229_inf__Int__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X3: A] : ( member2 @ A @ X3 @ R )
@ ^ [X3: A] : ( member2 @ A @ X3 @ S ) )
= ( ^ [X3: A] : ( member2 @ A @ X3 @ ( inf_inf @ ( set @ A ) @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_3230_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X3: A] : ( member2 @ A @ X3 @ A7 )
@ ^ [X3: A] : ( member2 @ A @ X3 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_3231_Int__Collect,axiom,
! [A: $tType,X: A,A5: set @ A,P: A > $o] :
( ( member2 @ A @ X @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) )
= ( ( member2 @ A @ X @ A5 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_3232_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ A7 )
& ( member2 @ A @ X3 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_3233_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X3: A,Y3: B] : ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
@ ^ [X3: A,Y3: B] : ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: A,Y3: B] : ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_3234_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C3: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X3: A] : ( product_Pair @ B @ A @ ( C3 @ X3 ) @ X3 )
@ S ) ).
% inj_Pair(2)
thf(fact_3235_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C3: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( C3 @ X3 ) )
@ S ) ).
% inj_Pair(1)
thf(fact_3236_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X3: A] : X3 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_3237_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H2: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_3238_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ A7 )
& ~ ( member2 @ A @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_3239_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X3: A] : ( member2 @ A @ X3 @ A7 )
@ ^ [X3: A] : ( member2 @ A @ X3 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_3240_list_Odisc__eq__case_I1_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
= ( nil @ A ) )
= ( case_list @ $o @ A @ $true
@ ^ [Uu3: A,Uv: list @ A] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_3241_list_Odisc__eq__case_I2_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
= ( case_list @ $o @ A @ $false
@ ^ [Uu3: A,Uv: list @ A] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_3242_tl__def,axiom,
! [A: $tType] :
( ( tl @ A )
= ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [X213: A,X223: list @ A] : X223 ) ) ).
% tl_def
thf(fact_3243_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H: B > A,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( G @ X3 ) @ ( H @ X3 ) )
@ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ H @ A5 ) ) ) ) ).
% prod.distrib
thf(fact_3244_Shift__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Shift @ A )
= ( ^ [Kl4: set @ ( list @ A ),K2: A] :
( collect @ ( list @ A )
@ ^ [Kl3: list @ A] : ( member2 @ ( list @ A ) @ ( cons @ A @ K2 @ Kl3 ) @ Kl4 ) ) ) ) ).
% Shift_def
thf(fact_3245_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member2 @ B @ I2 @ I5 )
& ( ( X @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member2 @ B @ I2 @ I5 )
& ( ( Y @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member2 @ B @ I2 @ I5 )
& ( ( times_times @ A @ ( X @ I2 ) @ ( Y @ I2 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3246_image__constant__conv,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X3: B] : C3
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X3: B] : C3
@ A5 )
= ( insert2 @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_3247_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A5: set @ A,C3: B] :
( ( member2 @ A @ X @ A5 )
=> ( ( image2 @ A @ B
@ ^ [X3: A] : C3
@ A5 )
= ( insert2 @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_3248_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S4: set @ A,T4: set @ A] :
( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( inf_inf @ ( set @ A ) @ S4 @ T4 ) )
= ( inf_inf @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ S4 )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ T4 ) ) ) ) ).
% translation_subtract_Int
thf(fact_3249_SUP__inf__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,A5: set @ B,G: C > A,B4: set @ C] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A4: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [B3: C] : ( inf_inf @ A @ ( F @ A4 ) @ ( G @ B3 ) )
@ B4 ) )
@ A5 ) ) ) ) ).
% SUP_inf_distrib2
thf(fact_3250_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F: B > A,B4: set @ B] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ B4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B3: B] : ( inf_inf @ A @ A3 @ ( F @ B3 ) )
@ B4 ) ) ) ) ).
% inf_SUP
thf(fact_3251_Sup__inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B4: set @ A,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B4 ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [B3: A] : ( inf_inf @ A @ B3 @ A3 )
@ B4 ) ) ) ) ).
% Sup_inf
thf(fact_3252_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,B4: set @ B,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ B4 ) ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B3: B] : ( inf_inf @ A @ ( F @ B3 ) @ A3 )
@ B4 ) ) ) ) ).
% SUP_inf
thf(fact_3253_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X3: B] :
( ( member2 @ B @ X3 @ A5 )
& ( P @ X3 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( G @ X3 ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_filter
thf(fact_3254_foldl__length__aux,axiom,
! [A: $tType,A3: nat,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I2: nat,X3: A] : ( suc @ I2 )
@ A3
@ L )
= ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldl_length_aux
thf(fact_3255_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ A5 )
=> ( ( B4 @ X3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_3256_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ A5 )
=> ( ( B4 @ X3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_3257_UN__empty,axiom,
! [B: $tType,A: $tType,B4: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_3258_UN__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( bot_bot @ ( set @ A ) )
@ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_3259_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X3: B,Y3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_3260_UN__extend__simps_I5_J,axiom,
! [I7: $tType,J5: $tType,A5: set @ I7,B4: J5 > ( set @ I7 ),C4: set @ J5] :
( ( inf_inf @ ( set @ I7 ) @ A5 @ ( complete_Sup_Sup @ ( set @ I7 ) @ ( image2 @ J5 @ ( set @ I7 ) @ B4 @ C4 ) ) )
= ( complete_Sup_Sup @ ( set @ I7 )
@ ( image2 @ J5 @ ( set @ I7 )
@ ^ [X3: J5] : ( inf_inf @ ( set @ I7 ) @ A5 @ ( B4 @ X3 ) )
@ C4 ) ) ) ).
% UN_extend_simps(5)
thf(fact_3261_UN__extend__simps_I4_J,axiom,
! [H3: $tType,G4: $tType,A5: G4 > ( set @ H3 ),C4: set @ G4,B4: set @ H3] :
( ( inf_inf @ ( set @ H3 ) @ ( complete_Sup_Sup @ ( set @ H3 ) @ ( image2 @ G4 @ ( set @ H3 ) @ A5 @ C4 ) ) @ B4 )
= ( complete_Sup_Sup @ ( set @ H3 )
@ ( image2 @ G4 @ ( set @ H3 )
@ ^ [X3: G4] : ( inf_inf @ ( set @ H3 ) @ ( A5 @ X3 ) @ B4 )
@ C4 ) ) ) ).
% UN_extend_simps(4)
thf(fact_3262_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: B > ( set @ A ),I5: set @ B] :
( ( inf_inf @ ( set @ A ) @ B4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] : ( inf_inf @ ( set @ A ) @ B4 @ ( A5 @ I2 ) )
@ I5 ) ) ) ).
% Int_UN_distrib
thf(fact_3263_Int__UN__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B,B4: C > ( set @ A ),J4: set @ C] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B4 @ J4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J2: C] : ( inf_inf @ ( set @ A ) @ ( A5 @ I2 ) @ ( B4 @ J2 ) )
@ J4 ) )
@ I5 ) ) ) ).
% Int_UN_distrib2
thf(fact_3264_Int__Union,axiom,
! [A: $tType,A5: set @ A,B4: set @ ( set @ A )] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 ) @ B4 ) ) ) ).
% Int_Union
thf(fact_3265_Int__Union2,axiom,
! [A: $tType,B4: set @ ( set @ A ),A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) @ A5 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [C5: set @ A] : ( inf_inf @ ( set @ A ) @ C5 @ A5 )
@ B4 ) ) ) ).
% Int_Union2
thf(fact_3266_zip__same__conv__map,axiom,
! [A: $tType,Xs: list @ A] :
( ( zip @ A @ A @ Xs @ Xs )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ Xs ) ) ).
% zip_same_conv_map
thf(fact_3267_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S: set @ A,S5: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [S6: set @ A] :
( ( minus_minus @ ( set @ A ) @ S6 @ S )
= ( minus_minus @ ( set @ A ) @ S5 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_3268_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F: A > B,X: A,Y: A,Zs: list @ A] :
( ( arg_min_list @ A @ B @ F @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F @ X ) @ ( F @ ( arg_min_list @ A @ B @ F @ ( cons @ A @ Y @ Zs ) ) ) ) @ X @ ( arg_min_list @ A @ B @ F @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_3269_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F3: B > A > A,Xs3: list @ B,A4: A] :
( foldl @ A @ B
@ ^ [X3: A,Y3: B] : ( F3 @ Y3 @ X3 )
@ A4
@ ( rev @ B @ Xs3 ) ) ) ) ).
% foldr_conv_foldl
thf(fact_3270_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F3: A > B > A,A4: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X3: B,Y3: A] : ( F3 @ Y3 @ X3 )
@ ( rev @ B @ Xs3 )
@ A4 ) ) ) ).
% foldl_conv_foldr
thf(fact_3271_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_3272_set__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( set2 @ A
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs ) )
= ( insert2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ).
% set_insort_insert
thf(fact_3273_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,X: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs ) ) ) ) ).
% sorted_insort_insert
thf(fact_3274_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 ) )
@ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys3 )
@ Xs3 ) ) ) ) ).
% product_concat_map
thf(fact_3275_map__add__upt_H,axiom,
! [Ofs: nat,A3: nat,B2: nat] :
( ( map @ nat @ nat
@ ^ [I2: nat] : ( plus_plus @ nat @ I2 @ Ofs )
@ ( upt @ A3 @ B2 ) )
= ( upt @ ( plus_plus @ nat @ A3 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_3276_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_3277_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_3278_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A5 ) )
= ( bot_bot @ A ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% SUP_constant
thf(fact_3279_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_3280_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_3281_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A,B4: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( if @ A @ ( member2 @ B @ X3 @ B4 ) @ ( G @ X3 ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_restrict
thf(fact_3282_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [X3: B] :
( ( G @ X3 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_3283_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ B,A3: A,B4: B > ( set @ A )] :
( ( ( C4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C4 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( insert2 @ A @ A3 @ ( B4 @ X3 ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_3284_UN__extend__simps_I3_J,axiom,
! [E4: $tType,F6: $tType,C4: set @ F6,A5: set @ E4,B4: F6 > ( set @ E4 )] :
( ( ( C4
= ( bot_bot @ ( set @ F6 ) ) )
=> ( ( sup_sup @ ( set @ E4 ) @ A5 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image2 @ F6 @ ( set @ E4 ) @ B4 @ C4 ) ) )
= A5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ F6 ) ) )
=> ( ( sup_sup @ ( set @ E4 ) @ A5 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image2 @ F6 @ ( set @ E4 ) @ B4 @ C4 ) ) )
= ( complete_Sup_Sup @ ( set @ E4 )
@ ( image2 @ F6 @ ( set @ E4 )
@ ^ [X3: F6] : ( sup_sup @ ( set @ E4 ) @ A5 @ ( B4 @ X3 ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3285_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C4: set @ C,A5: C > ( set @ D ),B4: set @ D] :
( ( ( C4
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C4 ) ) @ B4 )
= B4 ) )
& ( ( C4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C4 ) ) @ B4 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X3: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X3 ) @ B4 )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3286_length__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
@ Xs
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_3287_foldr__length__aux,axiom,
! [A: $tType,L: list @ A,A3: nat] :
( ( foldr @ A @ nat
@ ^ [X3: A] : suc
@ L
@ A3 )
= ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldr_length_aux
thf(fact_3288_Succ__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Succ @ A )
= ( ^ [Kl4: set @ ( list @ A ),Kl3: list @ A] :
( collect @ A
@ ^ [K2: A] : ( member2 @ ( list @ A ) @ ( append @ A @ Kl3 @ ( cons @ A @ K2 @ ( nil @ A ) ) ) @ Kl4 ) ) ) ) ).
% Succ_def
thf(fact_3289_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs: list @ B,F: B > ( list @ A )] :
( ( set2 @ A @ ( bind @ B @ A @ Xs @ F ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( set2 @ A @ ( F @ X3 ) )
@ ( set2 @ B @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_3290_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X3: A,Y3: B] : ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) )
= ( ^ [X3: list @ A,Y3: list @ B] : ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X3 @ Y3 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_3291_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( n_lists @ A @ ( suc @ N ) @ Xs )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys3: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y3: A] : ( cons @ A @ Y3 @ Ys3 )
@ Xs )
@ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_3292_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F: nat > A,M2: nat] :
( ( enumerate @ A @ N @ ( map @ nat @ A @ F @ ( upt @ N @ M2 ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K2: nat] : ( product_Pair @ nat @ A @ K2 @ ( F @ K2 ) )
@ ( upt @ N @ M2 ) ) ) ).
% enumerate_map_upt
thf(fact_3293_map__add__upt,axiom,
! [N: nat,M2: nat] :
( ( map @ nat @ nat
@ ^ [I2: nat] : ( plus_plus @ nat @ I2 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M2 ) )
= ( upt @ N @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% map_add_upt
thf(fact_3294_min__list_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A] :
( ( min_list @ A @ ( cons @ A @ X @ Xs ) )
= ( case_list @ A @ A @ X
@ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X @ ( min_list @ A @ Xs ) )
@ Xs ) ) ) ).
% min_list.simps
thf(fact_3295_product__lists_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,Xss2: list @ ( list @ A )] :
( ( product_lists @ A @ ( cons @ ( list @ A ) @ Xs @ Xss2 ) )
= ( concat @ ( list @ A )
@ ( map @ A @ ( list @ ( list @ A ) )
@ ^ [X3: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( product_lists @ A @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_3296_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( insert2 @ A @ ( F @ X3 ) @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= ( image2 @ B @ A @ F @ A5 ) ) ).
% UNION_singleton_eq_range
thf(fact_3297_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,G: A > B,B4: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( inj_on @ A @ B @ G @ B4 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ G @ B4 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member2 @ A @ X3 @ A5 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_3298_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F @ S ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_3299_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F @ S ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_3300_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ M2 )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_3301_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,P: B > $o,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( H @ X3 ) @ ( G @ X3 ) )
@ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_3302_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_3303_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3304_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( subseqs @ A @ Xs ) ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_3305_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X21: A,X22: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X21 @ X22 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X22 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_3306_map__upt__Suc,axiom,
! [A: $tType,F: nat > A,N: nat] :
( ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( cons @ A @ ( F @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I2: nat] : ( F @ ( suc @ I2 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_3307_finite__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3308_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A4: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X3: B,B3: A] : ( plus_plus @ A @ ( F3 @ X3 ) @ ( times_times @ A @ A4 @ B3 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_3309_map__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( map @ nat @ A @ ( nth @ A @ Xs ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_3310_slice__prepend,axiom,
! [A: $tType,I: nat,K: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ I @ K )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( slice @ A @ I @ K @ Xs )
= ( slice @ A @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( plus_plus @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( append @ A @ Ys @ Xs ) ) ) ) ) ).
% slice_prepend
thf(fact_3311_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_3312_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_3313_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,Zs2: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z3 @ Y ) @ ( zip @ A @ B @ Zs2 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_3314_set__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) )
= ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys3 )
= N )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_3315_map__decr__upt,axiom,
! [M2: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( upt @ M2 @ N ) ) ).
% map_decr_upt
thf(fact_3316_atLeastAtMost__upto,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I2: int,J2: int] : ( set2 @ int @ ( upto @ I2 @ J2 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_3317_map__by__foldl,axiom,
! [B: $tType,A: $tType,F: A > B,L: list @ A] :
( ( foldl @ ( list @ B ) @ A
@ ^ [L4: list @ B,X3: A] : ( append @ B @ L4 @ ( cons @ B @ ( F @ X3 ) @ ( nil @ B ) ) )
@ ( nil @ B )
@ L )
= ( map @ A @ B @ F @ L ) ) ).
% map_by_foldl
thf(fact_3318_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member2 @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B2 @ K2 ) @ ( C3 @ K2 ) )
@ S )
= ( times_times @ A @ ( B2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member2 @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B2 @ K2 ) @ ( C3 @ K2 ) )
@ S )
= ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3319_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,A5: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ I5 )
=> ( finite_finite2 @ C @ ( A5 @ X2 ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ I5 )
=> ! [Xa3: B] :
( ( member2 @ B @ Xa3 @ I5 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X2 ) @ ( A5 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A5 @ X3 ) )
@ I5 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_3320_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_3321_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_3322_is__unit__division__segment,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( one_one @ A ) ) ) ).
% is_unit_division_segment
thf(fact_3323_finite__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3324_foldl__set,axiom,
! [A: $tType,L: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ L )
= ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [X3: set @ A] : ( member2 @ ( set @ A ) @ X3 @ ( set2 @ ( set @ A ) @ L ) ) ) ) ) ).
% foldl_set
thf(fact_3325_Pow__set_I2_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( pow @ B @ ( set2 @ B @ ( cons @ B @ X @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow @ B @ ( set2 @ B @ Xs ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert2 @ B @ X ) @ ( pow @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_3326_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C3 ) @ ( times_times @ A @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C3 ) @ ( times_times @ A @ X @ C3 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_3327_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_3328_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_3329_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_3330_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_3331_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A,C3: A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member2 @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B2 @ K2 ) @ C3 )
@ S )
= ( times_times @ A @ ( B2 @ A3 ) @ ( power_power @ A @ C3 @ ( minus_minus @ nat @ ( finite_card @ B @ S ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member2 @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B2 @ K2 ) @ C3 )
@ S )
= ( power_power @ A @ C3 @ ( finite_card @ B @ S ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_3332_mergesort__remdups__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( mergesort_remdups @ A )
= ( ^ [Xs3: list @ A] :
( merge_list @ A @ ( nil @ ( list @ A ) )
@ ( map @ A @ ( list @ A )
@ ^ [X3: A] : ( cons @ A @ X3 @ ( nil @ A ) )
@ Xs3 ) ) ) ) ) ).
% mergesort_remdups_def
thf(fact_3333_card__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A5 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_3334_transpose_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss2: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss2 ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H2: A,T2: list @ A] : ( cons @ A @ H2 @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_3335_transpose_Oelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ( Y
!= ( transpose @ A @ Xss ) ) )
=> ~ ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) )
=> ( Y
!= ( cons @ ( list @ A )
@ ( cons @ A @ X2
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H2: A,T2: list @ A] : ( cons @ A @ H2 @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_3336_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_3337_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_3338_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A4: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A4 ) @ N3 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A4 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A4 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_3339_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_3340_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_3341_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: nat,K: nat,G: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ ( one_one @ A ) @ ( H @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_3342_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I3 ) )
= N ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I2: nat] :
( map @ nat @ A
@ ^ [J2: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J2 ) @ I2 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_3343_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A5: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A5 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_3344_card__lists__distinct__length__eq,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A5 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_3345_set__Cons__sing__Nil,axiom,
! [A: $tType,A5: set @ A] :
( ( set_Cons @ A @ A5 @ ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image2 @ A @ ( list @ A )
@ ^ [X3: A] : ( cons @ A @ X3 @ ( nil @ A ) )
@ A5 ) ) ).
% set_Cons_sing_Nil
thf(fact_3346_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A4: A,K2: nat] :
( if @ A
@ ( K2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L4: nat] : ( times_times @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ L4 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_3347_product__code,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_3348_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_3349_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A4: A,N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_3350_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_3351_inf1I,axiom,
! [A: $tType,A5: A > $o,X: A,B4: A > $o] :
( ( A5 @ X )
=> ( ( B4 @ X )
=> ( inf_inf @ ( A > $o ) @ A5 @ B4 @ X ) ) ) ).
% inf1I
thf(fact_3352_inf2I,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,X: A,Y: B,B4: A > B > $o] :
( ( A5 @ X @ Y )
=> ( ( B4 @ X @ Y )
=> ( inf_inf @ ( A > B > $o ) @ A5 @ B4 @ X @ Y ) ) ) ).
% inf2I
thf(fact_3353_filter__filter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs: list @ A] :
( ( filter2 @ A @ P @ ( filter2 @ A @ Q @ Xs ) )
= ( filter2 @ A
@ ^ [X3: A] :
( ( Q @ X3 )
& ( P @ X3 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_3354_sup1CI,axiom,
! [A: $tType,B4: A > $o,X: A,A5: A > $o] :
( ( ~ ( B4 @ X )
=> ( A5 @ X ) )
=> ( sup_sup @ ( A > $o ) @ A5 @ B4 @ X ) ) ).
% sup1CI
thf(fact_3355_filter__True,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( filter2 @ A @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_3356_filter__append,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( filter2 @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( filter2 @ A @ P @ Xs ) @ ( filter2 @ A @ P @ Ys ) ) ) ).
% filter_append
thf(fact_3357_remove1__filter__not,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ~ ( P @ X )
=> ( ( remove1 @ A @ X @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% remove1_filter_not
thf(fact_3358_removeAll__filter__not,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ~ ( P @ X )
=> ( ( removeAll @ A @ X @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% removeAll_filter_not
thf(fact_3359_set__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P @ Xs ) )
= ( collect @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% set_filter
thf(fact_3360_filter__False,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X2 ) )
=> ( ( filter2 @ A @ P @ Xs )
= ( nil @ A ) ) ) ).
% filter_False
thf(fact_3361_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_3362_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_3363_sup1I2,axiom,
! [A: $tType,B4: A > $o,X: A,A5: A > $o] :
( ( B4 @ X )
=> ( sup_sup @ ( A > $o ) @ A5 @ B4 @ X ) ) ).
% sup1I2
thf(fact_3364_sup1I1,axiom,
! [A: $tType,A5: A > $o,X: A,B4: A > $o] :
( ( A5 @ X )
=> ( sup_sup @ ( A > $o ) @ A5 @ B4 @ X ) ) ).
% sup1I1
thf(fact_3365_sup1E,axiom,
! [A: $tType,A5: A > $o,B4: A > $o,X: A] :
( ( sup_sup @ ( A > $o ) @ A5 @ B4 @ X )
=> ( ~ ( A5 @ X )
=> ( B4 @ X ) ) ) ).
% sup1E
thf(fact_3366_inf2D2,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,B4: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A5 @ B4 @ X @ Y )
=> ( B4 @ X @ Y ) ) ).
% inf2D2
thf(fact_3367_inf2D1,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,B4: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A5 @ B4 @ X @ Y )
=> ( A5 @ X @ Y ) ) ).
% inf2D1
thf(fact_3368_inf1D2,axiom,
! [A: $tType,A5: A > $o,B4: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A5 @ B4 @ X )
=> ( B4 @ X ) ) ).
% inf1D2
thf(fact_3369_inf1D1,axiom,
! [A: $tType,A5: A > $o,B4: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A5 @ B4 @ X )
=> ( A5 @ X ) ) ).
% inf1D1
thf(fact_3370_inf2E,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,B4: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A5 @ B4 @ X @ Y )
=> ~ ( ( A5 @ X @ Y )
=> ~ ( B4 @ X @ Y ) ) ) ).
% inf2E
thf(fact_3371_inf1E,axiom,
! [A: $tType,A5: A > $o,B4: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A5 @ B4 @ X )
=> ~ ( ( A5 @ X )
=> ~ ( B4 @ X ) ) ) ).
% inf1E
thf(fact_3372_filter_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( filter2 @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% filter.simps(1)
thf(fact_3373_rev__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( rev @ A @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ ( rev @ A @ Xs ) ) ) ).
% rev_filter
thf(fact_3374_filter_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( filter2 @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_3375_distinct__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( filter2 @ A @ P @ Xs ) ) ) ).
% distinct_filter
thf(fact_3376_filter__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( filter2 @ A @ P @ Xs )
= ( filter2 @ A @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_3377_filter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Xs )
= Xs )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% filter_id_conv
thf(fact_3378_sorted__wrt__filter,axiom,
! [A: $tType,F: A > A > $o,Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ F @ Xs )
=> ( sorted_wrt @ A @ F @ ( filter2 @ A @ P @ Xs ) ) ) ).
% sorted_wrt_filter
thf(fact_3379_filter__remove1,axiom,
! [A: $tType,Q: A > $o,X: A,Xs: list @ A] :
( ( filter2 @ A @ Q @ ( remove1 @ A @ X @ Xs ) )
= ( remove1 @ A @ X @ ( filter2 @ A @ Q @ Xs ) ) ) ).
% filter_remove1
thf(fact_3380_partition__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( member2 @ ( list @ A ) @ Xs
@ ( shuffles @ A @ ( filter2 @ A @ P @ Xs )
@ ( filter2 @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_3381_removeAll__filter__not__eq,axiom,
! [A: $tType] :
( ( removeAll @ A )
= ( ^ [X3: A] :
( filter2 @ A
@ ^ [Y3: A] : X3 != Y3 ) ) ) ).
% removeAll_filter_not_eq
thf(fact_3382_empty__filter__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( nil @ A )
= ( filter2 @ A @ P @ Xs ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% empty_filter_conv
thf(fact_3383_filter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Xs )
= ( nil @ A ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% filter_empty_conv
thf(fact_3384_filter__is__subset,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( filter2 @ A @ P @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% filter_is_subset
thf(fact_3385_length__filter__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_filter_le
thf(fact_3386_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_3387_distinct__map__filter,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,P: B > $o] :
( ( distinct @ A @ ( map @ B @ A @ F @ Xs ) )
=> ( distinct @ A @ ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_3388_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_3389_sum__length__filter__compl,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ Xs ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_3390_filter__concat,axiom,
! [A: $tType,P4: A > $o,Xs: list @ ( list @ A )] :
( ( filter2 @ A @ P4 @ ( concat @ A @ Xs ) )
= ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P4 ) @ Xs ) ) ) ).
% filter_concat
thf(fact_3391_inter__set__filter,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( set2 @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ A5 )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_3392_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X3: A] :
( X3
= ( G @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_3393_filter__shuffles,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( shuffles @ A @ ( filter2 @ A @ P @ Xs ) @ ( filter2 @ A @ P @ Ys ) ) ) ).
% filter_shuffles
thf(fact_3394_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_3395_union__coset__filter,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( coset @ A @ Xs ) @ A5 )
= ( coset @ A
@ ( filter2 @ A
@ ^ [X3: A] :
~ ( member2 @ A @ X3 @ A5 )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_3396_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_3397_filter__eq__Cons__iff,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X: A,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X @ Xs ) )
= ( ? [Us: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us @ ( cons @ A @ X @ Vs2 ) ) )
& ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Us ) )
=> ~ ( P @ X3 ) )
& ( P @ X )
& ( Xs
= ( filter2 @ A @ P @ Vs2 ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_3398_Cons__eq__filter__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( filter2 @ A @ P @ Ys ) )
= ( ? [Us: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us @ ( cons @ A @ X @ Vs2 ) ) )
& ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Us ) )
=> ~ ( P @ X3 ) )
& ( P @ X )
& ( Xs
= ( filter2 @ A @ P @ Vs2 ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_3399_filter__eq__ConsD,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X: A,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X @ Xs ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs3 ) ) )
& ! [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X6 ) )
& ( P @ X )
& ( Xs
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_3400_Cons__eq__filterD,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( filter2 @ A @ P @ Ys ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs3 ) ) )
& ! [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X6 ) )
& ( P @ X )
& ( Xs
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_3401_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,P: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ).
% sorted_filter
thf(fact_3402_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F: A > B,Y: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F @ ( insert2 @ A @ Y @ ( set2 @ A @ Xs ) ) )
=> ( ( filter2 @ A
@ ^ [X3: A] :
( ( F @ Y )
= ( F @ X3 ) )
@ Xs )
= ( filter2 @ A
@ ( ^ [Y5: A,Z5: A] : Y5 = Z5
@ Y )
@ Xs ) ) ) ).
% inj_on_filter_key_eq
thf(fact_3403_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,G: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F
@ ( filter2 @ B
@ ^ [X3: B] :
( ( F @ X3 )
= ( G @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_3404_length__filter__conv__card,axiom,
! [A: $tType,P4: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P4 @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P4 @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_3405_minus__coset__filter,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( coset @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ A5 )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_3406_filter__eq__snocD,axiom,
! [A: $tType,P: A > $o,L: list @ A,L3: list @ A,X: A] :
( ( ( filter2 @ A @ P @ L )
= ( append @ A @ L3 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ( ( member2 @ A @ X @ ( set2 @ A @ L ) )
& ( P @ X ) ) ) ).
% filter_eq_snocD
thf(fact_3407_filter__conv__foldr,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
( foldr @ A @ ( list @ A )
@ ^ [X3: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P3 @ X3 ) @ ( cons @ A @ X3 @ Xt ) @ Xt )
@ Xs3
@ ( nil @ A ) ) ) ) ).
% filter_conv_foldr
thf(fact_3408_set__minus__filter__out,axiom,
! [A: $tType,Xs: list @ A,Y: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X3: A] : X3 != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_3409_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ ( set2 @ A @ Ys ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_3410_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X3: A] :
~ ( member2 @ A @ X3 @ ( set2 @ A @ Ys ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_3411_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_3412_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member2 @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X3: A] :
~ ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_3413_filter__nth__ex__nth,axiom,
! [A: $tType,N: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) )
=> ? [M4: nat] :
( ( ord_less_eq @ nat @ N @ M4 )
& ( ord_less @ nat @ M4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ ( filter2 @ A @ P @ Xs ) @ N )
= ( nth @ A @ Xs @ M4 ) )
& ( ( filter2 @ A @ P @ ( take @ A @ M4 @ Xs ) )
= ( take @ A @ N @ ( filter2 @ A @ P @ Xs ) ) ) ) ) ).
% filter_nth_ex_nth
thf(fact_3414_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xss2: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs3: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) )
@ Xss2
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X3: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X3 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss2 )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_3415_distinct__length__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
= ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_length_filter
thf(fact_3416_transpose__max__length,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
@ ( transpose @ A @ Xs )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X3: list @ A] :
( X3
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_3417_transpose__aux__filter__tail,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) )
= ( map @ ( list @ A ) @ ( list @ A ) @ ( tl @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss2 ) ) ) ).
% transpose_aux_filter_tail
thf(fact_3418_transpose__aux__filter__head,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H2: A,T2: list @ A] : ( cons @ A @ H2 @ ( nil @ A ) ) )
@ Xss2 ) )
= ( map @ ( list @ A ) @ A @ ( hd @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss2 ) ) ) ).
% transpose_aux_filter_head
thf(fact_3419_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F: nat > A,A3: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A4: nat] : ( times_times @ A @ ( F @ A4 ) )
@ A3
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_3420_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs3: list @ A] : ( nth @ A @ Xs3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_3421_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] :
( ( member2 @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys2 ) )
=> ( ! [Ys2: list @ A,Zs3: list @ A] :
( ( member2 @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member2 @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys2 != Zs3 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys2 ) @ ( set2 @ A @ Zs3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat'
thf(fact_3422_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_3423_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ).
% transpose_column
thf(fact_3424_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_3425_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_3426_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_3427_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_3428_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M2: nat,X: A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_3429_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_3430_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y: A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : Y
@ A5 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ Y ) ) ) ).
% sum_constant
thf(fact_3431_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A5: set @ nat,C3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A5 )
& ( member2 @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C3 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
@ A5 )
= ( C3 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A5 )
& ( member2 @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C3 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
@ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_3432_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,F: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( F @ X3 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X3 ) ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_3433_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,P: B > $o,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X3 ) ) @ ( F @ X3 ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_3434_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A5: set @ nat,C3: nat > A,D3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A5 )
& ( member2 @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D3 @ I2 ) )
@ A5 )
= ( divide_divide @ A @ ( C3 @ ( zero_zero @ nat ) ) @ ( D3 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A5 )
& ( member2 @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D3 @ I2 ) )
@ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_3435_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,P: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( zero_neq_one_of_bool @ A @ ( P @ X3 ) )
@ A5 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_3436_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R3: A,F: B > A,A5: set @ B] :
( ( times_times @ A @ R3 @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N3: B] : ( times_times @ A @ R3 @ ( F @ N3 ) )
@ A5 ) ) ) ).
% sum_distrib_left
thf(fact_3437_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F: B > A,A5: set @ B,R3: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ R3 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N3: B] : ( times_times @ A @ ( F @ N3 ) @ R3 )
@ A5 ) ) ) ).
% sum_distrib_right
thf(fact_3438_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F: A > B,A5: set @ A,G: C > B,B4: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B4 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J2: C] : ( times_times @ B @ ( F @ I2 ) @ ( G @ J2 ) )
@ B4 )
@ A5 ) ) ) ).
% sum_product
thf(fact_3439_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A5: set @ B,F: B > A,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ A5 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_3440_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A,B4: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( if @ A @ ( member2 @ B @ X3 @ B4 ) @ ( G @ X3 ) @ ( zero_zero @ A ) )
@ A5 ) ) ) ) ).
% sum.inter_restrict
thf(fact_3441_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,I5: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M2 @ I2 ) )
@ I5 )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I5 ) ) ) ) ).
% sum_power_add
thf(fact_3442_filter__upt__take__conv,axiom,
! [A: $tType,P: A > $o,M2: nat,L: list @ A,N: nat] :
( ( filter2 @ nat
@ ^ [I2: nat] : ( P @ ( nth @ A @ ( take @ A @ M2 @ L ) @ I2 ) )
@ ( upt @ N @ M2 ) )
= ( filter2 @ nat
@ ^ [I2: nat] : ( P @ ( nth @ A @ L @ I2 ) )
@ ( upt @ N @ M2 ) ) ) ).
% filter_upt_take_conv
thf(fact_3443_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ I5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ I5 ) ) ) ) ) ) ).
% sum_pos
thf(fact_3444_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,F: B > A,K5: A] :
( ! [I3: B] :
( ( member2 @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ K5 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above
thf(fact_3445_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,K5: A,F: B > A] :
( ! [I3: B] :
( ( member2 @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ K5 @ ( F @ I3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) ) ) ) ).
% sum_bounded_below
thf(fact_3446_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T: set @ B,S: set @ B,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ T )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T @ S ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S @ T ) )
=> ( ( G @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( inf_inf @ ( set @ B ) @ S @ T ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_3447_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B4 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_3448_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A,B4: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_3449_upt__filter__extend,axiom,
! [U: nat,U4: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ U @ U4 )
=> ( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ U @ I3 )
& ( ord_less @ nat @ I3 @ U4 ) )
=> ~ ( P @ I3 ) )
=> ( ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
= ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U4 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_3450_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,P: B > $o,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( H @ X3 ) @ ( G @ X3 ) )
@ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_3451_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
=> ( ( G @ X2 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B4 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_3452_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A5 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_3453_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member2 @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_3454_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,A3: B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member2 @ B @ A3 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( F @ A3 ) ) ) )
& ( ~ ( member2 @ B @ A3 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_3455_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,B4: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_3456_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( ( inf_inf @ ( set @ B ) @ A5 @ B4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B4 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_3457_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A5: set @ A,B4: set @ A,F: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_3458_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B4 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_3459_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B2: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member2 @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B2 @ K2 ) @ ( C3 @ K2 ) )
@ S )
= ( plus_plus @ A @ ( B2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member2 @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B2 @ K2 ) @ ( C3 @ K2 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_3460_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A5: set @ B,F: B > A,B2: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A4: B] : ( divide_divide @ A @ ( F @ A4 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A4: B] : ( dvd_dvd @ A @ B2 @ ( F @ A4 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A4: B] :
~ ( dvd_dvd @ A @ B2 @ ( F @ A4 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_3461_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,A5: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ I5 )
=> ( finite_finite2 @ C @ ( A5 @ X2 ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ I5 )
=> ! [Xa3: B] :
( ( member2 @ B @ Xa3 @ I5 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X2 ) @ ( A5 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A5 @ X3 ) )
@ I5 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_3462_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_3463_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_3464_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: nat] :
( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_3465_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K2 ) ) @ K2 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_3466_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A5: set @ C,F: C > B] :
( ( member2 @ C @ I @ A5 )
=> ( ! [X2: C] :
( ( member2 @ C @ X2 @ ( minus_minus @ ( set @ C ) @ A5 @ ( insert2 @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X2 ) ) )
=> ( ( finite_finite2 @ C @ A5 )
=> ( ord_less_eq @ B @ ( F @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F @ A5 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_3467_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A5: set @ B,F: B > A,K5: A] :
( ! [I3: B] :
( ( member2 @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_3468_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,F: B > A,K5: A] :
( ! [I3: B] :
( ( member2 @ B @ I3 @ A5 )
=> ( ord_less @ A @ ( F @ I3 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A5 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_3469_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
( ! [I3: A] :
( ( member2 @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I5 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member2 @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I3 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] : ( times_times @ B @ ( A3 @ I2 ) @ ( X @ I2 ) )
@ I5 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_3470_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S: set @ A,R: set @ B,G: A > B,F: B > C] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ R )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G @ S ) @ R )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X3: A] : ( F @ ( G @ X3 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y3: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ S )
& ( ( G @ X3 )
= Y3 ) ) ) ) )
@ ( F @ Y3 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_3471_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_3472_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) ) @ ( power_power @ A @ X @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_3473_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_3474_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_3475_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_3476_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_3477_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_3478_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_3479_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M2 ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_3480_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) @ ( power_power @ A @ A3 @ K2 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_3481_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_3482_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A4: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F3 @ ( nth @ B @ Xs3 @ N3 ) ) @ ( power_power @ A @ A4 @ N3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_3483_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J2 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_3484_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K2 ) @ ( power_power @ A @ X @ K2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_3485_filter__upt__last,axiom,
! [A: $tType,P: A > $o,L: list @ A,Js2: list @ nat,J: nat,I: nat] :
( ( ( filter2 @ nat
@ ^ [K2: nat] : ( P @ ( nth @ A @ L @ K2 ) )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( append @ nat @ Js2 @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
=> ( ( ord_less @ nat @ J @ I )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% filter_upt_last
thf(fact_3486_double__gauss__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum
thf(fact_3487_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D3: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) ) ) ).
% double_arith_series
thf(fact_3488_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_3489_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K2 ) @ ( power_power @ A @ X @ K2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A3 ) @ ( one_one @ A ) ) @ K2 ) @ ( power_power @ A @ X @ K2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_3490_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_3491_card__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A5 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_3492_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_3493_gauss__sum,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum
thf(fact_3494_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_3495_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_3496_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_3497_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_3498_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R3: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ R3 @ K2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K2 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R3 @ ( suc @ M2 ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_3499_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I2: A] : ( plus_plus @ A @ I2 @ ( one_one @ A ) )
@ N3
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_3500_remove__rev__alt__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X3: A,Xs3: list @ A] :
( filter2 @ A
@ ^ [Y3: A] : Y3 != X3
@ ( rev @ A @ Xs3 ) ) ) ) ).
% remove_rev_alt_def
thf(fact_3501_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L4: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R4: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L4 ) @ R4 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R4 @ ( numeral_numeral @ A @ L4 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R4 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_3502_Pow__fold,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( pow @ A @ A5 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X3: A,A7: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A7 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X3 ) @ A7 ) )
@ ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A5 ) ) ) ).
% Pow_fold
thf(fact_3503_length__remdups__concat,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss2 ) ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xss2 ) ) ) ) ) ).
% length_remdups_concat
thf(fact_3504_remdups__upt,axiom,
! [M2: nat,N: nat] :
( ( remdups @ nat @ ( upt @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% remdups_upt
thf(fact_3505_fold__empty,axiom,
! [B: $tType,A: $tType,F: B > A > A,Z2: A] :
( ( finite_fold @ B @ A @ F @ Z2 @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ).
% fold_empty
thf(fact_3506_remdups__eq__nil__iff,axiom,
! [A: $tType,X: list @ A] :
( ( ( remdups @ A @ X )
= ( nil @ A ) )
= ( X
= ( nil @ A ) ) ) ).
% remdups_eq_nil_iff
thf(fact_3507_remdups__eq__nil__right__iff,axiom,
! [A: $tType,X: list @ A] :
( ( ( nil @ A )
= ( remdups @ A @ X ) )
= ( X
= ( nil @ A ) ) ) ).
% remdups_eq_nil_right_iff
thf(fact_3508_set__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_remdups
thf(fact_3509_length__remdups__eq,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ( remdups @ A @ Xs )
= Xs ) ) ).
% length_remdups_eq
thf(fact_3510_distinct__remdups,axiom,
! [A: $tType,Xs: list @ A] : ( distinct @ A @ ( remdups @ A @ Xs ) ) ).
% distinct_remdups
thf(fact_3511_remdups__id__iff__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups @ A @ Xs )
= Xs )
= ( distinct @ A @ Xs ) ) ).
% remdups_id_iff_distinct
thf(fact_3512_length__remdups__leq,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_remdups_leq
thf(fact_3513_nested__case__prod__simp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F3: B > C > D > A,X3: product_prod @ B @ C,Y3: D] :
( product_case_prod @ B @ C @ A
@ ^ [A4: B,B3: C] : ( F3 @ A4 @ B3 @ Y3 )
@ X3 ) ) ) ).
% nested_case_prod_simp
thf(fact_3514_remdups__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups @ A @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ Xs ) ) ).
% remdups_remdups
thf(fact_3515_remdups_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups.simps(1)
thf(fact_3516_remdups__map__remdups,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( remdups @ A @ ( map @ B @ A @ F @ ( remdups @ B @ Xs ) ) )
= ( remdups @ A @ ( map @ B @ A @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_3517_remdups__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( remdups @ A @ ( append @ A @ Xs @ ( remdups @ A @ Ys ) ) )
= ( remdups @ A @ ( append @ A @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_3518_distinct__remdups__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups @ A @ Xs )
= Xs ) ) ).
% distinct_remdups_id
thf(fact_3519_remdups__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( remdups @ A @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ ( remdups @ A @ Xs ) ) ) ).
% remdups_filter
thf(fact_3520_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [Y3: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X3: A,Z3: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C @ Y3 @ Z3 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_3521_insert__remdups,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( insert @ A @ X @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ ( insert @ A @ X @ Xs ) ) ) ).
% insert_remdups
thf(fact_3522_remdups_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X @ Xs ) )
= ( remdups @ A @ Xs ) ) )
& ( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( remdups @ A @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_3523_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 )
@ ^ [X3: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X3 ) )
@ ( zip @ B @ A @ Ys3 @ Xs3 ) ) ) ) ).
% zip_commute
thf(fact_3524_sorted__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups @ A @ Xs ) ) ) ) ).
% sorted_remdups
thf(fact_3525_map2__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,H: B > C > A,F: D > B,Xs: list @ D,G: D > C] :
( ( map @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ H ) @ ( zip @ B @ C @ ( map @ D @ B @ F @ Xs ) @ ( map @ D @ C @ G @ Xs ) ) )
= ( map @ D @ A
@ ^ [X3: D] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_3526_remove1__remdups,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ ( remove1 @ A @ X @ Xs ) ) ) ) ).
% remove1_remdups
thf(fact_3527_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,Xs: list @ C,Ys: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ Ys )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X3: C] : ( product_Pair @ A @ B @ ( F @ X3 ) ) )
@ ( zip @ C @ B @ Xs @ Ys ) ) ) ).
% zip_map1
thf(fact_3528_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F: C > B,Ys: list @ C] :
( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F @ Ys ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X3: A,Y3: C] : ( product_Pair @ A @ B @ X3 @ ( F @ Y3 ) ) )
@ ( zip @ A @ C @ Xs @ Ys ) ) ) ).
% zip_map2
thf(fact_3529_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: 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 )
@ ^ [X3: C,Y3: D] : ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ Y3 ) ) )
@ ( zip @ C @ D @ Xs @ Ys ) )
= ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ ( map @ D @ B @ G @ Ys ) ) ) ).
% map_prod_fun_zip
thf(fact_3530_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_3531_length__remdups__card__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) )
= ( finite_card @ A @ ( set2 @ A @ Xs ) ) ) ).
% length_remdups_card_conv
thf(fact_3532_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: ( product_prod @ B @ C ) > A,G: D > B,Xs: list @ D,Ys: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs ) @ Ys ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X3: D,Y3: C] : ( F @ ( product_Pair @ B @ C @ ( G @ X3 ) @ Y3 ) ) )
@ ( zip @ D @ C @ Xs @ Ys ) ) ) ).
% map_zip_map
thf(fact_3533_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: ( product_prod @ B @ C ) > A,Xs: list @ B,G: D > C,Ys: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G @ Ys ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X3: B,Y3: D] : ( F @ ( product_Pair @ B @ C @ X3 @ ( G @ Y3 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys ) ) ) ).
% map_zip_map2
thf(fact_3534_sum__diff1__nat,axiom,
! [A: $tType,A3: A,A5: set @ A,F: A > nat] :
( ( ( member2 @ A @ A3 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 ) @ ( F @ A3 ) ) ) )
& ( ~ ( member2 @ A @ A3 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 ) ) ) ) ).
% sum_diff1_nat
thf(fact_3535_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( complete_Sup_Sup @ A @ A5 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A5 ) ) ) ) ).
% Sup_fold_sup
thf(fact_3536_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert2 @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A5 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_3537_image__fold__insert,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( image2 @ A @ B @ F @ A5 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K2: A] : ( insert2 @ B @ ( F @ K2 ) )
@ ( bot_bot @ ( set @ B ) )
@ A5 ) ) ) ).
% image_fold_insert
thf(fact_3538_sum__Un__nat,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ B4 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_3539_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys: list @ A,Xs: list @ B,F: C > A > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldl @ C @ ( product_prod @ B @ A )
@ ^ [B3: C] :
( product_case_prod @ B @ A @ C
@ ^ [X3: B] : ( F @ B3 ) )
@ B2
@ ( zip @ B @ A @ Xs @ Ys ) )
= ( foldl @ C @ A @ F @ B2 @ Ys ) ) ) ).
% foldl_snd_zip
thf(fact_3540_prod_Oset__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,Xs: list @ B] :
( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups5270119922927024881d_list @ A @ ( map @ B @ A @ G @ ( remdups @ B @ Xs ) ) ) ) ) ).
% prod.set_conv_list
thf(fact_3541_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X5 )
=> ( ( finite_finite2 @ A @ X5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X5 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_3542_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B )] :
( ( finite_finite2 @ A @ I5 )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ I5 )
=> ( finite_finite2 @ B @ ( A5 @ X2 ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ I5 )
=> ! [Xa3: A] :
( ( member2 @ A @ Xa3 @ I5 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ ( A5 @ X2 ) @ ( A5 @ Xa3 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I2: A] : ( finite_card @ B @ ( A5 @ I2 ) )
@ I5 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_3543_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R4: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R4 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_3544_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R4: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R4 ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_3545_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_3546_transpose__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs ) )
= ( takeWhile @ ( list @ A )
@ ^ [X3: list @ A] :
( X3
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_3547_Set__filter__fold,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( filter3 @ A @ P @ A5 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X3: A,A14: set @ A] : ( if @ ( set @ A ) @ ( P @ X3 ) @ ( insert2 @ A @ X3 @ A14 ) @ A14 )
@ ( bot_bot @ ( set @ A ) )
@ A5 ) ) ) ).
% Set_filter_fold
thf(fact_3548_takeWhile__idem,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( takeWhile @ A @ P @ ( takeWhile @ A @ P @ Xs ) )
= ( takeWhile @ A @ P @ Xs ) ) ).
% takeWhile_idem
thf(fact_3549_member__filter,axiom,
! [A: $tType,X: A,P: A > $o,A5: set @ A] :
( ( member2 @ A @ X @ ( filter3 @ A @ P @ A5 ) )
= ( ( member2 @ A @ X @ A5 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_3550_takeWhile__eq__all__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P @ Xs )
= Xs )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% takeWhile_eq_all_conv
thf(fact_3551_takeWhile__append1,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% takeWhile_append1
thf(fact_3552_takeWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_3553_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P3: A > $o,A7: set @ A] :
( collect @ A
@ ^ [A4: A] :
( ( member2 @ A @ A4 @ A7 )
& ( P3 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_3554_distinct__takeWhile,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( takeWhile @ A @ P @ Xs ) ) ) ).
% distinct_takeWhile
thf(fact_3555_takeWhile_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( takeWhile @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% takeWhile.simps(1)
thf(fact_3556_set__takeWhileD,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ ( takeWhile @ A @ P @ Xs ) ) )
=> ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) ) ).
% set_takeWhileD
thf(fact_3557_takeWhile__cong,axiom,
! [A: $tType,L: list @ A,K: list @ A,P: A > $o,Q: A > $o] :
( ( L = K )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ L ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( takeWhile @ A @ P @ L )
= ( takeWhile @ A @ Q @ K ) ) ) ) ).
% takeWhile_cong
thf(fact_3558_inj__split__Cons,axiom,
! [A: $tType,X5: set @ ( product_prod @ ( list @ A ) @ A )] :
( inj_on @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs3: list @ A,N3: A] : ( cons @ A @ N3 @ Xs3 ) )
@ X5 ) ).
% inj_split_Cons
thf(fact_3559_takeWhile_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( takeWhile @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( takeWhile @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( takeWhile @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) ) ) ).
% takeWhile.simps(2)
thf(fact_3560_takeWhile__tail,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A,L: list @ A] :
( ~ ( P @ X )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ ( cons @ A @ X @ L ) ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ).
% takeWhile_tail
thf(fact_3561_length__takeWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_takeWhile_le
thf(fact_3562_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% sorted_takeWhile
thf(fact_3563_takeWhile__eq__take,axiom,
! [A: $tType] :
( ( takeWhile @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] : ( take @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs3 ) ) @ Xs3 ) ) ) ).
% takeWhile_eq_take
thf(fact_3564_takeWhile__eq__Nil__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ~ ( P @ ( hd @ A @ Xs ) ) ) ) ).
% takeWhile_eq_Nil_iff
thf(fact_3565_nth__length__takeWhile,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_3566_takeWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P @ Xs ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% takeWhile_nth
thf(fact_3567_drop__takeWhile,axiom,
! [A: $tType,I: nat,P: A > $o,L: list @ A] :
( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
=> ( ( drop @ A @ I @ ( takeWhile @ A @ P @ L ) )
= ( takeWhile @ A @ P @ ( drop @ A @ I @ L ) ) ) ) ).
% drop_takeWhile
thf(fact_3568_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) )
@ ^ [X3: A,Y3: B,Z3: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C @ Y3 @ Z3 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_3569_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X3: A,Y3: B] : ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R4 ) ) ) ) ) ) ).
% listrel_def
thf(fact_3570_takeWhile__not__last,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( takeWhile @ A
@ ^ [Y3: A] :
( Y3
!= ( last @ A @ Xs ) )
@ Xs )
= ( butlast @ A @ Xs ) ) ) ).
% takeWhile_not_last
thf(fact_3571_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_3572_eq__len__takeWhile__conv,axiom,
! [A: $tType,I: nat,P: A > $o,L: list @ A] :
( ( I
= ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
= ( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
& ! [J2: nat] :
( ( ord_less @ nat @ J2 @ I )
=> ( P @ ( nth @ A @ L @ J2 ) ) )
& ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% eq_len_takeWhile_conv
thf(fact_3573_less__length__takeWhile__conv,axiom,
! [A: $tType,I: nat,P: A > $o,L: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
= ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
& ! [J2: nat] :
( ( ord_less_eq @ nat @ J2 @ I )
=> ( P @ ( nth @ A @ L @ J2 ) ) ) ) ) ).
% less_length_takeWhile_conv
thf(fact_3574_filter__set,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( filter3 @ A @ P @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( filter2 @ A @ P @ Xs ) ) ) ).
% filter_set
thf(fact_3575_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) ) )
=> ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ N ) ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( take @ A @ N @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_3576_listset_Osimps_I2_J,axiom,
! [A: $tType,A5: set @ A,As4: list @ ( set @ A )] :
( ( listset @ A @ ( cons @ ( set @ A ) @ A5 @ As4 ) )
= ( set_Cons @ A @ A5 @ ( listset @ A @ As4 ) ) ) ).
% listset.simps(2)
thf(fact_3577_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] : ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_3578_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys: list @ A,Xs: list @ B,F: A > C > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldr @ ( product_prod @ B @ A ) @ C
@ ( product_case_prod @ B @ A @ ( C > C )
@ ^ [X3: B] : F )
@ ( zip @ B @ A @ Xs @ Ys )
@ B2 )
= ( foldr @ A @ C @ F @ Ys @ B2 ) ) ) ).
% foldr_snd_zip
thf(fact_3579_inter__Set__filter,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( filter3 @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ A5 )
@ B4 ) ) ) ).
% inter_Set_filter
thf(fact_3580_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lex @ A @ R4 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_3581_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,T4: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X3: B] : ( ord_less @ A @ T4 @ ( F @ X3 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X3: B] : ( ord_less @ A @ T4 @ ( F @ X3 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_3582_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R4: A > A > $o,Xs3: list @ A,Ys3: list @ A] : ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_3583_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R4: A > A > $o,Xs3: list @ A,Ys3: list @ A] : ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_3584_transpose_Opelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ( ( Y
= ( transpose @ A @ Xss ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) )
=> ( ( Y
= ( cons @ ( list @ A )
@ ( cons @ A @ X2
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H2: A,T2: list @ A] : ( cons @ A @ H2 @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_3585_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss2 ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss2 ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H2: A,T2: list @ A] : ( cons @ A @ H2 @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_3586_quicksort_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( linorder_quicksort @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y3: A] :
~ ( ord_less_eq @ A @ X @ Y3 )
@ Xs ) )
@ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X ) @ Xs ) ) ) ) ) ) ).
% quicksort.simps(2)
thf(fact_3587_set__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( set2 @ A @ ( linorder_quicksort @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ) ).
% set_quicksort
thf(fact_3588_transpose_Opsimps_I1_J,axiom,
! [A: $tType] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ) ).
% transpose.psimps(1)
thf(fact_3589_quicksort_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_quicksort @ A @ ( nil @ A ) )
= ( nil @ A ) ) ) ).
% quicksort.simps(1)
thf(fact_3590_transpose_Opsimps_I2_J,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
= ( transpose @ A @ Xss2 ) ) ) ).
% transpose.psimps(2)
thf(fact_3591_sorted__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linorder_quicksort @ A @ Xs ) ) ) ).
% sorted_quicksort
thf(fact_3592_transpose_Opinduct,axiom,
! [A: $tType,A0: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
=> ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( P @ ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ( ( P @ Xss )
=> ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) )
=> ( ( P
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H2: A,T2: list @ A] : ( cons @ ( list @ A ) @ T2 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) )
=> ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_3593_quicksort_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Y: list @ A] :
( ( ( linorder_quicksort @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( Y
!= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y3: A] :
~ ( ord_less_eq @ A @ X2 @ Y3 )
@ Xs2 ) )
@ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X2 ) @ Xs2 ) ) ) ) ) ) ) ) ) ).
% quicksort.elims
thf(fact_3594_quicksort_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Y: list @ A] :
( ( ( linorder_quicksort @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( nil @ A ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y3: A] :
~ ( ord_less_eq @ A @ X2 @ Y3 )
@ Xs2 ) )
@ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X2 ) @ Xs2 ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ).
% quicksort.pelims
thf(fact_3595_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F2: set @ A,I5: set @ A,F: A > B,I: A] :
( ( finite_finite2 @ A @ F2 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I2: A] :
( ( member2 @ A @ I2 @ I5 )
& ( ( F @ I2 )
!= ( zero_zero @ B ) ) ) )
@ F2 )
=> ( ( ( member2 @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) @ ( F @ I ) ) ) )
& ( ~ ( member2 @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_3596_sort__key__by__quicksort__code,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F3: B > A,Xs3: list @ B] :
( case_list @ ( list @ B ) @ B @ ( nil @ B )
@ ^ [X3: B] :
( case_list @ ( list @ B ) @ B @ Xs3
@ ^ [Y3: B] :
( case_list @ ( list @ B ) @ B @ ( if @ ( list @ B ) @ ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) @ Xs3 @ ( cons @ B @ Y3 @ ( cons @ B @ X3 @ ( nil @ B ) ) ) )
@ ^ [Ab: B,List2: list @ B] :
( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( list @ B )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( list @ B )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( append @ B @ ( linorder_sort_key @ B @ A @ F3 @ Lts ) @ ( append @ B @ Eqs @ ( linorder_sort_key @ B @ A @ F3 @ Gts ) ) ) )
@ ( linorder_part @ B @ A @ F3 @ ( F3 @ ( nth @ B @ Xs3 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Xs3 ) ) ) )
@ Xs3 ) ) ) ) ).
% sort_key_by_quicksort_code
thf(fact_3597_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A3: B,Xs: list @ B,F: B > A] :
( ( member2 @ B @ A3 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X3: B] :
( ( F @ A3 )
= ( F @ X3 ) )
@ Xs ) )
= A3 )
=> ( ( linorder_insort_key @ B @ A @ F @ A3 @ ( remove1 @ B @ A3 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_3598_sort__upt,axiom,
! [M2: nat,N: nat] :
( ( linorder_sort_key @ nat @ nat
@ ^ [X3: nat] : X3
@ ( upt @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% sort_upt
thf(fact_3599_sort__upto,axiom,
! [I: int,J: int] :
( ( linorder_sort_key @ int @ int
@ ^ [X3: int] : X3
@ ( upto @ I @ J ) )
= ( upto @ I @ J ) ) ).
% sort_upto
thf(fact_3600_sort__key__simps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A] :
( ( linorder_sort_key @ B @ A @ F @ ( nil @ B ) )
= ( nil @ B ) ) ) ).
% sort_key_simps(1)
thf(fact_3601_set__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( set2 @ B @ ( linorder_sort_key @ B @ A @ F @ Xs ) )
= ( set2 @ B @ Xs ) ) ) ).
% set_sort
thf(fact_3602_length__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_sort_key @ B @ A @ F @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ) ).
% length_sort
thf(fact_3603_distinct__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( distinct @ B @ ( linorder_sort_key @ B @ A @ F @ Xs ) )
= ( distinct @ B @ Xs ) ) ) ).
% distinct_sort
thf(fact_3604_remove1__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [X: B,F: B > A,Xs: list @ B] :
( ( remove1 @ B @ X @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
= Xs ) ) ).
% remove1_insort_key
thf(fact_3605_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_3606_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% length_insort
thf(fact_3607_sort__key__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Xs: list @ B] :
( ( linorder_sort_key @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F @ X @ ( linorder_sort_key @ B @ A @ F @ Xs ) ) ) ) ).
% sort_key_simps(2)
thf(fact_3608_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member2 @ A @ X @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_3609_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_3610_filter__insort__triv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,X: B,F: B > A,Xs: list @ B] :
( ~ ( P @ X )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
= ( filter2 @ B @ P @ Xs ) ) ) ) ).
% filter_insort_triv
thf(fact_3611_insort__key__left__comm,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Y: B,Xs: list @ B] :
( ( ( F @ X )
!= ( F @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F @ Y @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F @ X @ ( linorder_insort_key @ B @ A @ F @ Y @ Xs ) ) ) ) ) ).
% insort_key_left_comm
thf(fact_3612_insort__left__comm,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Xs: list @ A] :
( ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ Y
@ Xs ) )
= ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ Y
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs ) ) ) ) ).
% insort_left_comm
thf(fact_3613_sort__key__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [C3: B,Xs: list @ A] :
( ( linorder_sort_key @ A @ B
@ ^ [X3: A] : C3
@ Xs )
= Xs ) ) ).
% sort_key_const
thf(fact_3614_insort__not__Nil,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,A3: B,Xs: list @ B] :
( ( linorder_insort_key @ B @ A @ F @ A3 @ Xs )
!= ( nil @ B ) ) ) ).
% insort_not_Nil
thf(fact_3615_filter__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F: B > A,Xs: list @ B] :
( ( filter2 @ B @ P @ ( linorder_sort_key @ B @ A @ F @ Xs ) )
= ( linorder_sort_key @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) ) ) ) ).
% filter_sort
thf(fact_3616_sort__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F3: B > A,Xs3: list @ B] : ( foldr @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 ) @ Xs3 @ ( nil @ B ) ) ) ) ) ).
% sort_key_def
thf(fact_3617_sort__key__stable,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F: A > B,K: B,Xs: list @ A] :
( ( filter2 @ A
@ ^ [Y3: A] :
( ( F @ Y3 )
= K )
@ ( linorder_sort_key @ A @ B @ F @ Xs ) )
= ( filter2 @ A
@ ^ [Y3: A] :
( ( F @ Y3 )
= K )
@ Xs ) ) ) ).
% sort_key_stable
thf(fact_3618_sort__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 )
= ( linorder_quicksort @ A ) ) ) ).
% sort_quicksort
thf(fact_3619_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Y: B,Ys: list @ B] :
( ( ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F @ X @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ X @ ( cons @ B @ Y @ Ys ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F @ X @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F @ X @ Ys ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_3620_insort__key_Osimps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B] :
( ( linorder_insort_key @ B @ A @ F @ X @ ( nil @ B ) )
= ( cons @ B @ X @ ( nil @ B ) ) ) ) ).
% insort_key.simps(1)
thf(fact_3621_set__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Xs: list @ B] :
( ( set2 @ B @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
= ( insert2 @ B @ X @ ( set2 @ B @ Xs ) ) ) ) ).
% set_insort_key
thf(fact_3622_distinct__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Xs: list @ B] :
( ( distinct @ B @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
= ( ~ ( member2 @ B @ X @ ( set2 @ B @ Xs ) )
& ( distinct @ B @ Xs ) ) ) ) ).
% distinct_insort
thf(fact_3623_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_3624_sorted__sort__id,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ Xs )
= Xs ) ) ) ).
% sorted_sort_id
thf(fact_3625_sorted__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ Xs ) ) ) ).
% sorted_sort
thf(fact_3626_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( finite_fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3 )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.eq_fold
thf(fact_3627_sort__mergesort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 )
= ( mergesort @ A ) ) ) ).
% sort_mergesort
thf(fact_3628_sort__mergesort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% sort_mergesort_by_rel
thf(fact_3629_insort__insert__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs )
= ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ Xs ) ) ) ) ).
% insort_insert_insort
thf(fact_3630_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F: B > A,A3: B] :
( ! [X2: B] :
( ( member2 @ B @ X2 @ ( set2 @ B @ Xs ) )
=> ( ord_less_eq @ A @ ( F @ A3 ) @ ( F @ X2 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F @ A3 @ Xs )
= ( cons @ B @ A3 @ Xs ) ) ) ) ).
% insort_is_Cons
thf(fact_3631_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) ) ) ) ).
% sorted_insort_key
thf(fact_3632_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linorder_sort_key @ B @ A @ F @ Xs ) ) ) ) ).
% sorted_sort_key
thf(fact_3633_insort__insert__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord329482645794927042rt_key @ B @ A )
= ( ^ [F3: B > A,X3: B,Xs3: list @ B] : ( if @ ( list @ B ) @ ( member2 @ A @ ( F3 @ X3 ) @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs3 ) ) ) @ Xs3 @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs3 ) ) ) ) ) ).
% insort_insert_key_def
thf(fact_3634_insort__insert__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Xs: list @ B] :
( ~ ( member2 @ A @ ( F @ X ) @ ( image2 @ B @ A @ F @ ( set2 @ B @ Xs ) ) )
=> ( ( linord329482645794927042rt_key @ B @ A @ F @ X @ Xs )
= ( linorder_insort_key @ B @ A @ F @ X @ Xs ) ) ) ) ).
% insort_insert_insort_key
thf(fact_3635_sorted__list__of__set__sort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
= ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ ( remdups @ A @ Xs ) ) ) ) ).
% sorted_list_of_set_sort_remdups
thf(fact_3636_distinct__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) ) )
= ( ~ ( member2 @ A @ ( F @ X ) @ ( image2 @ B @ A @ F @ ( set2 @ B @ Xs ) ) )
& ( distinct @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% distinct_insort_key
thf(fact_3637_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,P: B > $o,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( ( P @ X )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F @ X @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ) ).
% filter_insort
thf(fact_3638_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,Xs: list @ A] :
( ( member2 @ A @ A3 @ ( set2 @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ A3
@ ( remove1 @ A @ A3 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_3639_part__code_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Pivot: A] :
( ( linorder_part @ B @ A @ F @ Pivot @ ( nil @ B ) )
= ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( nil @ B ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( nil @ B ) @ ( nil @ B ) ) ) ) ) ).
% part_code(1)
thf(fact_3640_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_3641_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X2 @ A3 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ A3
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_3642_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I5: set @ A,F: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I2: A] :
( ( member2 @ A @ I2 @ I5 )
& ( ( F @ I2 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member2 @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) @ ( F @ I ) ) ) )
& ( ~ ( member2 @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_3643_part__code_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Pivot: A,X: B,Xs: list @ B] :
( ( linorder_part @ B @ A @ F @ Pivot @ ( cons @ B @ X @ Xs ) )
= ( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ ( F @ X ) @ Pivot ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( cons @ B @ X @ Lts ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ Gts ) ) @ ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ Pivot @ ( F @ X ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ ( cons @ B @ X @ Gts ) ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( cons @ B @ X @ Eqs ) @ Gts ) ) ) ) )
@ ( linorder_part @ B @ A @ F @ Pivot @ Xs ) ) ) ) ).
% part_code(2)
thf(fact_3644_Bleast__code,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( bleast @ A @ ( set2 @ A @ Xs ) @ P )
= ( case_list @ A @ A @ ( abort_Bleast @ A @ ( set2 @ A @ Xs ) @ P )
@ ^ [X3: A,Xs3: list @ A] : X3
@ ( filter2 @ A @ P
@ ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ Xs ) ) ) ) ) ).
% Bleast_code
thf(fact_3645_Fpow__Pow__finite,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A7: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow @ A @ A7 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_3646_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 )
@ ^ [Uu3: product_prod @ A @ B] :
? [I2: nat] :
( ( Uu3
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I2 ) @ ( nth @ B @ Ys @ I2 ) ) )
& ( ord_less @ nat @ I2 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_3647_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [X3: product_prod @ A @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ X3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y3: A,Z3: B] : ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z3 ) @ R3 )
@ X3 ) ) ) ) ).
% listrel_iff_zip
thf(fact_3648_ball__empty,axiom,
! [A: $tType,P: A > $o,X6: A] :
( ( member2 @ A @ X6 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X6 ) ) ).
% ball_empty
thf(fact_3649_eq__or__mem__image__simp,axiom,
! [B: $tType,A: $tType,F: B > A,A3: B,B4: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [L4: B] :
( ( Uu3
= ( F @ L4 ) )
& ( ( L4 = A3 )
| ( member2 @ B @ L4 @ B4 ) ) ) )
= ( insert2 @ A @ ( F @ A3 )
@ ( collect @ A
@ ^ [Uu3: A] :
? [L4: B] :
( ( Uu3
= ( F @ L4 ) )
& ( member2 @ B @ L4 @ B4 ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_3650_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P3: A > $o] :
! [X3: A] :
( ( member2 @ A @ X3 @ A7 )
=> ( P3 @ X3 ) ) ) ) ).
% Ball_def
thf(fact_3651_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A7: set @ A,XS: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z3: list @ A] :
? [X3: A,Xs3: list @ A] :
( ( Z3
= ( cons @ A @ X3 @ Xs3 ) )
& ( member2 @ A @ X3 @ A7 )
& ( member2 @ ( list @ A ) @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_3652_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X3: B] :
( ( Uu3
= ( F @ X3 ) )
& ( P @ X3 ) ) )
= ( image2 @ B @ A @ F @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_3653_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X3: B] :
( ( Uu3
= ( F @ X3 ) )
& ( member2 @ B @ X3 @ A5 ) ) )
= ( image2 @ B @ A @ F @ A5 ) ) ).
% Setcompr_eq_image
thf(fact_3654_empty__in__Fpow,axiom,
! [A: $tType,A5: set @ A] : ( member2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A5 ) ) ).
% empty_in_Fpow
thf(fact_3655_Field__not__elem,axiom,
! [A: $tType,V: A,R: set @ ( product_prod @ A @ A )] :
( ~ ( member2 @ A @ V @ ( field2 @ A @ R ) )
=> ! [X6: product_prod @ A @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ X6 @ R )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z3: A] :
( ( Y3 != V )
& ( Z3 != V ) )
@ X6 ) ) ) ).
% Field_not_elem
thf(fact_3656_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
? [Us: list @ A,Z3: A,Z7: A,Vs2: list @ A] :
( ( Xs3
= ( append @ A @ Us @ ( cons @ A @ Z3 @ Vs2 ) ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Z7 ) @ R4 )
& ( Ys3
= ( append @ A @ Us @ ( cons @ A @ Z7 @ Vs2 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_3657_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X3: list @ A,Y3: list @ A] :
? [A4: A,V4: list @ A] :
( ( Y3
= ( append @ A @ X3 @ ( cons @ A @ A4 @ V4 ) ) )
| ? [U3: list @ A,B3: A,C6: A,W4: list @ A,Z3: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C6 ) @ R4 )
& ( X3
= ( append @ A @ U3 @ ( cons @ A @ B3 @ W4 ) ) )
& ( Y3
= ( append @ A @ U3 @ ( cons @ A @ C6 @ Z3 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_3658_sorted__wrt_Osimps_I2_J,axiom,
! [A: $tType,P: A > A > $o,X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ P @ ( cons @ A @ X @ Ys ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( P @ X @ X3 ) )
& ( sorted_wrt @ A @ P @ Ys ) ) ) ).
% sorted_wrt.simps(2)
thf(fact_3659_sorted__wrt_Oelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( sorted_wrt @ A @ X @ Xa )
=> ~ ! [X2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys2 ) )
=> ( ! [Xa3: A] :
( ( member2 @ A @ Xa3 @ ( set2 @ A @ Ys2 ) )
=> ( X @ X2 @ Xa3 ) )
& ( sorted_wrt @ A @ X @ Ys2 ) ) ) ) ).
% sorted_wrt.elims(3)
thf(fact_3660_Un__interval,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [B1: A,B22: A,B32: A,F: A > B] :
( ( ord_less_eq @ A @ B1 @ B22 )
=> ( ( ord_less_eq @ A @ B22 @ B32 )
=> ( ( sup_sup @ ( set @ B )
@ ( collect @ B
@ ^ [Uu3: B] :
? [I2: A] :
( ( Uu3
= ( F @ I2 ) )
& ( ord_less_eq @ A @ B1 @ I2 )
& ( ord_less @ A @ I2 @ B22 ) ) )
@ ( collect @ B
@ ^ [Uu3: B] :
? [I2: A] :
( ( Uu3
= ( F @ I2 ) )
& ( ord_less_eq @ A @ B22 @ I2 )
& ( ord_less @ A @ I2 @ B32 ) ) ) )
= ( collect @ B
@ ^ [Uu3: B] :
? [I2: A] :
( ( Uu3
= ( F @ I2 ) )
& ( ord_less_eq @ A @ B1 @ I2 )
& ( ord_less @ A @ I2 @ B32 ) ) ) ) ) ) ) ).
% Un_interval
thf(fact_3661_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys2: list @ A,X3: A,Y3: A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X3 @ Xs5 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_3662_takeWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) )
& ( ~ ! [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
=> ( P @ X6 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% takeWhile_append
thf(fact_3663_list__collect__set__def,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F3: B > ( set @ A ),L4: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [A4: B] :
( ( Uu3
= ( F3 @ A4 ) )
& ( member2 @ B @ A4 @ ( set2 @ B @ L4 ) ) ) ) ) ) ) ).
% list_collect_set_def
thf(fact_3664_sorted__wrt_Oelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( sorted_wrt @ A @ X @ Xa )
=> ( ( Xa
!= ( nil @ A ) )
=> ~ ! [X2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys2 ) )
=> ~ ( ! [Xa2: A] :
( ( member2 @ A @ Xa2 @ ( set2 @ A @ Ys2 ) )
=> ( X @ X2 @ Xa2 ) )
& ( sorted_wrt @ A @ X @ Ys2 ) ) ) ) ) ).
% sorted_wrt.elims(2)
thf(fact_3665_sorted__wrt_Oelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X @ Xa )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ Y )
=> ~ ! [X2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys2 ) )
=> ( Y
= ( ~ ( ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Ys2 ) )
=> ( X @ X2 @ Y3 ) )
& ( sorted_wrt @ A @ X @ Ys2 ) ) ) ) ) ) ) ).
% sorted_wrt.elims(1)
thf(fact_3666_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs3: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I2: nat] :
( ( Uu3
= ( nth @ A @ Xs3 @ I2 ) )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_3667_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A4: A] :
( ( Uu3
= ( inf_inf @ A @ X @ A4 ) )
& ( member2 @ A @ A4 @ A5 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_3668_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B4 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A4: A,B3: A] :
( ( Uu3
= ( inf_inf @ A @ A4 @ B3 ) )
& ( member2 @ A @ A4 @ A5 )
& ( member2 @ A @ B3 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_3669_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic7752659483105999362nf_fin @ A @ B4 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A4: A,B3: A] :
( ( Uu3
= ( sup_sup @ A @ A4 @ B3 ) )
& ( member2 @ A @ A4 @ A5 )
& ( member2 @ A @ B3 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_3670_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A4: A] :
( ( Uu3
= ( sup_sup @ A @ X @ A4 ) )
& ( member2 @ A @ A4 @ A5 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_3671_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z5: list @ A] : Y5 = Z5 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [X3: product_prod @ A @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ X3 @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Z5: A] : Y5 = Z5
@ X3 ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_3672_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ! [X2: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X2 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y3: list @ A,Z3: list @ A] :
( ( size_size @ ( list @ A ) @ Y3 )
= ( size_size @ ( list @ A ) @ Z3 ) )
@ X2 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
= ( Xs = Ys ) ) ) ) ).
% concat_eq_concat_iff
thf(fact_3673_concat__injective,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ! [X2: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X2 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y3: list @ A,Z3: list @ A] :
( ( size_size @ ( list @ A ) @ Y3 )
= ( size_size @ ( list @ A ) @ Z3 ) )
@ X2 ) )
=> ( Xs = Ys ) ) ) ) ).
% concat_injective
thf(fact_3674_list__collect__set__alt,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F3: B > ( set @ A ),L4: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [I2: nat] :
( ( Uu3
= ( F3 @ ( nth @ B @ L4 @ I2 ) ) )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ L4 ) ) ) ) ) ) ) ).
% list_collect_set_alt
thf(fact_3675_set__drop__conv,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( set2 @ A @ ( drop @ A @ N @ L ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I2: nat] :
( ( Uu3
= ( nth @ A @ L @ I2 ) )
& ( ord_less_eq @ nat @ N @ I2 )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) ) ) ) ) ).
% set_drop_conv
thf(fact_3676_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),N3: 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 )
= N3 )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N3 )
& ? [Xys2: list @ A,X3: A,Y3: A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X3 @ Xs5 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R4 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_3677_finite__inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A3: A,A5: set @ A] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ A5 ) )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [B3: A] :
( ( Uu3
= ( inf_inf @ A @ A3 @ B3 ) )
& ( member2 @ A @ B3 @ A5 ) ) ) ) ) ) ).
% finite_inf_Sup
thf(fact_3678_pairself__image__eq,axiom,
! [B: $tType,A: $tType,F: B > A,P: B > B > $o] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F ) @ ( collect @ ( product_prod @ B @ B ) @ ( product_case_prod @ B @ B @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [A4: B,B3: B] :
( ( Uu3
= ( product_Pair @ A @ A @ ( F @ A4 ) @ ( F @ B3 ) ) )
& ( P @ A4 @ B3 ) ) ) ) ).
% pairself_image_eq
thf(fact_3679_brk__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( brk_rel @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B )] :
( sup_sup @ ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu3: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X3: A,Y3: B] :
( ( Uu3
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $false @ X3 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) )
& ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R6 ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu3: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X3: A,Y3: B] :
( Uu3
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $true @ X3 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) ) ) ) ) ) ).
% brk_rel_def
thf(fact_3680_lexn_Osimps_I1_J,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R3 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_3681_pairself_Osimps,axiom,
! [B: $tType,A: $tType,F: A > B,A3: A,B2: A] :
( ( pairself @ A @ B @ F @ ( product_Pair @ A @ A @ A3 @ B2 ) )
= ( product_Pair @ B @ B @ ( F @ A3 ) @ ( F @ B2 ) ) ) ).
% pairself.simps
thf(fact_3682_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,B6: A] :
( ( Xa
= ( product_Pair @ A @ A @ A6 @ B6 ) )
=> ( Y
!= ( product_Pair @ B @ B @ ( X @ A6 ) @ ( X @ B6 ) ) ) ) ) ).
% pairself.elims
thf(fact_3683_Pow__Compl,axiom,
! [A: $tType,A5: set @ A] :
( ( pow @ A @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [B5: set @ A] :
( ( Uu3
= ( uminus_uminus @ ( set @ A ) @ B5 ) )
& ( member2 @ ( set @ A ) @ A5 @ ( pow @ A @ B5 ) ) ) ) ) ).
% Pow_Compl
thf(fact_3684_lexn__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),N: nat] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexn @ A @ R3 @ N ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= N )
& ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ) ).
% lexn_length
thf(fact_3685_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,B6: A] :
( ( Xa
= ( product_Pair @ A @ A @ A6 @ B6 ) )
=> ( ( Y
= ( product_Pair @ B @ B @ ( X @ A6 ) @ ( X @ B6 ) ) )
=> ~ ( 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 @ B6 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_3686_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X25: list @ A] :
( ? [Y3: A,Ys3: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X3 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ord_less @ A @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X3 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( ord_less @ A @ X3 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X3 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_3687_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S: set @ A,F: A > B > B,A5: set @ A,B4: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F @ Z2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( finite_fold @ A @ B @ F @ ( finite_fold @ A @ B @ F @ Z2 @ A5 ) @ B4 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_3688_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F3: B > A,A4: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( times_times @ A @ A4 ) @ ( F3 @ ( nth @ B @ Xs3 @ N3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_3689_funpow__times__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [F: A > nat,X: A] :
( ( compow @ ( A > A ) @ ( F @ X ) @ ( times_times @ A @ X ) )
= ( times_times @ A @ ( power_power @ A @ X @ ( F @ X ) ) ) ) ) ).
% funpow_times_power
thf(fact_3690_rotate__def,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N3: nat] : ( compow @ ( ( list @ A ) > ( list @ A ) ) @ N3 @ ( rotate1 @ A ) ) ) ) ).
% rotate_def
thf(fact_3691_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_3692_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,A3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A3 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_3693_butlast__power,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( compow @ ( ( list @ A ) > ( list @ A ) ) @ N @ ( butlast @ A ) @ Xs )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) @ Xs ) ) ).
% butlast_power
thf(fact_3694_numeral__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( numeral_numeral @ A )
= ( ^ [K2: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K2 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% numeral_unfold_funpow
thf(fact_3695_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S: set @ A,F: A > B > B,X: A,A5: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F @ Z2 @ ( insert2 @ A @ X @ A5 ) )
= ( F @ X @ ( finite_fold @ A @ B @ F @ Z2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_3696_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S: set @ A,F: A > B > B,A5: set @ A,X: A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( finite_fold @ A @ B @ F @ Z2 @ A5 )
= ( F @ X @ ( finite_fold @ A @ B @ F @ Z2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_3697_ord_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o] :
( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X25: list @ A] :
( ? [Y3: A,Ys3: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X3 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X3 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord.lexordp_def
thf(fact_3698_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S
@ ( insert2 @ A
@ ( ord_Least @ A
@ ^ [N3: A] : ( member2 @ A @ N3 @ S ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N ) ) ) ).
% enumerate_Suc
thf(fact_3699_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_3700_min__list_Oelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( Y
!= ( case_list @ A @ A @ X2
@ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X2 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ).
% min_list.elims
thf(fact_3701_ord_Olexordp__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp2 @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp2 @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_3702_ord_Olexordp__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] :
( ( lexordp2 @ A @ Less @ ( nil @ A ) @ Ys )
= ( Ys
!= ( nil @ A ) ) ) ).
% ord.lexordp_simps(1)
thf(fact_3703_ord_Olexordp__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
~ ( lexordp2 @ A @ Less @ Xs @ ( nil @ A ) ) ).
% ord.lexordp_simps(2)
thf(fact_3704_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_3705_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_3706_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_3707_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_3708_ord_Olexordp__irreflexive,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ! [X2: A] :
~ ( Less @ X2 @ X2 )
=> ~ ( lexordp2 @ A @ Less @ Xs @ Xs ) ) ).
% ord.lexordp_irreflexive
thf(fact_3709_ord_Olexordp_Ocong,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( lexordp2 @ A ) ) ).
% ord.lexordp.cong
thf(fact_3710_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F @ Z2 @ Less_eq @ Less )
=> ( ( Z2
= ( F @ A3 @ B2 ) )
= ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_3711_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F @ Z2 @ Less_eq @ Less )
=> ( ( ( F @ A3 @ B2 )
= Z2 )
= ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_3712_ord_Olexordp_OCons,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp2 @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_3713_ord_Olexordp_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp2 @ A @ Less @ Xs @ Ys )
=> ( lexordp2 @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_3714_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B11: A] :
( ( P @ B11 )
=> ( ord_less_eq @ A @ A6 @ B11 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_3715_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B11: A] :
( ( P @ B11 )
=> ( ord_less_eq @ A @ A6 @ B11 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_3716_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_3717_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X2 @ Y4 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_3718_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 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y2 @ Ya2 ) ) )
=> ( Y2 = X6 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z2 ) ) ) ) ).
% Least1_le
thf(fact_3719_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 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y2 @ Ya2 ) ) )
=> ( Y2 = X6 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_3720_ord_Olexordp__append__leftI,axiom,
! [A: $tType,Less: A > A > $o,Us2: list @ A,Vs: list @ A,Xs: list @ A] :
( ( lexordp2 @ A @ Less @ Us2 @ Vs )
=> ( lexordp2 @ A @ Less @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs ) ) ) ).
% ord.lexordp_append_leftI
thf(fact_3721_ord_Olexordp__append__leftD,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A,Us2: list @ A,Vs: list @ A] :
( ( lexordp2 @ A @ Less @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs ) )
=> ( ! [A6: A] :
~ ( Less @ A6 @ A6 )
=> ( lexordp2 @ A @ Less @ Us2 @ Vs ) ) ) ).
% ord.lexordp_append_leftD
thf(fact_3722_ord_Olexordp__into__lexordp__eq,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( lexordp2 @ A @ Less @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less @ Xs @ Ys ) ) ).
% ord.lexordp_into_lexordp_eq
thf(fact_3723_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_3724_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_3725_ord_Olexordp_ONil,axiom,
! [A: $tType,Less: A > A > $o,Y: A,Ys: list @ A] : ( lexordp2 @ A @ Less @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ).
% ord.lexordp.Nil
thf(fact_3726_ord_Olexordp_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp2 @ A @ Less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y2: A,Ys2: list @ A] :
( A22
!= ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [X2: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( Less @ X2 @ Y2 ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X2 @ Y2 )
=> ( ~ ( Less @ Y2 @ X2 )
=> ~ ( lexordp2 @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_3727_ord_Olexordp_Osimps,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Y3: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp2 @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_3728_ord_Olexordp__append__left__rightI,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Us2: list @ A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp2 @ A @ Less @ ( append @ A @ Us2 @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us2 @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% ord.lexordp_append_left_rightI
thf(fact_3729_ord_Olexordp__append__rightI,axiom,
! [A: $tType,Ys: list @ A,Less: A > A > $o,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( lexordp2 @ A @ Less @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ).
% ord.lexordp_append_rightI
thf(fact_3730_hd__def,axiom,
! [A: $tType] :
( ( hd @ A )
= ( case_list @ A @ A @ ( undefined @ A )
@ ^ [X213: A,X223: list @ A] : X213 ) ) ).
% hd_def
thf(fact_3731_Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( bleast @ A )
= ( ^ [S7: set @ A,P3: A > $o] :
( ord_Least @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ S7 )
& ( P3 @ X3 ) ) ) ) ) ) ).
% Bleast_def
thf(fact_3732_abort__Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( abort_Bleast @ A )
= ( ^ [S7: set @ A,P3: A > $o] :
( ord_Least @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ S7 )
& ( P3 @ X3 ) ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_3733_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,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B6: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B6 @ Bs2 ) )
=> ( Y
!= ( cons @ C @ ( X @ A6 @ B6 ) @ ( zipf @ A @ B @ C @ X @ As @ Bs2 ) ) ) ) )
=> ( ( ? [V2: A,Va2: list @ A] :
( Xa
= ( cons @ A @ V2 @ Va2 ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V2: B,Va2: list @ B] :
( Xb
= ( cons @ B @ V2 @ Va2 ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) ) ) ) ) ) ).
% zipf.elims
thf(fact_3734_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa )
= Y )
=> ( ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( Y != X2 ) )
=> ( ! [X2: A,Y2: A,Zs3: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs3 ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X @ X2 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs3 ) ) ) ) @ X2 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs3 ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_3735_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X )
=> ( ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( case_list @ A @ A @ X2
@ ^ [A4: A,List2: list @ A] : ( ord_min @ A @ X2 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_3736_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,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B6: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B6 @ Bs2 ) )
=> ( ( Y
= ( cons @ C @ ( X @ A6 @ B6 ) @ ( zipf @ A @ B @ C @ X @ As @ 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 @ As ) @ ( cons @ B @ B6 @ Bs2 ) ) ) ) ) ) )
=> ( ! [V2: A,Va2: list @ A] :
( ( Xa
= ( cons @ A @ V2 @ Va2 ) )
=> ( ( 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 @ V2 @ Va2 ) @ ( nil @ B ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V2: B,Va2: list @ B] :
( ( Xb
= ( cons @ B @ V2 @ Va2 ) )
=> ( ( 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 @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% zipf.pelims
thf(fact_3737_filter__rev__alt,axiom,
! [A: $tType] :
( ( filter_rev @ A )
= ( ^ [P3: A > $o,L4: list @ A] : ( filter2 @ A @ P3 @ ( rev @ A @ L4 ) ) ) ) ).
% filter_rev_alt
thf(fact_3738_filter__rev__aux__alt,axiom,
! [A: $tType] :
( ( filter_rev_aux @ A )
= ( ^ [A4: list @ A,P3: A > $o,L4: list @ A] : ( append @ A @ ( filter2 @ A @ P3 @ ( rev @ A @ L4 ) ) @ A4 ) ) ) ).
% filter_rev_aux_alt
thf(fact_3739_filter__rev__def,axiom,
! [A: $tType] :
( ( filter_rev @ A )
= ( filter_rev_aux @ A @ ( nil @ A ) ) ) ).
% filter_rev_def
thf(fact_3740_filter__rev__aux_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,A3: list @ A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( filter_rev_aux @ A @ A3 @ P @ ( cons @ A @ X @ Xs ) )
= ( filter_rev_aux @ A @ ( cons @ A @ X @ A3 ) @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( filter_rev_aux @ A @ A3 @ P @ ( cons @ A @ X @ Xs ) )
= ( filter_rev_aux @ A @ A3 @ P @ Xs ) ) ) ) ).
% filter_rev_aux.simps(2)
thf(fact_3741_filter__rev__aux_Osimps_I1_J,axiom,
! [A: $tType,A3: list @ A,P: A > $o] :
( ( filter_rev_aux @ A @ A3 @ P @ ( nil @ A ) )
= A3 ) ).
% filter_rev_aux.simps(1)
thf(fact_3742_precise__extr__pure_I2_J,axiom,
! [B: $tType,A: $tType,R: A > B > assn,P: $o] :
( ( precise @ A @ B
@ ^ [X3: A,Y3: B] : ( times_times @ assn @ ( R @ X3 @ Y3 ) @ ( pure_assn @ P ) ) )
= ( P
=> ( precise @ A @ B @ R ) ) ) ).
% precise_extr_pure(2)
thf(fact_3743_precise__extr__pure_I1_J,axiom,
! [B: $tType,A: $tType,P: $o,R: A > B > assn] :
( ( precise @ A @ B
@ ^ [X3: A,Y3: B] : ( times_times @ assn @ ( pure_assn @ P ) @ ( R @ X3 @ Y3 ) ) )
= ( P
=> ( precise @ A @ B @ R ) ) ) ).
% precise_extr_pure(1)
thf(fact_3744_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member2 @ B @ X3 @ I5 )
& ( ( P4 @ X3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member2 @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert2 @ B @ I @ I5 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P4 @ I5 ) ) )
& ( ~ ( member2 @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert2 @ B @ I @ I5 ) )
= ( times_times @ A @ ( P4 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P4 @ I5 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_3745_set__nths,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I5 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I2: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I2 ) )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member2 @ nat @ I2 @ I5 ) ) ) ) ).
% set_nths
thf(fact_3746_nths__nil,axiom,
! [A: $tType,A5: set @ nat] :
( ( nths @ A @ ( nil @ A ) @ A5 )
= ( nil @ A ) ) ).
% nths_nil
thf(fact_3747_nths__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_3748_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_3749_nths__upt__eq__take,axiom,
! [A: $tType,L: list @ A,N: nat] :
( ( nths @ A @ L @ ( set_ord_lessThan @ nat @ N ) )
= ( take @ A @ N @ L ) ) ).
% nths_upt_eq_take
thf(fact_3750_nths__singleton,axiom,
! [A: $tType,A5: set @ nat,X: A] :
( ( ( member2 @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A5 )
= ( cons @ A @ X @ ( nil @ A ) ) ) )
& ( ~ ( member2 @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A5 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_3751_notin__set__nthsI,axiom,
! [A: $tType,X: A,Xs: list @ A,I5: set @ nat] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member2 @ A @ X @ ( set2 @ A @ ( nths @ A @ Xs @ I5 ) ) ) ) ).
% notin_set_nthsI
thf(fact_3752_in__set__nthsD,axiom,
! [A: $tType,X: A,Xs: list @ A,I5: set @ nat] :
( ( member2 @ A @ X @ ( set2 @ A @ ( nths @ A @ Xs @ I5 ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_3753_nths__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,I5: set @ nat] :
( ( nths @ A @ ( map @ B @ A @ F @ Xs ) @ I5 )
= ( map @ B @ A @ F @ ( nths @ B @ Xs @ I5 ) ) ) ).
% nths_map
thf(fact_3754_distinct__nthsI,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( nths @ A @ Xs @ I5 ) ) ) ).
% distinct_nthsI
thf(fact_3755_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,I5: set @ B] :
( ( groups1962203154675924110t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X3: B] :
( ( member2 @ B @ X3 @ I5 )
& ( ( G @ X3 )
!= ( one_one @ A ) ) ) ) )
= ( groups1962203154675924110t_prod @ B @ A @ G @ I5 ) ) ) ).
% prod.non_neutral'
thf(fact_3756_set__nths__subset,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( nths @ A @ Xs @ I5 ) ) @ ( set2 @ A @ Xs ) ) ).
% set_nths_subset
thf(fact_3757_nths__all,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member2 @ nat @ I3 @ I5 ) )
=> ( ( nths @ A @ Xs @ I5 )
= Xs ) ) ).
% nths_all
thf(fact_3758_sorted__nths,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I5: set @ nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs @ I5 ) ) ) ) ).
% sorted_nths
thf(fact_3759_nths__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I5: set @ nat] :
( ( nths @ A @ ( drop @ A @ N @ Xs ) @ I5 )
= ( nths @ A @ Xs @ ( image2 @ nat @ nat @ ( plus_plus @ nat @ N ) @ I5 ) ) ) ).
% nths_drop
thf(fact_3760_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I2: B] : ( times_times @ A @ ( G @ I2 ) @ ( H @ I2 ) )
@ I5 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I5 ) @ ( groups1962203154675924110t_prod @ B @ A @ H @ I5 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_3761_snga__prec,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( precise @ ( list @ A ) @ ( array @ A )
@ ^ [X3: list @ A,P6: array @ A] : ( snga_assn @ A @ P6 @ X3 ) ) ) ).
% snga_prec
thf(fact_3762_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N3: nat,Xs3: list @ A] : ( nths @ A @ Xs3 @ ( collect @ nat @ ( ord_less_eq @ nat @ N3 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_3763_sngr__prec,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( precise @ A @ ( ref @ A )
@ ^ [X3: A,P6: ref @ A] : ( sngr_assn @ A @ P6 @ X3 ) ) ) ).
% sngr_prec
thf(fact_3764_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T @ S ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S )
= ( groups1962203154675924110t_prod @ B @ A @ G @ T ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_3765_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T @ S ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T )
= ( groups1962203154675924110t_prod @ B @ A @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_3766_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T: set @ B,H: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T @ S ) )
=> ( ( H @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S )
= ( groups1962203154675924110t_prod @ B @ A @ H @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_3767_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T: set @ B,G: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T @ S ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T )
= ( groups1962203154675924110t_prod @ B @ A @ H @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_3768_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member2 @ B @ X3 @ I5 )
& ( ( G @ X3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member2 @ B @ X3 @ I5 )
& ( ( H @ X3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I2: B] : ( times_times @ A @ ( G @ I2 ) @ ( H @ I2 ) )
@ I5 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I5 ) @ ( groups1962203154675924110t_prod @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_3769_nths__append,axiom,
! [A: $tType,L: list @ A,L3: list @ A,A5: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L3 ) @ A5 )
= ( append @ A @ ( nths @ A @ L @ A5 )
@ ( nths @ A @ L3
@ ( collect @ nat
@ ^ [J2: nat] : ( member2 @ nat @ ( plus_plus @ nat @ J2 @ ( size_size @ ( list @ A ) @ L ) ) @ A5 ) ) ) ) ) ).
% nths_append
thf(fact_3770_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P6: B > A,I8: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member2 @ B @ X3 @ I8 )
& ( ( P6 @ X3 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P6
@ ( collect @ B
@ ^ [X3: B] :
( ( member2 @ B @ X3 @ I8 )
& ( ( P6 @ X3 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_3771_filter__in__nths,axiom,
! [A: $tType,Xs: list @ A,S4: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ( filter2 @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs @ S4 ) ) )
@ Xs )
= ( nths @ A @ Xs @ S4 ) ) ) ).
% filter_in_nths
thf(fact_3772_length__nths,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I5 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member2 @ nat @ I2 @ I5 ) ) ) ) ) ).
% length_nths
thf(fact_3773_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
( nths @ A @ Xs3
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P3 @ ( nth @ A @ Xs3 @ I2 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_3774_nths__Cons,axiom,
! [A: $tType,X: A,L: list @ A,A5: set @ nat] :
( ( nths @ A @ ( cons @ A @ X @ L ) @ A5 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member2 @ nat @ ( zero_zero @ nat ) @ A5 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J2: nat] : ( member2 @ nat @ ( suc @ J2 ) @ A5 ) ) ) ) ) ).
% nths_Cons
thf(fact_3775_sorted__list__of__set__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( linord144544945434240204of_set @ A @ A
@ ^ [X3: A] : X3 ) ) ) ).
% sorted_list_of_set_def
thf(fact_3776_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M2: nat,A3: A] :
( ( enumerate @ A @ N @ ( replicate @ A @ M2 @ A3 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A3 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M2 ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_3777_finite__def,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P6: ( set @ A ) > $o,X3: set @ A] :
( ( X3
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,A4: A] :
( ( X3
= ( insert2 @ A @ A4 @ A7 ) )
& ( P6 @ A7 ) ) ) ) ) ).
% finite_def
thf(fact_3778_listrel1__subset__listrel,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R8: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R8 )
=> ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R8 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R3 ) @ ( listrel @ A @ A @ R8 ) ) ) ) ).
% listrel1_subset_listrel
thf(fact_3779_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X3: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_3780_UNIV__I,axiom,
! [A: $tType,X: A] : ( member2 @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_3781_inf__top__left,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% inf_top_left
thf(fact_3782_inf__top__right,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% inf_top_right
thf(fact_3783_inf__eq__top__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( top_top @ A ) )
= ( ( X
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% inf_eq_top_iff
thf(fact_3784_top__eq__inf__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ X @ Y ) )
= ( ( X
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% top_eq_inf_iff
thf(fact_3785_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B2: A] :
( ( ( inf_inf @ A @ A3 @ B2 )
= ( top_top @ A ) )
= ( ( A3
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_3786_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ A3 )
= A3 ) ) ).
% inf_top.left_neutral
thf(fact_3787_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B2: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ A3 @ B2 ) )
= ( ( A3
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_3788_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ ( top_top @ A ) )
= A3 ) ) ).
% inf_top.right_neutral
thf(fact_3789_boolean__algebra_Odisj__one__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_right
thf(fact_3790_boolean__algebra_Odisj__one__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_left
thf(fact_3791_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_3792_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_3793_Int__UNIV,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( top_top @ ( set @ A ) ) )
= ( ( A5
= ( top_top @ ( set @ A ) ) )
& ( B4
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_3794_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_3795_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_3796_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% min_top
thf(fact_3797_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% min_top2
thf(fact_3798_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_3799_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_3800_replicate__eq__replicate,axiom,
! [A: $tType,M2: nat,X: A,N: nat,Y: A] :
( ( ( replicate @ A @ M2 @ X )
= ( replicate @ A @ N @ Y ) )
= ( ( M2 = N )
& ( ( M2
!= ( zero_zero @ nat ) )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_3801_length__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_3802_concat__replicate__trivial,axiom,
! [A: $tType,I: nat] :
( ( concat @ A @ ( replicate @ ( list @ A ) @ I @ ( nil @ A ) ) )
= ( nil @ A ) ) ).
% concat_replicate_trivial
thf(fact_3803_map__replicate,axiom,
! [A: $tType,B: $tType,F: B > A,N: nat,X: B] :
( ( map @ B @ A @ F @ ( replicate @ B @ N @ X ) )
= ( replicate @ A @ N @ ( F @ X ) ) ) ).
% map_replicate
thf(fact_3804_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_3805_rev__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( rev @ A @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) ).
% rev_replicate
thf(fact_3806_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S6: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S6: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_3807_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_3808_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_3809_sup__compl__top__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( sup_sup @ A @ X @ Y ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left1
thf(fact_3810_sup__compl__top__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left2
thf(fact_3811_boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_left
thf(fact_3812_boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_right
thf(fact_3813_Diff__UNIV,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_3814_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_3815_inj__map__eq__map,axiom,
! [B: $tType,A: $tType,F: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map @ A @ B @ F @ Xs )
= ( map @ A @ B @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_3816_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_3817_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_3818_in__set__replicate,axiom,
! [A: $tType,X: A,N: nat,Y: A] :
( ( member2 @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ Y ) ) )
= ( ( X = Y )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_3819_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A3 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_3820_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_3821_Gcd__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Gcd_UNIV
thf(fact_3822_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_3823_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_3824_drop__replicate,axiom,
! [A: $tType,I: nat,K: nat,X: A] :
( ( drop @ A @ I @ ( replicate @ A @ K @ X ) )
= ( replicate @ A @ ( minus_minus @ nat @ K @ I ) @ X ) ) ).
% drop_replicate
thf(fact_3825_takeWhile__replicate,axiom,
! [A: $tType,P: A > $o,X: A,N: nat] :
( ( ( P @ X )
=> ( ( takeWhile @ A @ P @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) )
& ( ~ ( P @ X )
=> ( ( takeWhile @ A @ P @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) ) ) ).
% takeWhile_replicate
thf(fact_3826_last__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_3827_foldr__replicate,axiom,
! [A: $tType,B: $tType,F: B > A > A,N: nat,X: B] :
( ( foldr @ B @ A @ F @ ( replicate @ B @ N @ X ) )
= ( compow @ ( A > A ) @ N @ ( F @ X ) ) ) ).
% foldr_replicate
thf(fact_3828_take__replicate,axiom,
! [A: $tType,I: nat,K: nat,X: A] :
( ( take @ A @ I @ ( replicate @ A @ K @ X ) )
= ( replicate @ A @ ( ord_min @ nat @ I @ K ) @ X ) ) ).
% take_replicate
thf(fact_3829_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_3830_tl__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( tl @ A @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ).
% tl_replicate
thf(fact_3831_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I @ X ) @ ( replicate @ B @ J @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_3832_range__constant,axiom,
! [B: $tType,A: $tType,X: A] :
( ( image2 @ B @ A
@ ^ [Uu3: B] : X
@ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_3833_inj__mapI,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ) ).
% inj_mapI
thf(fact_3834_inj__map,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
= ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_map
thf(fact_3835_set__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_3836_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X: B] : ( member2 @ A @ ( F @ X ) @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_3837_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X: B] :
( ( B2
= ( F @ X ) )
=> ( member2 @ A @ B2 @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_3838_Un__UNIV__left,axiom,
! [A: $tType,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B4 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_3839_Un__UNIV__right,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_3840_Int__UNIV__left,axiom,
! [A: $tType,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_3841_Int__UNIV__right,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) )
= A5 ) ).
% Int_UNIV_right
thf(fact_3842_UNIV__eq__I,axiom,
! [A: $tType,A5: set @ A] :
( ! [X2: A] : ( member2 @ A @ X2 @ A5 )
=> ( ( top_top @ ( set @ A ) )
= A5 ) ) ).
% UNIV_eq_I
thf(fact_3843_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : ( member2 @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_3844_append__replicate__commute,axiom,
! [A: $tType,N: nat,X: A,K: nat] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ K @ X ) )
= ( append @ A @ ( replicate @ A @ K @ X ) @ ( replicate @ A @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_3845_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert2 @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_3846_total__lexord,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lexord @ A @ R3 ) ) ) ).
% total_lexord
thf(fact_3847_subset__UNIV,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_3848_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_3849_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( A3
!= ( top_top @ A ) )
= ( ord_less @ A @ A3 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_3850_boolean__algebra_Oconj__one__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% boolean_algebra.conj_one_right
thf(fact_3851_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_3852_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_3853_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_3854_total__lenlex,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lenlex @ A @ R3 ) ) ) ).
% total_lenlex
thf(fact_3855_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_3856_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_3857_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $true ) ) ).
% UNIV_def
thf(fact_3858_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
( ( image2 @ B @ A
@ ^ [X3: B] : ( F @ ( G @ X3 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image2 @ C @ A @ F @ ( image2 @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_3859_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A] :
( ( member2 @ A @ B2 @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X2: B] :
( B2
!= ( F @ X2 ) ) ) ).
% rangeE
thf(fact_3860_infinite__UNIV__listI,axiom,
! [A: $tType] :
~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% infinite_UNIV_listI
thf(fact_3861_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X3: A] : X3
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_3862_inj__mapD,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
=> ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_mapD
thf(fact_3863_replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( replicate @ A @ ( suc @ N ) @ X )
= ( cons @ A @ X @ ( replicate @ A @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_3864_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_3865_replicate__app__Cons__same,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( append @ A @ ( replicate @ A @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_3866_replicate__length__same,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_3867_replicate__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N )
=> ( ! [Y2: A] :
( ( member2 @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate @ A @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_3868_sup__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left1
thf(fact_3869_sup__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( sup_sup @ A @ X @ B2 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left2
thf(fact_3870_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: nat,X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X ) ) ) ).
% sorted_replicate
thf(fact_3871_replicate__add,axiom,
! [A: $tType,N: nat,M2: nat,X: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N @ M2 ) @ X )
= ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ M2 @ X ) ) ) ).
% replicate_add
thf(fact_3872_filter__replicate,axiom,
! [A: $tType,P: A > $o,X: A,N: nat] :
( ( ( P @ X )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) )
& ( ~ ( P @ X )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) ) ) ).
% filter_replicate
thf(fact_3873_comm__append__are__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [M4: nat,N7: nat,Zs3: list @ A] :
( ( ( concat @ A @ ( replicate @ ( list @ A ) @ M4 @ Zs3 ) )
= Xs )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N7 @ Zs3 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_3874_range__subsetD,axiom,
! [B: $tType,A: $tType,F: B > A,B4: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) @ B4 )
=> ( member2 @ A @ ( F @ I ) @ B4 ) ) ).
% range_subsetD
thf(fact_3875_map__injective,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F @ Xs )
= ( map @ B @ A @ F @ Ys ) )
=> ( ( inj_on @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_3876_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_3877_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_3878_Compl__partition,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_3879_Compl__partition2,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ A5 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_3880_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_3881_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X3: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_3882_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z5: A] : Y5 = Z5
@ X )
@ Xs ) )
@ X )
= ( filter2 @ A
@ ( ^ [Y5: A,Z5: A] : Y5 = Z5
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_3883_UNIV__coset,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( coset @ A @ ( nil @ A ) ) ) ).
% UNIV_coset
thf(fact_3884_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F: B > A] :
( ( collect @ A
@ ^ [U3: A] :
? [X3: B] :
( U3
= ( F @ X3 ) ) )
= ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_3885_finite__Collect,axiom,
! [A: $tType,B: $tType,S: set @ A,F: B > A] :
( ( finite_finite2 @ A @ S )
=> ( ( inj_on @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [A4: B] : ( member2 @ A @ ( F @ A4 ) @ S ) ) ) ) ) ).
% finite_Collect
thf(fact_3886_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% sup_shunt
thf(fact_3887_replicate__append__same,axiom,
! [A: $tType,I: nat,X: A] :
( ( append @ A @ ( replicate @ A @ I @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( replicate @ A @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_3888_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A3: A,X: A,Y: A] :
( ( ( inf_inf @ A @ A3 @ X )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ X )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A3 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ Y )
= ( top_top @ A ) )
=> ( X = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_3889_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F: B > A,A3: A,X: B] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F @ X )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_3890_image__Int,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ F @ B4 ) ) ) ) ).
% image_Int
thf(fact_3891_lex__def,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( complete_Sup_Sup @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ nat @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( lexn @ A @ R4 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% lex_def
thf(fact_3892_map__removeAll__inj,axiom,
! [B: $tType,A: $tType,F: A > B,X: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( map @ A @ B @ F @ ( removeAll @ A @ X @ Xs ) )
= ( removeAll @ B @ ( F @ X ) @ ( map @ A @ B @ F @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_3893_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] :
( ( map @ nat @ A
@ ^ [I2: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X ) ) ).
% map_replicate_trivial
thf(fact_3894_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_3895_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_3896_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N: nat,X: A] :
( ( replicate @ A @ ( suc @ N ) @ X )
= ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% replicate_Suc_conv_snoc
thf(fact_3897_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
@ ^ [X3: A] : X3 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_3898_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_3899_inf__top_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( semila1105856199041335345_order @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_3900_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X: A,Ys: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N @ X ) @ Ys )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N @ Ys ) ) ) ).
% zip_replicate1
thf(fact_3901_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_3902_map__zip1,axiom,
! [A: $tType,B: $tType,K: B,L: list @ A] :
( ( map @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ K )
@ L )
= ( zip @ A @ B @ L @ ( replicate @ B @ ( size_size @ ( list @ A ) @ L ) @ K ) ) ) ).
% map_zip1
thf(fact_3903_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,N: nat,Y: B] :
( ( zip @ A @ B @ Xs @ ( replicate @ B @ N @ Y ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ Y )
@ ( take @ A @ N @ Xs ) ) ) ).
% zip_replicate2
thf(fact_3904_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat,Y: A] :
( ( ( cons @ A @ X @ Xs )
= ( replicate @ A @ N @ Y ) )
= ( ( X = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_3905_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [N7: nat,Zs3: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N7 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N7 @ Zs3 ) )
= ( append @ A @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_3906_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_3907_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_3908_sorted__key__list__of__set__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord144544945434240204of_set @ B @ A )
= ( ^ [F3: B > A] : ( finite_folding_F @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 ) @ ( nil @ B ) ) ) ) ) ).
% sorted_key_list_of_set_def
thf(fact_3909_map__snd__mk__snd,axiom,
! [B: $tType,A: $tType,K: A,L: list @ B] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( map @ B @ ( product_prod @ B @ A )
@ ^ [X3: B] : ( product_Pair @ B @ A @ X3 @ K )
@ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_snd_mk_snd
thf(fact_3910_Id__on__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( id_on @ A @ ( set2 @ A @ Xs ) )
= ( set2 @ ( product_prod @ A @ A )
@ ( map @ A @ ( product_prod @ A @ A )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_3911_merge__true__star,axiom,
( ( times_times @ assn @ ( top_top @ assn ) @ ( top_top @ assn ) )
= ( top_top @ assn ) ) ).
% merge_true_star
thf(fact_3912_assn__basic__inequalities_I1_J,axiom,
( ( top_top @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(1)
thf(fact_3913_assn__basic__inequalities_I5_J,axiom,
( ( top_top @ assn )
!= ( bot_bot @ assn ) ) ).
% assn_basic_inequalities(5)
thf(fact_3914_map__snd__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ A @ ( product_snd @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) )
= Xs ) ).
% map_snd_enumerate
thf(fact_3915_img__fst,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( member2 @ A @ A3 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) ) ) ).
% img_fst
thf(fact_3916_img__snd,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( member2 @ B @ B2 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S ) ) ) ).
% img_snd
thf(fact_3917_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_3918_map__fst__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_3919_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_3920_zip__map__fst__snd,axiom,
! [B: $tType,A: $tType,Zs: list @ ( product_prod @ A @ B )] :
( ( zip @ A @ B @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs ) @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs ) )
= Zs ) ).
% zip_map_fst_snd
thf(fact_3921_map__fst__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) )
= ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% map_fst_enumerate
thf(fact_3922_sorted__wrt__map__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ X3 ) @ ( product_fst @ A @ B @ Y3 ) )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ).
% sorted_wrt_map_linord
thf(fact_3923_map__fst__mk__snd,axiom,
! [B: $tType,A: $tType,K: B,L: list @ A] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ K )
@ L ) )
= L ) ).
% map_fst_mk_snd
thf(fact_3924_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_3925_sorted__wrt__map__rev__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ Y3 ) @ ( product_fst @ A @ B @ X3 ) )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ) ).
% sorted_wrt_map_rev_linord
thf(fact_3926_one__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% one_div_numeral
thf(fact_3927_one__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% one_mod_numeral
thf(fact_3928_norm__assertion__simps_I12_J,axiom,
! [X: assn] :
( ( sup_sup @ assn @ X @ ( top_top @ assn ) )
= ( top_top @ assn ) ) ).
% norm_assertion_simps(12)
thf(fact_3929_norm__assertion__simps_I11_J,axiom,
! [X: assn] :
( ( sup_sup @ assn @ ( top_top @ assn ) @ X )
= ( top_top @ assn ) ) ).
% norm_assertion_simps(11)
thf(fact_3930_norm__assertion__simps_I3_J,axiom,
! [X: assn] :
( ( inf_inf @ assn @ ( top_top @ assn ) @ X )
= X ) ).
% norm_assertion_simps(3)
thf(fact_3931_norm__assertion__simps_I4_J,axiom,
! [X: assn] :
( ( inf_inf @ assn @ X @ ( top_top @ assn ) )
= X ) ).
% norm_assertion_simps(4)
thf(fact_3932_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_3933_merge__true__star__ctx,axiom,
! [P: assn] :
( ( times_times @ assn @ ( top_top @ assn ) @ ( times_times @ assn @ ( top_top @ assn ) @ P ) )
= ( times_times @ assn @ ( top_top @ assn ) @ P ) ) ).
% merge_true_star_ctx
thf(fact_3934_fs__contract,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B > C,S: set @ C] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [P6: product_prod @ A @ B] :
( ( Uu3 = P6 )
& ( member2 @ C @ ( F @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) @ S ) ) ) )
= ( collect @ A
@ ^ [A4: A] :
? [B3: B] : ( member2 @ C @ ( F @ A4 @ B3 ) @ S ) ) ) ).
% fs_contract
thf(fact_3935_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_3936_All__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ! [A4: A,X8: B] : ( P @ A4 @ X8 ) )
= ( ! [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% All_prod_contract
thf(fact_3937_Ex__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ? [A4: A,X8: B] : ( P @ A4 @ X8 ) )
= ( ? [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% Ex_prod_contract
thf(fact_3938_sndE,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: B > $o] :
( ( X
= ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( ( P @ ( product_snd @ A @ B @ X ) )
=> ( P @ B2 ) ) ) ).
% sndE
thf(fact_3939_fstE,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: A > $o] :
( ( X
= ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( ( P @ ( product_fst @ A @ B @ X ) )
=> ( P @ A3 ) ) ) ).
% fstE
thf(fact_3940_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: nat > $o,Xs: list @ A,Is: list @ nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P6 ) ) )
@ ( zip @ A @ nat @ Xs @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P6 ) )
@ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_3941_fn__snd__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > C] :
( ( ^ [X3: product_prod @ A @ B] : ( F @ ( product_snd @ A @ B @ X3 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [Uu3: A] : F ) ) ).
% fn_snd_conv
thf(fact_3942_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ ( product_prod @ A @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( zip @ A @ B @ Xs @ Ys )
= Zs )
= ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs )
= Xs )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_3943_fn__fst__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C] :
( ( ^ [X3: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ X3 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [A4: A,Uu3: B] : ( F @ A4 ) ) ) ).
% fn_fst_conv
thf(fact_3944_update__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,I: nat,Xy: product_prod @ A @ B] :
( ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ Xy )
= ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ ( product_fst @ A @ B @ Xy ) ) @ ( list_update @ B @ Ys @ I @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% update_zip
thf(fact_3945_nths__shift__lemma,axiom,
! [A: $tType,A5: set @ nat,Xs: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member2 @ nat @ ( product_snd @ A @ nat @ P6 ) @ A5 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member2 @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A5 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_3946_in__snd__imageE,axiom,
! [A: $tType,B: $tType,Y: A,S: set @ ( product_prod @ B @ A )] :
( ( member2 @ A @ Y @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S ) )
=> ~ ! [X2: B] :
~ ( member2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ S ) ) ).
% in_snd_imageE
thf(fact_3947_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X: A,S: set @ ( product_prod @ A @ B )] :
( ( member2 @ A @ X @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) )
=> ~ ! [Y2: B] :
~ ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S ) ) ).
% in_fst_imageE
thf(fact_3948_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs3: list @ A,A7: set @ nat] :
( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member2 @ nat @ ( product_snd @ A @ nat @ P6 ) @ A7 )
@ ( zip @ A @ nat @ Xs3 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ) ).
% nths_def
thf(fact_3949_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_3950_in__set__zip,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,Xs: list @ A,Ys: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ P4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( ? [N3: nat] :
( ( ( nth @ A @ Xs @ N3 )
= ( product_fst @ A @ B @ P4 ) )
& ( ( nth @ B @ Ys @ N3 )
= ( product_snd @ A @ B @ P4 ) )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_3951_entails__solve__finalize_I1_J,axiom,
! [M: list @ ( product_prod @ assn @ assn ),P: assn] : ( fI_RESULT @ M @ P @ ( one_one @ assn ) @ ( top_top @ assn ) ) ).
% entails_solve_finalize(1)
thf(fact_3952_snd__image__mp,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ B @ A ),B4: set @ A,X: B,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A5 ) @ B4 )
=> ( ( member2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A5 )
=> ( member2 @ A @ Y @ B4 ) ) ) ).
% snd_image_mp
thf(fact_3953_fst__image__mp,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B ),B4: set @ A,X: A,Y: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 ) @ B4 )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ A5 )
=> ( member2 @ A @ X @ B4 ) ) ) ).
% fst_image_mp
thf(fact_3954_snd__in__Field,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_snd @ A @ A ) @ R ) @ ( field2 @ A @ R ) ) ).
% snd_in_Field
thf(fact_3955_fst__in__Field,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_fst @ A @ A ) @ R ) @ ( field2 @ A @ R ) ) ).
% fst_in_Field
thf(fact_3956_in__set__enumerate__eq,axiom,
! [A: $tType,P4: product_prod @ nat @ A,N: nat,Xs: list @ A] :
( ( member2 @ ( product_prod @ nat @ A ) @ P4 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P4 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P4 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P4 ) @ N ) )
= ( product_snd @ nat @ A @ P4 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_3957_sorted__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% sorted_enumerate
thf(fact_3958_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > C > A,G: A > B > C > B,A3: A,B2: B,Xs: list @ C] :
( ( product_fst @ A @ B
@ ( foldl @ ( product_prod @ A @ B ) @ C
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ B ) )
@ ^ [A4: A,B3: B,X3: C] : ( product_Pair @ A @ B @ ( F @ A4 @ X3 ) @ ( G @ A4 @ B3 @ X3 ) ) )
@ ( product_Pair @ A @ B @ A3 @ B2 )
@ Xs ) )
= ( foldl @ A @ C @ F @ A3 @ Xs ) ) ).
% fst_foldl
thf(fact_3959_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 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( Y = Z2 ) ) ) ) ).
% distinct_map_fstD
thf(fact_3960_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_3961_map__fst__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) @ Xs ) ) ).
% map_fst_zip_take
thf(fact_3962_mergesort__by__rel__split__length,axiom,
! [A: $tType,Xs1: list @ A,Xs23: 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 @ Xs23 ) @ 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 @ Xs23 ) @ Xs ) ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs23 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% mergesort_by_rel_split_length
thf(fact_3963_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S: set @ A,F: A > B > B,A5: set @ A,X: A,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( finite_folding_F @ A @ B @ F @ Z2 @ A5 )
= ( F @ X @ ( finite_folding_F @ A @ B @ F @ Z2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% folding_on.remove
thf(fact_3964_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S: set @ A,F: A > B > B,X: A,A5: set @ A,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_folding_F @ A @ B @ F @ Z2 @ ( insert2 @ A @ X @ A5 ) )
= ( F @ X @ ( finite_folding_F @ A @ B @ F @ Z2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% folding_on.insert_remove
thf(fact_3965_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_3966_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs1: list @ A,Xs23: list @ A,X12: A,X24: A,Xs: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs23 ) @ ( cons @ A @ X12 @ ( cons @ A @ X24 @ Xs ) ) )
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ Xs1 ) @ ( cons @ A @ X24 @ Xs23 ) ) @ Xs ) ) ).
% mergesort_by_rel_split.simps(3)
thf(fact_3967_mergesort__by__rel__split_Osimps_I1_J,axiom,
! [A: $tType,Xs1: list @ A,Xs23: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs23 ) @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs23 ) ) ).
% mergesort_by_rel_split.simps(1)
thf(fact_3968_folding__on_Oempty,axiom,
! [A: $tType,B: $tType,S: set @ A,F: A > B > B,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F )
=> ( ( finite_folding_F @ A @ B @ F @ Z2 @ ( bot_bot @ ( set @ A ) ) )
= Z2 ) ) ).
% folding_on.empty
thf(fact_3969_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite_folding_on @ A @ ( list @ A ) @ ( top_top @ ( set @ A ) )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3 ) ) ) ).
% sorted_list_of_set.fold_insort_key.folding_on_axioms
thf(fact_3970_mergesort__by__rel__split_Oelims,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X @ Xa )
= Y )
=> ( ! [Xs12: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs22 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs22 ) ) ) )
=> ( ! [Xs12: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs22 ) )
=> ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs12 ) @ Xs22 ) ) ) )
=> ~ ! [Xs12: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs22 ) )
=> ! [X1: A,X23: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) )
=> ( Y
!= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs12 ) @ ( cons @ A @ X23 @ Xs22 ) ) @ Xs2 ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
thf(fact_3971_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list @ A,Xs23: list @ A,X: A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs23 ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs1 ) @ Xs23 ) ) ).
% mergesort_by_rel_split.simps(2)
thf(fact_3972_mergesort__by__rel_Osimps,axiom,
! [A: $tType] :
( ( mergesort_by_rel @ A )
= ( ^ [R6: A > A > $o,Xs3: list @ A] : ( if @ ( list @ A ) @ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Xs3 @ ( merges9089515139780605204_merge @ A @ R6 @ ( mergesort_by_rel @ A @ R6 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) @ ( mergesort_by_rel @ A @ R6 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) ) ) ) ) ).
% mergesort_by_rel.simps
thf(fact_3973_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_3974_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R: A > A > $o,X12: A,X24: A,Xs: list @ A] :
( ( mergesort_by_rel @ A @ R @ ( cons @ A @ X12 @ ( cons @ A @ X24 @ Xs ) ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs13: list @ A,Xs24: list @ A] : ( merges9089515139780605204_merge @ A @ R @ ( mergesort_by_rel @ A @ R @ Xs13 ) @ ( mergesort_by_rel @ A @ R @ Xs24 ) )
@ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ ( nil @ A ) ) @ ( cons @ A @ X24 @ ( nil @ A ) ) ) @ Xs ) ) ) ).
% mergesort_by_rel_simps(3)
thf(fact_3975_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Xy2: product_prod @ A @ C] :
( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Yz: product_prod @ C @ B] :
( if @ ( list @ ( product_prod @ A @ B ) )
@ ( ( product_snd @ A @ C @ Xy2 )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy2 ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys ) ) ) ) ).
% set_relcomp
thf(fact_3976_mergesort__by__rel__merge__simps_I3_J,axiom,
! [A: $tType,R: A > A > $o,Ys: list @ A] :
( ( merges9089515139780605204_merge @ A @ R @ ( nil @ A ) @ Ys )
= Ys ) ).
% mergesort_by_rel_merge_simps(3)
thf(fact_3977_sorted__wrt__mergesort__by__rel__merge,axiom,
! [A: $tType,R: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ! [X2: A,Y2: A] :
( ( R @ X2 @ Y2 )
| ( R @ Y2 @ X2 ) )
=> ( ! [X2: A,Y2: A,Z4: A] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z4 )
=> ( R @ X2 @ Z4 ) ) )
=> ( ( sorted_wrt @ A @ R @ ( merges9089515139780605204_merge @ A @ R @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ R @ Xs )
& ( sorted_wrt @ A @ R @ Ys ) ) ) ) ) ).
% sorted_wrt_mergesort_by_rel_merge
thf(fact_3978_set__mergesort__by__rel__merge,axiom,
! [A: $tType,R: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( merges9089515139780605204_merge @ A @ R @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_mergesort_by_rel_merge
thf(fact_3979_mergesort__by__rel__merge__simps_I1_J,axiom,
! [A: $tType,R: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ( R @ X @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ X @ ( merges9089515139780605204_merge @ A @ R @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ) )
& ( ~ ( R @ X @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ Y @ ( merges9089515139780605204_merge @ A @ R @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_simps(1)
thf(fact_3980_mergesort__by__rel__merge__simps_I2_J,axiom,
! [A: $tType,R: A > A > $o,Xs: list @ A] :
( ( merges9089515139780605204_merge @ A @ R @ Xs @ ( nil @ A ) )
= Xs ) ).
% mergesort_by_rel_merge_simps(2)
thf(fact_3981_mergesort__by__rel__merge_Osimps_I3_J,axiom,
! [A: $tType,R: A > A > $o,V: A,Va: list @ A] :
( ( merges9089515139780605204_merge @ A @ R @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
= ( cons @ A @ V @ Va ) ) ).
% mergesort_by_rel_merge.simps(3)
thf(fact_3982_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 )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A,Ys2: list @ A] :
( ( Xb
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( ( ( X @ X2 @ Y2 )
=> ( Y
= ( cons @ A @ X2 @ ( merges9089515139780605204_merge @ A @ X @ Xs2 @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) )
& ( ~ ( X @ X2 @ Y2 )
=> ( Y
= ( cons @ A @ Y2 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X2 @ Xs2 ) @ Ys2 ) ) ) ) ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V2: A,Va2: list @ A] :
( ( Xb
= ( cons @ A @ V2 @ Va2 ) )
=> ( Y
!= ( cons @ A @ V2 @ Va2 ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.elims
thf(fact_3983_trancl__Int__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 ) @ S4 ) @ R3 ) @ S4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 ) @ S4 ) ) ) ).
% trancl_Int_subset
thf(fact_3984_mergesort__by__rel__merge_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A,Ys2: list @ A] :
( ( Xb
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ( ( ( X @ X2 @ Y2 )
=> ( Y
= ( cons @ A @ X2 @ ( merges9089515139780605204_merge @ A @ X @ Xs2 @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) )
& ( ~ ( X @ X2 @ Y2 )
=> ( Y
= ( cons @ A @ Y2 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X2 @ Xs2 ) @ 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 @ X2 @ Xs2 ) @ ( 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 ) )
=> ! [V2: A,Va2: list @ A] :
( ( Xb
= ( cons @ A @ V2 @ Va2 ) )
=> ( ( Y
= ( cons @ A @ V2 @ Va2 ) )
=> ~ ( 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 @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
thf(fact_3985_mergesort__by__rel_Opsimps,axiom,
! [A: $tType,R: A > A > $o,Xs: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R @ Xs ) )
=> ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R @ Xs )
= Xs ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R @ Xs )
= ( merges9089515139780605204_merge @ A @ R @ ( mergesort_by_rel @ A @ R @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) @ ( mergesort_by_rel @ A @ R @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) ) ) ) ) ) ).
% mergesort_by_rel.psimps
thf(fact_3986_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_3987_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 ) )
=> ( ! [R5: A > A > $o,Xs2: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R5 @ Xs2 ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R5 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R5 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
=> ( P @ R5 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% mergesort_by_rel.pinduct
thf(fact_3988_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa ) )
=> ( ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ( Y = X2 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X2: A,Y2: A,Zs3: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs3 ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X @ X2 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs3 ) ) ) ) @ X2 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y2 @ Zs3 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs3 ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_3989_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys2 ) )
=> ( ( Y
= ( ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ Ys2 ) )
=> ( X @ X2 @ Y3 ) )
& ( sorted_wrt @ A @ X @ Ys2 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys2 ) ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
thf(fact_3990_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
=> ~ ! [X2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys2 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys2 ) ) )
=> ~ ( ! [Xa2: A] :
( ( member2 @ A @ Xa2 @ ( set2 @ A @ Ys2 ) )
=> ( X @ X2 @ Xa2 ) )
& ( sorted_wrt @ A @ X @ Ys2 ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
thf(fact_3991_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ! [X2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys2 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys2 ) ) )
=> ( ! [Xa3: A] :
( ( member2 @ A @ Xa3 @ ( set2 @ A @ Ys2 ) )
=> ( X @ X2 @ Xa3 ) )
& ( sorted_wrt @ A @ X @ Ys2 ) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
thf(fact_3992_finite__range__prod,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > ( product_prod @ A @ C )] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ ( comp @ ( product_prod @ A @ C ) @ A @ B @ ( product_fst @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ( finite_finite2 @ C @ ( image2 @ B @ C @ ( comp @ ( product_prod @ A @ C ) @ C @ B @ ( product_snd @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ C ) @ ( image2 @ B @ ( product_prod @ A @ C ) @ F @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% finite_range_prod
thf(fact_3993_remdups__adj__singleton__iff,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( Xs
!= ( nil @ A ) )
& ( Xs
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ ( hd @ A @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_3994_INT__simps_I4_J,axiom,
! [G4: $tType,H3: $tType,C4: set @ H3,A5: set @ G4,B4: H3 > ( set @ G4 )] :
( ( ( C4
= ( bot_bot @ ( set @ H3 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G4 )
@ ( image2 @ H3 @ ( set @ G4 )
@ ^ [X3: H3] : ( minus_minus @ ( set @ G4 ) @ A5 @ ( B4 @ X3 ) )
@ C4 ) )
= ( top_top @ ( set @ G4 ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ H3 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G4 )
@ ( image2 @ H3 @ ( set @ G4 )
@ ^ [X3: H3] : ( minus_minus @ ( set @ G4 ) @ A5 @ ( B4 @ X3 ) )
@ C4 ) )
= ( minus_minus @ ( set @ G4 ) @ A5 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image2 @ H3 @ ( set @ G4 ) @ B4 @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_3995_card__quotient__disjoint,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [X3: A] : ( equiv_quotient @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
@ A5 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R3 ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_3996_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,Xs: list @ C] :
( ( map @ B @ A @ F @ ( map @ C @ B @ G @ Xs ) )
= ( map @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_3997_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,List: list @ A] :
( ( map @ B @ C @ F @ ( map @ A @ B @ G @ List ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_3998_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: list @ A] :
( ( map @ B @ C @ G @ ( map @ A @ B @ F @ V ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_3999_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F ) @ ( map @ A @ B @ G ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% List.map.comp
thf(fact_4000_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C] :
( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F ) @ ( map @ A @ C @ G ) )
= ( map @ A @ B @ ( comp @ C @ B @ A @ F @ G ) ) ) ).
% map_comp_map
thf(fact_4001_remdups__adj__Nil__iff,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% remdups_adj_Nil_iff
thf(fact_4002_remdups__adj__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups_adj @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% remdups_adj_set
thf(fact_4003_remdups__adj__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups_adj @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_rev
thf(fact_4004_hd__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% hd_remdups_adj
thf(fact_4005_last__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( last @ A @ ( remdups_adj @ A @ Xs ) )
= ( last @ A @ Xs ) ) ).
% last_remdups_adj
thf(fact_4006_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( ( complete_Inf_Inf @ A @ A5 )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
=> ? [Y3: A] :
( ( member2 @ A @ Y3 @ A5 )
& ( ord_less @ A @ Y3 @ X3 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_4007_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_4008_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_4009_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_4010_Inf__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: A,A5: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ A5 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A5 ) ) ) ) ).
% Inf_insert
thf(fact_4011_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_4012_remdups__adj__Cons__alt,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_4013_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : F
@ A5 ) )
= F ) ) ) ).
% INF_const
thf(fact_4014_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% cINF_const
thf(fact_4015_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A5: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) )
= ( bot_bot @ A ) )
= ( ! [X3: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X3 )
=> ? [Y3: B] :
( ( member2 @ B @ Y3 @ A5 )
& ( ord_less @ A @ ( F @ Y3 ) @ X3 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_4016_INT__constant,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A5 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% INT_constant
thf(fact_4017_INT__insert,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A3: B,A5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( insert2 @ B @ A3 @ A5 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B4 @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% INT_insert
thf(fact_4018_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_4019_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ D,A5: set @ C,B4: D > ( set @ C )] :
( ( ( C4
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X3: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B4 @ X3 ) )
@ C4 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X3: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B4 @ X3 ) )
@ C4 ) )
= ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B4 @ C4 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_4020_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ A,A5: A > ( set @ B ),B4: set @ B] :
( ( ( C4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X3: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X3 ) @ B4 )
@ C4 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X3: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X3 ) @ B4 )
@ C4 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C4 ) ) @ B4 ) ) ) ) ).
% INT_simps(1)
thf(fact_4021_INT__simps_I3_J,axiom,
! [E4: $tType,F6: $tType,C4: set @ E4,A5: E4 > ( set @ F6 ),B4: set @ F6] :
( ( ( C4
= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F6 )
@ ( image2 @ E4 @ ( set @ F6 )
@ ^ [X3: E4] : ( minus_minus @ ( set @ F6 ) @ ( A5 @ X3 ) @ B4 )
@ C4 ) )
= ( top_top @ ( set @ F6 ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F6 )
@ ( image2 @ E4 @ ( set @ F6 )
@ ^ [X3: E4] : ( minus_minus @ ( set @ F6 ) @ ( A5 @ X3 ) @ B4 )
@ C4 ) )
= ( minus_minus @ ( set @ F6 ) @ ( complete_Inf_Inf @ ( set @ F6 ) @ ( image2 @ E4 @ ( set @ F6 ) @ A5 @ C4 ) ) @ B4 ) ) ) ) ).
% INT_simps(3)
thf(fact_4022_remdups__adj_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups_adj @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups_adj.simps(1)
thf(fact_4023_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X: A > B,Y: A > B,F: C > A] :
( ( X = Y )
=> ( ( comp @ A @ B @ C @ X @ F )
= ( comp @ A @ B @ C @ Y @ F ) ) ) ).
% comp_cong_left
thf(fact_4024_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > B,Y: A > B,F: B > C] :
( ( X = Y )
=> ( ( comp @ B @ C @ A @ F @ X )
= ( comp @ B @ C @ A @ F @ Y ) ) ) ).
% comp_cong_right
thf(fact_4025_fun__comp__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C,Fg: A > B] :
( ( ( comp @ C @ B @ A @ F @ G )
= Fg )
= ( ! [X3: A] :
( ( F @ ( G @ X3 ) )
= ( Fg @ X3 ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_4026_remdups__adj__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups_adj @ A @ Xs )
= Xs ) ) ).
% remdups_adj_distinct
thf(fact_4027_remdups__adj_Osimps_I3_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
= ( cons @ A @ X @ ( remdups_adj @ A @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_4028_distinct__adj__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] : ( distinct_adj @ A @ ( remdups_adj @ A @ Xs ) ) ).
% distinct_adj_remdups_adj
thf(fact_4029_distinct__adj__altdef,axiom,
! [A: $tType] :
( ( distinct_adj @ A )
= ( ^ [Xs3: list @ A] :
( ( remdups_adj @ A @ Xs3 )
= Xs3 ) ) ) ).
% distinct_adj_altdef
thf(fact_4030_INF__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: $o > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ $o @ A @ A5 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ A @ ( A5 @ $true ) @ ( A5 @ $false ) ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_4031_INT__bool__eq,axiom,
! [A: $tType,A5: $o > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ $o @ ( set @ A ) @ A5 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ ( set @ A ) @ ( A5 @ $true ) @ ( A5 @ $false ) ) ) ).
% INT_bool_eq
thf(fact_4032_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,U: A] :
( ! [V2: A] :
( ( member2 @ A @ V2 @ A5 )
=> ( ord_less_eq @ A @ V2 @ U ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_4033_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A3: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ X5 )
=> ( ord_less_eq @ A @ A3 @ X2 ) )
=> ( ! [Y2: A] :
( ! [X6: A] :
( ( member2 @ A @ X6 @ X5 )
=> ( ord_less_eq @ A @ Y2 @ X6 ) )
=> ( ord_less_eq @ A @ Y2 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X5 )
= A3 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4034_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,Z2: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ X5 )
=> ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ X5 ) ) ) ) ) ).
% cInf_greatest
thf(fact_4035_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,Z2: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X5 ) @ Z2 )
=> ? [X2: A] :
( ( member2 @ A @ X2 @ X5 )
& ( ord_less @ A @ X2 @ Z2 ) ) ) ) ) ).
% cInf_lessD
thf(fact_4036_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F: B > A,X: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ I5 )
=> ( ( F @ I3 )
= X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I5 ) )
= X ) ) ) ) ).
% INF_eq_const
thf(fact_4037_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_4038_Inf__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A3: A,A5: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ A5 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A5 ) ) ) ) ).
% Inf_finite_insert
thf(fact_4039_Inf__union__distrib,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B4 ) ) ) ) ).
% Inf_union_distrib
thf(fact_4040_foldr_Ofoldr__Cons,axiom,
! [B: $tType,A: $tType,F: A > B > B,X: A,Xs: list @ A] :
( ( foldr @ A @ B @ F @ ( cons @ A @ X @ Xs ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( foldr @ A @ B @ F @ Xs ) ) ) ).
% foldr.foldr_Cons
thf(fact_4041_remdups__adj_Oelims,axiom,
! [A: $tType,X: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ( ! [X2: A] :
( ( X
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( Y
!= ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ~ ( ( ( X2 = Y2 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X2 @ Xs2 ) ) ) )
& ( ( X2 != Y2 )
=> ( Y
= ( cons @ A @ X2 @ ( remdups_adj @ A @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_4042_remdups__adj_Osimps_I2_J,axiom,
! [A: $tType,X: A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% remdups_adj.simps(2)
thf(fact_4043_remdups__adj__length,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% remdups_adj_length
thf(fact_4044_sorted__remdups__adj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% sorted_remdups_adj
thf(fact_4045_INF__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K: B,I5: set @ B,A5: B > A] :
( ( member2 @ B @ K @ I5 )
=> ( ( inf_inf @ A @ ( A5 @ K ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ A5 @ I5 ) ) )
= ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ A5 @ I5 ) ) ) ) ) ).
% INF_absorb
thf(fact_4046_INF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A5: set @ B,G: B > A] :
( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A4: B] : ( inf_inf @ A @ ( F @ A4 ) @ ( G @ A4 ) )
@ A5 ) ) ) ) ).
% INF_inf_distrib
thf(fact_4047_Inter__insert,axiom,
! [A: $tType,A3: set @ A,B4: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( insert2 @ ( set @ A ) @ A3 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B4 ) ) ) ).
% Inter_insert
thf(fact_4048_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_4049_Inter__Un__distrib,axiom,
! [A: $tType,A5: set @ ( set @ A ),B4: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B4 ) ) ) ).
% Inter_Un_distrib
thf(fact_4050_INT__absorb,axiom,
! [B: $tType,A: $tType,K: A,I5: set @ A,A5: A > ( set @ B )] :
( ( member2 @ A @ K @ I5 )
=> ( ( inf_inf @ ( set @ B ) @ ( A5 @ K ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) )
= ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) ) ) ).
% INT_absorb
thf(fact_4051_INT__Int__distrib,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B4: B > ( set @ A ),I5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] : ( inf_inf @ ( set @ A ) @ ( A5 @ I2 ) @ ( B4 @ I2 ) )
@ I5 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ I5 ) ) ) ) ).
% INT_Int_distrib
thf(fact_4052_Int__Inter__image,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B4: B > ( set @ A ),C4: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X3: B] : ( inf_inf @ ( set @ A ) @ ( A5 @ X3 ) @ ( B4 @ X3 ) )
@ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ C4 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C4 ) ) ) ) ).
% Int_Inter_image
thf(fact_4053_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,M2: A,F: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ A5 )
=> ( ord_less_eq @ A @ M2 @ ( F @ X2 ) ) )
=> ( ord_less_eq @ A @ M2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_4054_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F: B > A,C3: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ C3 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I5 ) )
= C3 )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ I5 )
=> ( ( F @ X3 )
= C3 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4055_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,A3: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X5 ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ X5 )
=> ( ord_less @ A @ A3 @ X3 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_4056_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ).
% Inf_le_Sup
thf(fact_4057_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] :
( ( member2 @ A @ X2 @ A5 )
=> ( ( member2 @ A @ Y2 @ A5 )
=> ( member2 @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ A5 ) ) )
=> ( member2 @ A @ ( complete_Inf_Inf @ A @ A5 ) @ A5 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_4058_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S ) ) @ A3 ) ) ) ) ).
% cInf_abs_ge
thf(fact_4059_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B4 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_4060_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_4061_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_4062_inf__Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A5 ) @ B4 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ B4 @ A5 ) ) ) ) ).
% inf_Inf_fold_inf
thf(fact_4063_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ).
% Min_Inf
thf(fact_4064_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X5 )
= ( lattic643756798350308766er_Min @ A @ X5 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_4065_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_4066_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X5 )
= ( lattic7752659483105999362nf_fin @ A @ X5 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_4067_remdups__adj__append__two,axiom,
! [A: $tType,Xs: list @ A,X: A,Y: A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X = Y ) @ ( nil @ A ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_append_two
thf(fact_4068_remdups__adj__map__injective,axiom,
! [B: $tType,A: $tType,F: A > B,Xs: list @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( remdups_adj @ B @ ( map @ A @ B @ F @ Xs ) )
= ( map @ A @ B @ F @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% remdups_adj_map_injective
thf(fact_4069_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_4070_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A5 ) )
= ( top_top @ A ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y3: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% INF_constant
thf(fact_4071_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,X: A,F: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( inf_inf @ A @ X @ ( F @ I2 ) )
@ I5 ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I5 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_4072_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F: B > A,X: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( inf_inf @ A @ ( F @ I2 ) @ X )
@ I5 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I5 ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_4073_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A3: B,A5: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( insert2 @ B @ A3 @ A5 ) ) )
= ( inf_inf @ A @ ( F @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ).
% INF_insert
thf(fact_4074_INF__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M: B > A,A5: set @ B,B4: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M @ B4 ) ) ) ) ) ).
% INF_union
thf(fact_4075_INT__empty,axiom,
! [B: $tType,A: $tType,B4: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_4076_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ D,A5: set @ C,B4: D > ( set @ C )] :
( ( ( C4
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B4 @ C4 ) ) )
= A5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B4 @ C4 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X3: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B4 @ X3 ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_4077_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C4: set @ A,A5: A > ( set @ B ),B4: set @ B] :
( ( ( C4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C4 ) ) @ B4 )
= B4 ) )
& ( ( C4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C4 ) ) @ B4 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X3: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X3 ) @ B4 )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_4078_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,F: B > C,A5: A > ( set @ B )] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member2 @ A @ I3 @ I5 )
=> ( inj_on @ B @ C @ F @ ( A5 @ I3 ) ) )
=> ( inj_on @ B @ C @ F @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_4079_INT__Un,axiom,
! [A: $tType,B: $tType,M: B > ( set @ A ),A5: set @ B,B4: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M @ A5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M @ B4 ) ) ) ) ).
% INT_Un
thf(fact_4080_Inter__Un__subset,axiom,
! [A: $tType,A5: set @ ( set @ A ),B4: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B4 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A5 @ B4 ) ) ) ).
% Inter_Un_subset
thf(fact_4081_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,H: B > C,G: C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X2: B,Y2: B] :
( ( member2 @ B @ X2 @ A5 )
=> ( ( member2 @ B @ Y2 @ A5 )
=> ( ( X2 != Y2 )
=> ( ( ( H @ X2 )
= ( H @ Y2 ) )
=> ( ( G @ ( H @ X2 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A5 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_4082_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B12: set @ ( set @ A ),A5: set @ A] :
( ( ( B12
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ A5 )
= A5 ) )
& ( ( B12
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ A5 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 )
@ B12 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_4083_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B12: set @ ( set @ A ),A5: set @ A] :
( ( ( B12
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) )
= A5 ) )
& ( ( B12
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 ) @ B12 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_4084_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_4085_remdups__adj__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( case_list @ ( list @ A ) @ A @ ( cons @ A @ X @ ( nil @ A ) )
@ ^ [Y3: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( X = Y3 ) @ ( cons @ A @ Y3 @ Xs3 ) @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs3 ) ) )
@ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_4086_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S ) @ L ) ) @ E3 ) ) ) ) ).
% cInf_asclose
thf(fact_4087_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_4088_Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( complete_Inf_Inf @ A @ A5 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ A5 ) ) ) ) ).
% Inf_fold_inf
thf(fact_4089_remdups__adj__adjacent,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I )
!= ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_4090_remdups__adj__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_replicate
thf(fact_4091_remdups__adj__singleton,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( remdups_adj @ A @ Xs )
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( Xs
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_4092_remdups__adj__append,axiom,
! [A: $tType,Xs_1: list @ A,X: A,Xs_2: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ Xs_2 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_4093_INT__extend__simps_I3_J,axiom,
! [F6: $tType,E4: $tType,C4: set @ E4,A5: E4 > ( set @ F6 ),B4: set @ F6] :
( ( ( C4
= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( minus_minus @ ( set @ F6 ) @ ( complete_Inf_Inf @ ( set @ F6 ) @ ( image2 @ E4 @ ( set @ F6 ) @ A5 @ C4 ) ) @ B4 )
= ( minus_minus @ ( set @ F6 ) @ ( top_top @ ( set @ F6 ) ) @ B4 ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( minus_minus @ ( set @ F6 ) @ ( complete_Inf_Inf @ ( set @ F6 ) @ ( image2 @ E4 @ ( set @ F6 ) @ A5 @ C4 ) ) @ B4 )
= ( complete_Inf_Inf @ ( set @ F6 )
@ ( image2 @ E4 @ ( set @ F6 )
@ ^ [X3: E4] : ( minus_minus @ ( set @ F6 ) @ ( A5 @ X3 ) @ B4 )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_4094_INT__extend__simps_I4_J,axiom,
! [G4: $tType,H3: $tType,C4: set @ H3,A5: set @ G4,B4: H3 > ( set @ G4 )] :
( ( ( C4
= ( bot_bot @ ( set @ H3 ) ) )
=> ( ( minus_minus @ ( set @ G4 ) @ A5 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image2 @ H3 @ ( set @ G4 ) @ B4 @ C4 ) ) )
= A5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ H3 ) ) )
=> ( ( minus_minus @ ( set @ G4 ) @ A5 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image2 @ H3 @ ( set @ G4 ) @ B4 @ C4 ) ) )
= ( complete_Inf_Inf @ ( set @ G4 )
@ ( image2 @ H3 @ ( set @ G4 )
@ ^ [X3: H3] : ( minus_minus @ ( set @ G4 ) @ A5 @ ( B4 @ X3 ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_4095_remdups__adj__append_H,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ A ) )
| ( ( last @ A @ Xs )
!= ( hd @ A @ Ys ) ) )
=> ( ( remdups_adj @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remdups_adj @ A @ Xs ) @ ( remdups_adj @ A @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_4096_remdups__adj__length__ge1,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_4097_quotient__empty,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( bot_bot @ ( set @ A ) ) @ R3 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% quotient_empty
thf(fact_4098_quotient__is__empty,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ( equiv_quotient @ A @ A5 @ R3 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty
thf(fact_4099_quotient__is__empty2,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ( bot_bot @ ( set @ ( set @ A ) ) )
= ( equiv_quotient @ A @ A5 @ R3 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty2
thf(fact_4100_quotient__diff1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,A3: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A4: A] : ( equiv_quotient @ A @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
@ A5 )
=> ( ( member2 @ A @ A3 @ A5 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ R3 )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A5 @ R3 ) @ ( equiv_quotient @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) @ R3 ) ) ) ) ) ).
% quotient_diff1
thf(fact_4101_size__list__map,axiom,
! [A: $tType,B: $tType,F: A > nat,G: B > A,Xs: list @ B] :
( ( size_list @ A @ F @ ( map @ B @ A @ G @ Xs ) )
= ( size_list @ B @ ( comp @ A @ nat @ B @ F @ G ) @ Xs ) ) ).
% size_list_map
thf(fact_4102_subset__mset_OcINF__const,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X3: B] : C3
@ A5 ) )
= C3 ) ) ).
% subset_mset.cINF_const
thf(fact_4103_length__filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F @ Xs ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_4104_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: B > nat,G: A > B] :
( ( comp @ ( list @ B ) @ nat @ ( list @ A ) @ ( size_list @ B @ F ) @ ( map @ A @ B @ G ) )
= ( size_list @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% list.size_gen_o_map
thf(fact_4105_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: A > C,G: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) )
@ F )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image2 @ D @ B @ G )
@ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_4106_UNIV__bool,axiom,
( ( top_top @ ( set @ $o ) )
= ( insert2 @ $o @ $false @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% UNIV_bool
thf(fact_4107_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ Y )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ X )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3
@ Y ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
thf(fact_4108_filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs: list @ B] :
( ( filter2 @ A @ P @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs ) ) ) ).
% filter_map
thf(fact_4109_takeWhile__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs: list @ B] :
( ( takeWhile @ A @ P @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( takeWhile @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs ) ) ) ).
% takeWhile_map
thf(fact_4110_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: B > A > A,F: C > B,Xs: list @ C,A3: A] :
( ( foldr @ B @ A @ G @ ( map @ C @ B @ F @ Xs ) @ A3 )
= ( foldr @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F ) @ Xs @ A3 ) ) ).
% foldr_map
thf(fact_4111_rotate__add,axiom,
! [A: $tType,M2: nat,N: nat] :
( ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( rotate @ A @ M2 ) @ ( rotate @ A @ N ) ) ) ).
% rotate_add
thf(fact_4112_folding__insort__key_Oinsort__key__commute,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A,X: B,Y: B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( ( member2 @ B @ X @ S )
=> ( ( member2 @ B @ Y @ S )
=> ( ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F @ Y ) @ ( insort_key @ A @ B @ Less_eq @ F @ X ) )
= ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F @ X ) @ ( insort_key @ A @ B @ Less_eq @ F @ Y ) ) ) ) ) ) ).
% folding_insort_key.insort_key_commute
thf(fact_4113_set__oo__map__alt,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ ( list @ B ) @ ( set @ B ) @ ( list @ A ) @ ( set2 @ B ) @ ( map @ A @ B @ F ) )
= ( ^ [L4: list @ A] : ( image2 @ A @ B @ F @ ( set2 @ A @ L4 ) ) ) ) ).
% set_oo_map_alt
thf(fact_4114_zip__takeWhile__fst,axiom,
! [A: $tType,B: $tType,P: A > $o,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( takeWhile @ A @ P @ Xs ) @ Ys )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ A @ $o @ ( product_prod @ A @ B ) @ P @ ( product_fst @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_takeWhile_fst
thf(fact_4115_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_4116_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B4: set @ ( set @ B ),G: B > A] :
( ! [X2: set @ B] :
( ( member2 @ ( set @ B ) @ X2 @ B4 )
=> ( finite_finite2 @ B @ X2 ) )
=> ( ! [A15: set @ B] :
( ( member2 @ ( set @ B ) @ A15 @ B4 )
=> ! [A24: set @ B] :
( ( member2 @ ( set @ B ) @ A24 @ B4 )
=> ( ( A15 != A24 )
=> ! [X2: B] :
( ( member2 @ B @ X2 @ A15 )
=> ( ( member2 @ B @ X2 @ A24 )
=> ( ( G @ X2 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B4 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_4117_remove__rev__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X3: A] :
( filter_rev @ A
@ ( comp @ $o @ $o @ A @ (~)
@ ( ^ [Y5: A,Z5: A] : Y5 = Z5
@ X3 ) ) ) ) ) ).
% remove_rev_def
thf(fact_4118_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G2 ) @ ( one_one @ A ) ) ) ) ) ).
% prod.eq_fold
thf(fact_4119_remdup__sort__mergesort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( remdups @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 ) )
= ( mergesort_remdups @ A ) ) ) ).
% remdup_sort_mergesort_remdups
thf(fact_4120_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C4: set @ ( set @ B ),G: B > A] :
( ! [X2: set @ B] :
( ( member2 @ ( set @ B ) @ X2 @ C4 )
=> ( finite_finite2 @ B @ X2 ) )
=> ( ! [X2: set @ B] :
( ( member2 @ ( set @ B ) @ X2 @ C4 )
=> ! [Xa3: set @ B] :
( ( member2 @ ( set @ B ) @ Xa3 @ C4 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X2 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ C4 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_4121_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ ( set @ B ),G: B > A] :
( ! [X2: set @ B] :
( ( member2 @ ( set @ B ) @ X2 @ C4 )
=> ( finite_finite2 @ B @ X2 ) )
=> ( ! [X2: set @ B] :
( ( member2 @ ( set @ B ) @ X2 @ C4 )
=> ! [Xa3: set @ B] :
( ( member2 @ ( set @ B ) @ Xa3 @ C4 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X2 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ C4 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_4122_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B4: A,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( inf_inf @ A @ B4 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F ) @ B4 @ A5 ) ) ) ) ).
% inf_INF_fold_inf
thf(fact_4123_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_4124_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_4125_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F ) @ ( bot_bot @ A ) @ A5 ) ) ) ) ).
% SUP_fold_sup
thf(fact_4126_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F ) @ ( top_top @ A ) @ A5 ) ) ) ) ).
% INF_fold_inf
thf(fact_4127_remdups__adj_Opelims,axiom,
! [A: $tType,X: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( nil @ A ) ) ) )
=> ( ! [X2: A] :
( ( X
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ( Y
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( ( ( ( X2 = Y2 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X2 @ Xs2 ) ) ) )
& ( ( X2 != Y2 )
=> ( Y
= ( cons @ A @ X2 @ ( remdups_adj @ A @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_4128_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys )
= ( ? [F3: nat > nat] :
( ( order_mono @ nat @ nat @ F3 )
& ( ( image2 @ nat @ nat @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ ( F3 @ I2 ) ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
= ( ( F3 @ I2 )
= ( F3 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_4129_range__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( product_prod @ A @ B ) @ F @ ( top_top @ ( set @ C ) ) )
@ ( product_Sigma @ A @ B @ ( image2 @ C @ A @ ( comp @ ( product_prod @ A @ B ) @ A @ C @ ( product_fst @ A @ B ) @ F ) @ ( top_top @ ( set @ C ) ) )
@ ^ [Uu3: A] : ( image2 @ C @ B @ ( comp @ ( product_prod @ A @ B ) @ B @ C @ ( product_snd @ A @ B ) @ F ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% range_prod
thf(fact_4130_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_4131_Cons__in__lists__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,A5: set @ A] :
( ( member2 @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( lists @ A @ A5 ) )
= ( ( member2 @ A @ X @ A5 )
& ( member2 @ ( list @ A ) @ Xs @ ( lists @ A @ A5 ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_4132_in__listsI,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( member2 @ A @ X2 @ A5 ) )
=> ( member2 @ ( list @ A ) @ Xs @ ( lists @ A @ A5 ) ) ) ).
% in_listsI
thf(fact_4133_lists__Int__eq,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( lists @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ A5 ) @ ( lists @ A @ B4 ) ) ) ).
% lists_Int_eq
thf(fact_4134_append__in__lists__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,A5: set @ A] :
( ( member2 @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( lists @ A @ A5 ) )
= ( ( member2 @ ( list @ A ) @ Xs @ ( lists @ A @ A5 ) )
& ( member2 @ ( list @ A ) @ Ys @ ( lists @ A @ A5 ) ) ) ) ).
% append_in_lists_conv
thf(fact_4135_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,A5: set @ A,X5: set @ B] :
( ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : X5 )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_UNIV_cancel
thf(fact_4136_lists__UNIV,axiom,
! [A: $tType] :
( ( lists @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% lists_UNIV
thf(fact_4137_set__product,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( product_Sigma @ A @ B @ ( set2 @ A @ Xs )
@ ^ [Uu3: A] : ( set2 @ B @ Ys ) ) ) ).
% set_product
thf(fact_4138_pairself__image__cart,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ B,B4: set @ B] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F )
@ ( product_Sigma @ B @ B @ A5
@ ^ [Uu3: B] : B4 ) )
= ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F @ A5 )
@ ^ [Uu3: A] : ( image2 @ B @ A @ F @ B4 ) ) ) ).
% pairself_image_cart
thf(fact_4139_lists_ONil,axiom,
! [A: $tType,A5: set @ A] : ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( lists @ A @ A5 ) ) ).
% lists.Nil
thf(fact_4140_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% mono_add
thf(fact_4141_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,M2: A,N: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_min @ B @ ( F @ M2 ) @ ( F @ N ) )
= ( F @ ( ord_min @ A @ M2 @ N ) ) ) ) ) ).
% min_of_mono
thf(fact_4142_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,M2: A,N: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_max @ B @ ( F @ M2 ) @ ( F @ N ) )
= ( F @ ( ord_max @ A @ M2 @ N ) ) ) ) ) ).
% max_of_mono
thf(fact_4143_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% mono_strict_invE
thf(fact_4144_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% monoD
thf(fact_4145_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% monoE
thf(fact_4146_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( order_mono @ A @ B @ F ) ) ) ).
% monoI
thf(fact_4147_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F3: A > B] :
! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) ) ) ).
% mono_def
thf(fact_4148_lists__IntI,axiom,
! [A: $tType,L: list @ A,A5: set @ A,B4: set @ A] :
( ( member2 @ ( list @ A ) @ L @ ( lists @ A @ A5 ) )
=> ( ( member2 @ ( list @ A ) @ L @ ( lists @ A @ B4 ) )
=> ( member2 @ ( list @ A ) @ L @ ( lists @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ).
% lists_IntI
thf(fact_4149_listsE,axiom,
! [A: $tType,X: A,L: list @ A,A5: set @ A] :
( ( member2 @ ( list @ A ) @ ( cons @ A @ X @ L ) @ ( lists @ A @ A5 ) )
=> ~ ( ( member2 @ A @ X @ A5 )
=> ~ ( member2 @ ( list @ A ) @ L @ ( lists @ A @ A5 ) ) ) ) ).
% listsE
thf(fact_4150_lists_OCons,axiom,
! [A: $tType,A3: A,A5: set @ A,L: list @ A] :
( ( member2 @ A @ A3 @ A5 )
=> ( ( member2 @ ( list @ A ) @ L @ ( lists @ A @ A5 ) )
=> ( member2 @ ( list @ A ) @ ( cons @ A @ A3 @ L ) @ ( lists @ A @ A5 ) ) ) ) ).
% lists.Cons
thf(fact_4151_listrel__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R3 )
@ ( product_Sigma @ ( list @ A ) @ ( list @ A ) @ ( lists @ A @ A5 )
@ ^ [Uu3: list @ A] : ( lists @ A @ A5 ) ) ) ) ).
% listrel_subset
thf(fact_4152_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I2: nat] : ( compow @ ( A > A ) @ I2 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_4153_in__lists__conv__set,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( member2 @ ( list @ A ) @ Xs @ ( lists @ A @ A5 ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( member2 @ A @ X3 @ A5 ) ) ) ) ).
% in_lists_conv_set
thf(fact_4154_in__listsD,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( member2 @ ( list @ A ) @ Xs @ ( lists @ A @ A5 ) )
=> ! [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
=> ( member2 @ A @ X6 @ A5 ) ) ) ).
% in_listsD
thf(fact_4155_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mono_invE
thf(fact_4156_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F: A > B,A5: A,B4: A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( F @ ( inf_inf @ A @ A5 @ B4 ) ) @ ( inf_inf @ B @ ( F @ A5 ) @ ( F @ B4 ) ) ) ) ) ).
% mono_inf
thf(fact_4157_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F: A > B,A5: A,B4: A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F @ A5 ) @ ( F @ B4 ) ) @ ( F @ ( sup_sup @ A @ A5 @ B4 ) ) ) ) ) ).
% mono_sup
thf(fact_4158_lists_Osimps,axiom,
! [A: $tType,A3: list @ A,A5: set @ A] :
( ( member2 @ ( list @ A ) @ A3 @ ( lists @ A @ A5 ) )
= ( ( A3
= ( nil @ A ) )
| ? [A4: A,L4: list @ A] :
( ( A3
= ( cons @ A @ A4 @ L4 ) )
& ( member2 @ A @ A4 @ A5 )
& ( member2 @ ( list @ A ) @ L4 @ ( lists @ A @ A5 ) ) ) ) ) ).
% lists.simps
thf(fact_4159_lists_Ocases,axiom,
! [A: $tType,A3: list @ A,A5: set @ A] :
( ( member2 @ ( list @ A ) @ A3 @ ( lists @ A @ A5 ) )
=> ( ( A3
!= ( nil @ A ) )
=> ~ ! [A6: A,L2: list @ A] :
( ( A3
= ( cons @ A @ A6 @ L2 ) )
=> ( ( member2 @ A @ A6 @ A5 )
=> ~ ( member2 @ ( list @ A ) @ L2 @ ( lists @ A @ A5 ) ) ) ) ) ) ).
% lists.cases
thf(fact_4160_lists__image__witness,axiom,
! [A: $tType,B: $tType,X: list @ A,F: B > A,Q: set @ B] :
( ( member2 @ ( list @ A ) @ X @ ( lists @ A @ ( image2 @ B @ A @ F @ Q ) ) )
=> ~ ! [Xo: list @ B] :
( ( member2 @ ( list @ B ) @ Xo @ ( lists @ B @ Q ) )
=> ( X
!= ( map @ B @ A @ F @ Xo ) ) ) ) ).
% lists_image_witness
thf(fact_4161_in__prod__fst__sndI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: set @ A,B4: set @ B] :
( ( member2 @ A @ ( product_fst @ A @ B @ X ) @ A5 )
=> ( ( member2 @ B @ ( product_snd @ A @ B @ X ) @ B4 )
=> ( member2 @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 ) ) ) ) ).
% in_prod_fst_sndI
thf(fact_4162_R__subset__Field,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
@ ^ [Uu3: A] : ( field2 @ A @ R ) ) ) ).
% R_subset_Field
thf(fact_4163_listrel__refl__on,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A5 @ R3 )
=> ( refl_on @ ( list @ A ) @ ( lists @ A @ A5 ) @ ( listrel @ A @ A @ R3 ) ) ) ).
% listrel_refl_on
thf(fact_4164_lists__eq__set,axiom,
! [A: $tType] :
( ( lists @ A )
= ( ^ [A7: set @ A] :
( collect @ ( list @ A )
@ ^ [Xs3: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A7 ) ) ) ) ).
% lists_eq_set
thf(fact_4165_Rings_Omono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A3 ) ) ) ) ).
% Rings.mono_mult
thf(fact_4166_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F: A > A,P4: A,K: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ ( F @ P4 ) @ P4 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) @ P4 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_4167_lists__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( lists @ A @ A5 ) @ ( lists @ A @ B4 ) ) ) ).
% lists_mono
thf(fact_4168_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B4
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A5 )
& ( finite_finite2 @ B @ B4 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_4169_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B4 ) ) ) ).
% finite_cartesian_productD2
thf(fact_4170_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_cartesian_productD1
thf(fact_4171_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ A5 )
& ( ( B4 @ X3 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A6: A] :
( ( member2 @ A @ A6 @ A5 )
=> ( finite_finite2 @ B @ ( B4 @ A6 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) ) ) ) ).
% finite_SigmaI2
thf(fact_4172_Restr__trancl__mono,axiom,
! [A: $tType,V: A,W: A,E5: set @ ( product_prod @ A @ A ),U2: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E5
@ ( product_Sigma @ A @ A @ U2
@ ^ [Uu3: A] : U2 ) ) ) )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ E5 ) ) ) ).
% Restr_trancl_mono
thf(fact_4173_set__zip__cart,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,L: list @ A,L3: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ L @ L3 ) ) )
=> ( member2 @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ ( set2 @ A @ L )
@ ^ [Uu3: A] : ( set2 @ B @ L3 ) ) ) ) ).
% set_zip_cart
thf(fact_4174_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,S: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ? [X6: A] :
( ( member2 @ A @ X6 @ S )
& ! [Xa3: A] :
( ( member2 @ A @ Xa3 @ S )
=> ( ord_less_eq @ A @ X6 @ Xa3 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y3: B] : ( member2 @ B @ Y3 @ ( image2 @ A @ B @ F @ S ) ) )
= ( F
@ ( ord_Least @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ S ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_4175_lists__image,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( lists @ A @ ( image2 @ B @ A @ F @ A5 ) )
= ( image2 @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F ) @ ( lists @ B @ A5 ) ) ) ).
% lists_image
thf(fact_4176_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M2: nat,N: nat,F: A > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( order_mono @ A @ A @ F )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M2 @ F @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_4177_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ B @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_4178_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_4179_lfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,K: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F @ ( bot_bot @ A ) )
= ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) )
=> ( ( complete_lattice_lfp @ A @ F )
= ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_4180_lists__of__len__fin2,axiom,
! [A: $tType,P: set @ A,N: nat] :
( ( finite_finite2 @ A @ P )
=> ( finite_finite2 @ ( list @ A )
@ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
@ ( collect @ ( list @ A )
@ ^ [L4: list @ A] :
( N
= ( size_size @ ( list @ A ) @ L4 ) ) ) ) ) ) ).
% lists_of_len_fin2
thf(fact_4181_lists__of__len__fin1,axiom,
! [A: $tType,P: set @ A,N: nat] :
( ( finite_finite2 @ A @ P )
=> ( finite_finite2 @ ( list @ A )
@ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
@ ( collect @ ( list @ A )
@ ^ [L4: list @ A] :
( ( size_size @ ( list @ A ) @ L4 )
= N ) ) ) ) ) ).
% lists_of_len_fin1
thf(fact_4182_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X: A,A5: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu3: A] : A5 ) )
= ( finite_card @ B @ A5 ) ) ).
% card_cartesian_product_singleton
thf(fact_4183_lists__length__Suc__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N ) ) ) )
= ( image2 @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs3: list @ A,N3: A] : ( cons @ A @ N3 @ Xs3 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) )
@ ^ [Uu3: list @ A] : A5 ) ) ) ).
% lists_length_Suc_eq
thf(fact_4184_snd__image__times,axiom,
! [B: $tType,A: $tType,A5: set @ B,B4: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu3: B] : B4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu3: B] : B4 ) )
= B4 ) ) ) ).
% snd_image_times
thf(fact_4185_fst__image__times,axiom,
! [B: $tType,A: $tType,B4: set @ B,A5: set @ A] :
( ( ( B4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 ) )
= A5 ) ) ) ).
% fst_image_times
thf(fact_4186_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_4187_Times__empty,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B4
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_4188_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_4189_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,F: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F ) )
=> ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F )
=> ( ! [A6: A,B6: B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B6 ) @ ( F @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) ) ) )
=> ( P @ A6 @ B6 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% lfp_induct2
thf(fact_4190_mono__Int,axiom,
! [B: $tType,A: $tType,F: ( set @ A ) > ( set @ B ),A5: set @ A,B4: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F )
=> ( ord_less_eq @ ( set @ B ) @ ( F @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) @ ( inf_inf @ ( set @ B ) @ ( F @ A5 ) @ ( F @ B4 ) ) ) ) ).
% mono_Int
thf(fact_4191_mono__Un,axiom,
! [B: $tType,A: $tType,F: ( set @ A ) > ( set @ B ),A5: set @ A,B4: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F )
=> ( ord_less_eq @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F @ A5 ) @ ( F @ B4 ) ) @ ( F @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% mono_Un
thf(fact_4192_ord_Olexordp_Omono,axiom,
! [A: $tType,Less: A > A > $o] :
( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X25: list @ A] :
( ? [Y3: A,Ys3: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X3 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ( Less @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X3 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( Less @ X3 @ Y3 )
& ~ ( Less @ Y3 @ X3 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ).
% ord.lexordp.mono
thf(fact_4193_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P6: ( set @ A ) > $o,X3: set @ A] :
( ( X3
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,A4: A] :
( ( X3
= ( insert2 @ A @ A4 @ A7 ) )
& ( P6 @ A7 ) ) ) ) ).
% finite.mono
thf(fact_4194_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X25: list @ A] :
( ? [Y3: A,Ys3: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X3 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ord_less @ A @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X3 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( ord_less @ A @ X3 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X3 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp.mono
thf(fact_4195_times__eq__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B,C4: set @ A,D2: set @ B] :
( ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 )
= ( product_Sigma @ A @ B @ C4
@ ^ [Uu3: A] : D2 ) )
= ( ( ( A5 = C4 )
& ( B4 = D2 ) )
| ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B4
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C4
= ( bot_bot @ ( set @ A ) ) )
| ( D2
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_4196_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J4: set @ A,C4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I5 @ J4 ) @ C4 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C4 ) @ ( product_Sigma @ A @ B @ J4 @ C4 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_4197_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I5: set @ A,X5: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I5 @ X5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ I5 )
=> ( ( X5 @ X3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_4198_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,C4: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A5 @ B4 )
@ ^ [Uu3: A] : C4 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : C4 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu3: A] : C4 ) ) ) ).
% Times_Int_distrib1
thf(fact_4199_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I2: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ I2 ) @ ( B4 @ I2 ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) @ ( product_Sigma @ A @ B @ I5 @ B4 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_4200_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B,C4: set @ A,D2: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 )
@ ( product_Sigma @ A @ B @ C4
@ ^ [Uu3: A] : D2 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A5 @ C4 )
@ ^ [Uu3: A] : ( inf_inf @ ( set @ B ) @ B4 @ D2 ) ) ) ).
% Times_Int_Times
thf(fact_4201_times__subset__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,C4: set @ B,B4: set @ A,D2: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : C4 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu3: A] : D2 ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( C4
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
& ( ord_less_eq @ ( set @ B ) @ C4 @ D2 ) ) ) ) ).
% times_subset_iff
thf(fact_4202_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
= ( collect @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ A5 )
& ( ( B4 @ X3 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_4203_Refl__Field__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) )
= ( inf_inf @ ( set @ A ) @ ( field2 @ A @ R3 ) @ A5 ) ) ) ).
% Refl_Field_Restr
thf(fact_4204_Refl__Field__Restr2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R3 ) )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) )
= A5 ) ) ) ).
% Refl_Field_Restr2
thf(fact_4205_Total__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( total_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( total_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) ) ) ).
% Total_Restr
thf(fact_4206_total__on__imp__Total__Restr,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A5 @ R3 )
=> ( total_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) ) ) ).
% total_on_imp_Total_Restr
thf(fact_4207_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q: A > B > C,F: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q )
=> ( order_mono @ A @ ( D > C )
@ ^ [I2: A,X3: D] : ( Q @ I2 @ ( F @ X3 ) ) ) ) ) ).
% mono_compose
thf(fact_4208_Restr__subset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu3: A] : B4 ) )
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) ) ) ).
% Restr_subset
thf(fact_4209_Restr__Field,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R3 )
@ ^ [Uu3: A] : ( field2 @ A @ R3 ) ) )
= R3 ) ).
% Restr_Field
thf(fact_4210_Field__Restr__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ord_less_eq @ ( set @ A )
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) )
@ A5 ) ).
% Field_Restr_subset
thf(fact_4211_Refl__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( refl_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) ) ) ).
% Refl_Restr
thf(fact_4212_def__lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: A,F: A > A,P: A] :
( ( A5
= ( complete_lattice_lfp @ A @ F ) )
=> ( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ ( F @ ( inf_inf @ A @ A5 @ P ) ) @ P )
=> ( ord_less_eq @ A @ A5 @ P ) ) ) ) ) ).
% def_lfp_induct
thf(fact_4213_lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,P: A] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ ( F @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F ) @ P ) ) @ P )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ P ) ) ) ) ).
% lfp_induct
thf(fact_4214_def__lfp__induct__set,axiom,
! [A: $tType,A5: set @ A,F: ( set @ A ) > ( set @ A ),A3: A,P: A > $o] :
( ( A5
= ( complete_lattice_lfp @ ( set @ A ) @ F ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ( member2 @ A @ A3 @ A5 )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( F @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ) ) ).
% def_lfp_induct_set
thf(fact_4215_lfp__induct__set,axiom,
! [A: $tType,A3: A,F: ( set @ A ) > ( set @ A ),P: A > $o] :
( ( member2 @ A @ A3 @ ( complete_lattice_lfp @ ( set @ A ) @ F ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( F @ ( inf_inf @ ( set @ A ) @ ( complete_lattice_lfp @ ( set @ A ) @ F ) @ ( collect @ A @ P ) ) ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ) ).
% lfp_induct_set
thf(fact_4216_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F3: A > $o,Xs3: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F3 @ Xs3 ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F3 ) @ Xs3 ) ) ) ) ).
% partition_filter_conv
thf(fact_4217_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_4218_rtrancl__last__visit__node,axiom,
! [A: $tType,S4: A,S5: A,R: set @ ( product_prod @ A @ A ),Sh: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S4 @ S5 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( ( S4 != Sh )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S4 @ S5 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) )
| ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S4 @ Sh ) @ ( transitive_rtrancl @ A @ R ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Sh @ S5 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit_node
thf(fact_4219_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > A,G: C > B,A5: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X3: C] : ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ X3 ) )
@ A5 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A5 )
@ ^ [X3: A] : ( image2 @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_4220_vimageI,axiom,
! [B: $tType,A: $tType,F: B > A,A3: B,B2: A,B4: set @ A] :
( ( ( F @ A3 )
= B2 )
=> ( ( member2 @ A @ B2 @ B4 )
=> ( member2 @ B @ A3 @ ( vimage @ B @ A @ F @ B4 ) ) ) ) ).
% vimageI
thf(fact_4221_vimage__eq,axiom,
! [A: $tType,B: $tType,A3: A,F: A > B,B4: set @ B] :
( ( member2 @ A @ A3 @ ( vimage @ A @ B @ F @ B4 ) )
= ( member2 @ B @ ( F @ A3 ) @ B4 ) ) ).
% vimage_eq
thf(fact_4222_vimage__ident,axiom,
! [A: $tType,Y6: set @ A] :
( ( vimage @ A @ A
@ ^ [X3: A] : X3
@ Y6 )
= Y6 ) ).
% vimage_ident
thf(fact_4223_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F: A > B,P: B > $o] :
( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y3: A] : ( P @ ( F @ Y3 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_4224_vimage__empty,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( vimage @ A @ B @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_4225_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( vimage @ A @ B @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% vimage_UNIV
thf(fact_4226_vimage__Int,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ B,B4: set @ B] :
( ( vimage @ A @ B @ F @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A5 ) @ ( vimage @ A @ B @ F @ B4 ) ) ) ).
% vimage_Int
thf(fact_4227_vimage__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ B,B4: set @ B] :
( ( vimage @ A @ B @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F @ A5 ) @ ( vimage @ A @ B @ F @ B4 ) ) ) ).
% vimage_Un
thf(fact_4228_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] : ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) ) ).
% listrel_rtrancl_refl
thf(fact_4229_vimage__const,axiom,
! [B: $tType,A: $tType,C3: B,A5: set @ B] :
( ( ( member2 @ B @ C3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : C3
@ A5 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member2 @ B @ C3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : C3
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_4230_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ A] :
( ( image2 @ B @ A @ F @ ( vimage @ B @ A @ F @ A5 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_4231_vimage__if,axiom,
! [B: $tType,A: $tType,C3: B,A5: set @ B,D3: B,B4: set @ A] :
( ( ( member2 @ B @ C3 @ A5 )
=> ( ( ( member2 @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member2 @ A @ X3 @ B4 ) @ C3 @ D3 )
@ A5 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member2 @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member2 @ A @ X3 @ B4 ) @ C3 @ D3 )
@ A5 )
= B4 ) ) ) )
& ( ~ ( member2 @ B @ C3 @ A5 )
=> ( ( ( member2 @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member2 @ A @ X3 @ B4 ) @ C3 @ D3 )
@ A5 )
= ( uminus_uminus @ ( set @ A ) @ B4 ) ) )
& ( ~ ( member2 @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member2 @ A @ X3 @ B4 ) @ C3 @ D3 )
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_4232_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set @ A,F: A > B,G: A > B,Y: set @ B] :
( ! [W3: A] :
( ( member2 @ A @ W3 @ S )
=> ( ( F @ W3 )
= ( G @ W3 ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ Y ) @ S )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_4233_vimage__Diff,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ B,B4: set @ B] :
( ( vimage @ A @ B @ F @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F @ A5 ) @ ( vimage @ A @ B @ F @ B4 ) ) ) ).
% vimage_Diff
thf(fact_4234_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V: A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( U != V )
=> ~ ! [Vh: A] :
( ( U != Vh )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% converse_rtranclE'
thf(fact_4235_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ ( transitive_rtrancl @ A @ R3 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel1_rtrancl_subset_rtrancl_listrel1
thf(fact_4236_vimageD,axiom,
! [A: $tType,B: $tType,A3: A,F: A > B,A5: set @ B] :
( ( member2 @ A @ A3 @ ( vimage @ A @ B @ F @ A5 ) )
=> ( member2 @ B @ ( F @ A3 ) @ A5 ) ) ).
% vimageD
thf(fact_4237_vimageE,axiom,
! [A: $tType,B: $tType,A3: A,F: A > B,B4: set @ B] :
( ( member2 @ A @ A3 @ ( vimage @ A @ B @ F @ B4 ) )
=> ( member2 @ B @ ( F @ A3 ) @ B4 ) ) ).
% vimageE
thf(fact_4238_vimageI2,axiom,
! [B: $tType,A: $tType,F: B > A,A3: B,A5: set @ A] :
( ( member2 @ A @ ( F @ A3 ) @ A5 )
=> ( member2 @ B @ A3 @ ( vimage @ B @ A @ F @ A5 ) ) ) ).
% vimageI2
thf(fact_4239_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
( ! [X2: A] :
( ( P @ ( F @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_4240_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) )
= ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel_rtrancl_eq_rtrancl_listrel1
thf(fact_4241_vimage__Compl,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ B] :
( ( vimage @ A @ B @ F @ ( uminus_uminus @ ( set @ B ) @ A5 ) )
= ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F @ A5 ) ) ) ).
% vimage_Compl
thf(fact_4242_rtrancl__mono__rightI,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ S3 ) ) ) ).
% rtrancl_mono_rightI
thf(fact_4243_rtrancl__mono__mp,axiom,
! [A: $tType,U2: set @ ( product_prod @ A @ A ),V3: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V3 )
=> ( ( member2 @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ U2 ) )
=> ( member2 @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ V3 ) ) ) ) ).
% rtrancl_mono_mp
thf(fact_4244_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_4245_subset__vimage__iff,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B,B4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( vimage @ A @ B @ F @ B4 ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( member2 @ B @ ( F @ X3 ) @ B4 ) ) ) ) ).
% subset_vimage_iff
thf(fact_4246_vimage__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,F: B > A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F @ A5 ) @ ( vimage @ B @ A @ F @ B4 ) ) ) ).
% vimage_mono
thf(fact_4247_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F3: A > B,B5: set @ B] :
( collect @ A
@ ^ [X3: A] : ( member2 @ B @ ( F3 @ X3 ) @ B5 ) ) ) ) ).
% vimage_def
thf(fact_4248_in__rtrancl__insert,axiom,
! [A: $tType,X: product_prod @ A @ A,R: set @ ( product_prod @ A @ A ),R3: product_prod @ A @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ R ) )
=> ( member2 @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ R3 @ R ) ) ) ) ).
% in_rtrancl_insert
thf(fact_4249_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ A5 ) @ B4 )
= ( ord_less_eq @ ( set @ B ) @ A5 @ ( vimage @ B @ A @ F @ B4 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_4250_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ ( vimage @ B @ A @ F @ A5 ) ) @ A5 ) ).
% image_vimage_subset
thf(fact_4251_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A3: A,F: A > B,B2: B] :
( ( member2 @ A @ A3 @ ( vimage @ A @ B @ F @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F @ A3 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_4252_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_4253_rtrancl__sub__insert__rtrancl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ R ) ) ) ).
% rtrancl_sub_insert_rtrancl
thf(fact_4254_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),Zs: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_4255_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > ( product_prod @ B @ C ),A5: set @ B,B4: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu3: B] : B4 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F ) @ A5 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) @ B4 ) ) ) ).
% vimage_Times
thf(fact_4256_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A5: set @ B,F: B > ( set @ A )] :
( ( ( member2 @ B @ X @ A5 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A5 @ F ) )
= ( F @ X ) ) )
& ( ~ ( member2 @ B @ X @ A5 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A5 @ F ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_4257_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F @ A5 )
= ( bot_bot @ ( set @ B ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_4258_vimage__insert,axiom,
! [A: $tType,B: $tType,F: A > B,A3: B,B4: set @ B] :
( ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F @ B4 ) ) ) ).
% vimage_insert
thf(fact_4259_finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F2: set @ A,H: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ F2 )
=> ( ( inj_on @ B @ A @ H @ A5 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ F2 ) @ A5 ) ) ) ) ).
% finite_vimage_IntI
thf(fact_4260_inv__image__partition,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ! [Y2: A] :
( ( member2 @ 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_4261_trancl__union__outside,axiom,
! [A: $tType,V: A,W: A,E5: set @ ( product_prod @ A @ A ),U2: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E5 @ U2 ) ) )
=> ( ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ E5 ) )
=> ? [X2: A,Y2: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ X2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E5 @ U2 ) ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ U2 )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ W ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E5 @ U2 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_4262_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_4263_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_4264_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_4265_trancl__over__edgeE,axiom,
! [A: $tType,U: A,W: A,V1: A,V22: A,E5: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V1 @ V22 ) @ E5 ) ) )
=> ( ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W ) @ ( transitive_trancl @ A @ E5 ) )
=> ~ ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V1 ) @ ( transitive_rtrancl @ A @ E5 ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V22 @ W ) @ ( transitive_rtrancl @ A @ E5 ) ) ) ) ) ).
% trancl_over_edgeE
thf(fact_4266_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) )
=> ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_4267_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R3 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_4268_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ R3 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ).
% rtrancl_listrel1_if_listrel
thf(fact_4269_Restr__rtrancl__mono,axiom,
! [A: $tType,V: A,W: A,E5: set @ ( product_prod @ A @ A ),U2: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E5
@ ( product_Sigma @ A @ A @ U2
@ ^ [Uu3: A] : U2 ) ) ) )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_rtrancl @ A @ E5 ) ) ) ).
% Restr_rtrancl_mono
thf(fact_4270_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ? [X2: A] :
( ( member2 @ A @ X2 @ ( image2 @ B @ A @ F @ A5 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_4271_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ~ ! [Y2: A] :
( ( member2 @ A @ Y2 @ ( image2 @ B @ A @ F @ A5 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_4272_partition_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( partition @ A @ P @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ).
% partition.simps(1)
thf(fact_4273_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F2: set @ A,H: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ F2 )
=> ( ! [Y2: A] :
( ( member2 @ A @ Y2 @ F2 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ F2 ) @ A5 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_4274_partition__P,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Yes2: list @ A,No2: list @ A] :
( ( ( partition @ A @ P @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) )
=> ( ! [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Yes2 ) )
=> ( P @ X6 ) )
& ! [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ No2 ) )
=> ~ ( P @ X6 ) ) ) ) ).
% partition_P
thf(fact_4275_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,E5: set @ ( product_prod @ A @ A ),F: A > B] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ E5 ) )
=> ( member2 @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ A3 ) @ ( F @ B2 ) ) @ ( transitive_rtrancl @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( pairself @ A @ B @ F ) @ E5 ) ) ) ) ).
% rtrancl_mapI
thf(fact_4276_listrel__subset__rtrancl__listrel1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R3 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel_subset_rtrancl_listrel1
thf(fact_4277_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F3: A > B,B5: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : ( vimage @ A @ B @ F3 @ ( insert2 @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) )
@ B5 ) ) ) ) ).
% vimage_eq_UN
thf(fact_4278_partition_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( partition @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ^ [Yes3: list @ A,No3: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes3 ) @ No3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes3 @ ( cons @ A @ X @ No3 ) ) )
@ ( partition @ A @ P @ Xs ) ) ) ).
% partition.simps(2)
thf(fact_4279_partition__filter1,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ).
% partition_filter1
thf(fact_4280_rtrancl__last__touch,axiom,
! [A: $tType,Q3: A,Q6: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q6 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member2 @ A @ Q3 @ S )
=> ~ ! [Qt: A] :
( ( member2 @ A @ Qt @ S )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q6 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_touch
thf(fact_4281_rtrancl__last__visit_H,axiom,
! [A: $tType,Q3: A,Q6: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q6 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q6 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : S ) ) ) )
=> ~ ! [Qt: A] :
( ( member2 @ A @ Qt @ S )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q6 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit'
thf(fact_4282_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ? [X2: A] :
( ( member2 @ A @ X2 @ ( image2 @ B @ A @ F @ A5 ) )
& ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_4283_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ~ ! [Y2: A] :
( ( member2 @ A @ Y2 @ ( image2 @ B @ A @ F @ A5 ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_4284_partition__rev_Oelims,axiom,
! [A: $tType,X: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( partition_rev @ A @ X @ Xa @ Xb )
= Y )
=> ( ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ( Xb
= ( nil @ A ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) ) ) )
=> ~ ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ! [X2: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs2 ) )
=> ( Y
!= ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X2 @ No ) ) ) @ Xs2 ) ) ) ) ) ) ).
% partition_rev.elims
thf(fact_4285_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F: A > B,D2: set @ A,A5: set @ B] :
( ( inj_on @ A @ B @ F @ D2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A5 ) @ D2 ) ) @ ( finite_card @ B @ A5 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_4286_partition__set,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Yes2: list @ A,No2: list @ A] :
( ( ( partition @ A @ P @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) )
=> ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Yes2 ) @ ( set2 @ A @ No2 ) )
= ( set2 @ A @ Xs ) ) ) ).
% partition_set
thf(fact_4287_rtrancl__last__visit,axiom,
! [A: $tType,Q3: A,Q6: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q6 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q6 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : S ) ) ) )
=> ~ ! [Qt: A] :
( ( member2 @ A @ Qt @ S )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_trancl @ A @ R ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q6 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit
thf(fact_4288_trancl__multi__insert2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),M2: A,X5: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( insert2 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu3: A] : X5 ) ) ) )
=> ( ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ! [X2: A] :
( ( member2 @ A @ X2 @ X5 )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ M2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_4289_trancl__multi__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),X5: set @ A,M2: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ X5
@ ^ [Uu3: A] : ( insert2 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
=> ( ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ! [X2: A] :
( ( member2 @ A @ X2 @ X5 )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_4290_partition__filter2,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P @ Xs ) )
= ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs ) ) ).
% partition_filter2
thf(fact_4291_partition__rev_Opelims,axiom,
! [A: $tType,X: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( partition_rev @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ( Xb
= ( nil @ A ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) ) ) ) )
=> ~ ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ! [X2: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X2 @ No ) ) ) @ Xs2 ) )
=> ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% partition_rev.pelims
thf(fact_4292_quicksort__by__rel_Osimps_I2_J,axiom,
! [A: $tType,R: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
( ( quicksort_by_rel @ A @ R @ Sl2 @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( R @ Y3 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ).
% quicksort_by_rel.simps(2)
thf(fact_4293_quicksort__by__rel_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs2 ) )
=> ( Y
!= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( X @ Y3 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) ) ) ) ) ) ).
% quicksort_by_rel.elims
thf(fact_4294_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,A3: B] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ A5 )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ A5 )
& ( ( F @ X3 )
= A3 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_4295_set__quicksort__by__rel,axiom,
! [A: $tType,R: A > A > $o,Sl2: list @ A,Xs: list @ A] :
( ( set2 @ A @ ( quicksort_by_rel @ A @ R @ Sl2 @ Xs ) )
= ( set2 @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% set_quicksort_by_rel
thf(fact_4296_quicksort__by__rel_Osimps_I1_J,axiom,
! [A: $tType,R: A > A > $o,Sl2: list @ A] :
( ( quicksort_by_rel @ A @ R @ Sl2 @ ( nil @ A ) )
= Sl2 ) ).
% quicksort_by_rel.simps(1)
thf(fact_4297_quicksort__by__rel__remove__acc__guared,axiom,
! [A: $tType,Sl2: list @ A,R: A > A > $o,Xs: list @ A] :
( ( Sl2
!= ( nil @ A ) )
=> ( ( quicksort_by_rel @ A @ R @ Sl2 @ Xs )
= ( append @ A @ ( quicksort_by_rel @ A @ R @ ( nil @ A ) @ Xs ) @ Sl2 ) ) ) ).
% quicksort_by_rel_remove_acc_guared
thf(fact_4298_quicksort__by__rel__remove__acc,axiom,
! [A: $tType] :
( ( quicksort_by_rel @ A )
= ( ^ [R6: A > A > $o,Sl3: list @ A,Xs3: list @ A] : ( append @ A @ ( quicksort_by_rel @ A @ R6 @ ( nil @ A ) @ Xs3 ) @ Sl3 ) ) ) ).
% quicksort_by_rel_remove_acc
thf(fact_4299_sorted__wrt__quicksort__by__rel,axiom,
! [X7: $tType,R: X7 > X7 > $o,Xs: list @ X7] :
( ! [X2: X7,Y2: X7] :
( ( R @ X2 @ Y2 )
| ( R @ Y2 @ X2 ) )
=> ( ! [X2: X7,Y2: X7,Z4: X7] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z4 )
=> ( R @ X2 @ Z4 ) ) )
=> ( sorted_wrt @ X7 @ R @ ( quicksort_by_rel @ X7 @ R @ ( nil @ X7 ) @ Xs ) ) ) ) ).
% sorted_wrt_quicksort_by_rel
thf(fact_4300_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_4301_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X8: set @ A] :
( the @ A
@ ^ [X3: A] :
( X8
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_4302_sort__quicksort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 )
= ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ) ).
% sort_quicksort_by_rel
thf(fact_4303_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: A > B,A3: B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X3: A] :
( ( F @ X3 )
= A3 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_4304_quicksort__by__rel_Opsimps_I2_J,axiom,
! [A: $tType,R: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( cons @ A @ X @ Xs ) ) ) )
=> ( ( quicksort_by_rel @ A @ R @ Sl2 @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( R @ Y3 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ) ).
% quicksort_by_rel.psimps(2)
thf(fact_4305_quicksort__by__rel_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( X @ Y3 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% quicksort_by_rel.pelims
thf(fact_4306_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 ) ) )
=> ( ! [R5: 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 ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ( P @ R5 @ Sl @ ( nil @ A ) ) )
=> ( ! [R5: A > A > $o,Sl: list @ A,X2: A,Xs2: 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 ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y4: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z3: A] : ( R5 @ Z3 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y4 )
= Xa2 )
=> ( P @ R5 @ Sl @ Y4 ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y4: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z3: A] : ( R5 @ Z3 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y4 )
= Xa2 )
=> ( P @ R5 @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ R5 @ Sl @ Y4 ) ) @ Xb2 ) ) )
=> ( P @ R5 @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% quicksort_by_rel.pinduct
thf(fact_4307_quicksort__by__rel_Opsimps_I1_J,axiom,
! [A: $tType,R: A > A > $o,Sl2: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( nil @ A ) ) ) )
=> ( ( quicksort_by_rel @ A @ R @ Sl2 @ ( nil @ A ) )
= Sl2 ) ) ).
% quicksort_by_rel.psimps(1)
thf(fact_4308_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ E5
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : R ) ) ) )
=> ( ~ ( member2 @ A @ U @ R )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E5 @ R ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_4309_Preorder__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_preorder_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) ) ) ).
% Preorder_Restr
thf(fact_4310_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F: A > B,G: C > B,A5: set @ A,B4: set @ C] :
( ( inj_on @ ( product_prod @ A @ C ) @ B
@ ( product_case_prod @ A @ C @ B
@ ^ [A4: A,B3: C] : ( times_times @ B @ ( F @ A4 ) @ ( G @ B3 ) ) )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu3: A] : B4 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B4 ) )
= ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
@ ( collect @ B
@ ^ [Uu3: B] :
? [A4: A,B3: C] :
( ( Uu3
= ( times_times @ B @ ( F @ A4 ) @ ( G @ B3 ) ) )
& ( member2 @ A @ A4 @ A5 )
& ( member2 @ C @ B3 @ B4 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_4311_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S: set @ A,F: A > B > B,A5: set @ A,Z2: B,Y: B,A3: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S )
=> ( ( finite_fold_graph @ A @ B @ F @ Z2 @ A5 @ Y )
=> ( ( member2 @ A @ A3 @ A5 )
=> ? [Y7: B] :
( ( Y
= ( F @ A3 @ Y7 ) )
& ( finite_fold_graph @ A @ B @ F @ Z2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ Y7 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
thf(fact_4312_list_Omap__id0,axiom,
! [A: $tType] :
( ( map @ A @ A @ ( id @ A ) )
= ( id @ ( list @ A ) ) ) ).
% list.map_id0
thf(fact_4313_rel__restrict__empty,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( rel_restrict @ A @ R @ ( bot_bot @ ( set @ A ) ) )
= R ) ).
% rel_restrict_empty
thf(fact_4314_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_4315_fold__graph_OemptyI,axiom,
! [A: $tType,B: $tType,F: A > B > B,Z2: B] : ( finite_fold_graph @ A @ B @ F @ Z2 @ ( bot_bot @ ( set @ A ) ) @ Z2 ) ).
% fold_graph.emptyI
thf(fact_4316_empty__fold__graphE,axiom,
! [A: $tType,B: $tType,F: A > B > B,Z2: B,X: B] :
( ( finite_fold_graph @ A @ B @ F @ Z2 @ ( bot_bot @ ( set @ A ) ) @ X )
=> ( X = Z2 ) ) ).
% empty_fold_graphE
thf(fact_4317_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X: A,Y: A,A5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A5 @ R ) )
=> ~ ( member2 @ A @ Y @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_4318_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X: A,Y: A,A5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A5 @ R ) )
=> ~ ( member2 @ A @ X @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_4319_rel__restrictI,axiom,
! [A: $tType,X: A,R: set @ A,Y: A,E5: set @ ( product_prod @ A @ A )] :
( ~ ( member2 @ A @ X @ R )
=> ( ~ ( member2 @ A @ Y @ R )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E5 )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E5 @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_4320_rel__restrict__lift,axiom,
! [A: $tType,X: A,Y: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E5 @ R ) )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E5 ) ) ).
% rel_restrict_lift
thf(fact_4321_List_Omap_Oidentity,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X3: A] : X3 )
= ( id @ ( list @ A ) ) ) ).
% List.map.identity
thf(fact_4322_list_Omap__id,axiom,
! [A: $tType,T4: list @ A] :
( ( map @ A @ A @ ( id @ A ) @ T4 )
= T4 ) ).
% list.map_id
thf(fact_4323_rel__restrict__mono,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A5 @ R ) @ ( rel_restrict @ A @ B4 @ R ) ) ) ).
% rel_restrict_mono
thf(fact_4324_rel__restrict__sub,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A5 ) @ R ) ).
% rel_restrict_sub
thf(fact_4325_rel__restrict__union,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( rel_restrict @ A @ R @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( rel_restrict @ A @ ( rel_restrict @ A @ R @ A5 ) @ B4 ) ) ).
% rel_restrict_union
thf(fact_4326_finite__rel__restrict,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( rel_restrict @ A @ R @ A5 ) ) ) ).
% finite_rel_restrict
thf(fact_4327_foldr__Nil,axiom,
! [A: $tType,B: $tType,F: A > B > B] :
( ( foldr @ A @ B @ F @ ( nil @ A ) )
= ( id @ B ) ) ).
% foldr_Nil
thf(fact_4328_fold__graph_Ocases,axiom,
! [A: $tType,B: $tType,F: A > B > B,Z2: B,A1: set @ A,A22: B] :
( ( finite_fold_graph @ A @ B @ F @ Z2 @ A1 @ A22 )
=> ( ( ( A1
= ( bot_bot @ ( set @ A ) ) )
=> ( A22 != Z2 ) )
=> ~ ! [X2: A,A9: set @ A] :
( ( A1
= ( insert2 @ A @ X2 @ A9 ) )
=> ! [Y2: B] :
( ( A22
= ( F @ X2 @ Y2 ) )
=> ( ~ ( member2 @ A @ X2 @ A9 )
=> ~ ( finite_fold_graph @ A @ B @ F @ Z2 @ A9 @ Y2 ) ) ) ) ) ) ).
% fold_graph.cases
thf(fact_4329_fold__graph_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold_graph @ A @ B )
= ( ^ [F3: A > B > B,Z3: B,A12: set @ A,A23: B] :
( ( ( A12
= ( bot_bot @ ( set @ A ) ) )
& ( A23 = Z3 ) )
| ? [X3: A,A7: set @ A,Y3: B] :
( ( A12
= ( insert2 @ A @ X3 @ A7 ) )
& ( A23
= ( F3 @ X3 @ Y3 ) )
& ~ ( member2 @ A @ X3 @ A7 )
& ( finite_fold_graph @ A @ B @ F3 @ Z3 @ A7 @ Y3 ) ) ) ) ) ).
% fold_graph.simps
thf(fact_4330_foldr__filter,axiom,
! [A: $tType,B: $tType,F: B > A > A,P: B > $o,Xs: list @ B] :
( ( foldr @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) )
= ( foldr @ B @ A
@ ^ [X3: B] : ( if @ ( A > A ) @ ( P @ X3 ) @ ( F @ X3 ) @ ( id @ A ) )
@ Xs ) ) ).
% foldr_filter
thf(fact_4331_rel__restrict__trancl__mem,axiom,
! [A: $tType,A3: A,B2: A,A5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A5 @ R ) ) )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A5 ) @ R ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_4332_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V: A,W: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E5 @ R ) ) )
=> ~ ( member2 @ A @ V @ R ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_4333_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V: A,W: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E5 @ R ) ) )
=> ~ ( member2 @ A @ W @ R ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_4334_rel__restrict__mono2,axiom,
! [A: $tType,R: set @ A,S: set @ A,A5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ R @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A5 @ S ) @ ( rel_restrict @ A @ A5 @ R ) ) ) ).
% rel_restrict_mono2
thf(fact_4335_rel__restrict__trancl__sub,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),R: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A5 @ R ) ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A5 ) @ R ) ) ).
% rel_restrict_trancl_sub
thf(fact_4336_rel__restrict__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [V4: A,W4: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V4 @ W4 ) @ R6 )
& ~ ( member2 @ A @ V4 @ A7 )
& ~ ( member2 @ A @ W4 @ A7 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_4337_rel__restrict__Int__empty,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ ( field2 @ A @ R ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( rel_restrict @ A @ R @ A5 )
= R ) ) ).
% rel_restrict_Int_empty
thf(fact_4338_Field__rel__restrict,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ ( rel_restrict @ A @ R @ A5 ) ) @ ( minus_minus @ ( set @ A ) @ ( field2 @ A @ R ) @ A5 ) ) ).
% Field_rel_restrict
thf(fact_4339_rel__restrict__compl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A5 ) @ ( rel_restrict @ A @ R @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% rel_restrict_compl
thf(fact_4340_homo__rel__restrict__mono,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B4: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu3: A] : B4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A5 )
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ B4 @ A5 )
@ ^ [Uu3: A] : ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% homo_rel_restrict_mono
thf(fact_4341_rel__restrict__alt__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A ),A7: set @ A] :
( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R6
@ ( product_Sigma @ A @ A @ ( uminus_uminus @ ( set @ A ) @ A7 )
@ ^ [Uu3: A] : ( uminus_uminus @ ( set @ A ) @ A7 ) ) ) ) ) ).
% rel_restrict_alt_def
thf(fact_4342_rel__restrict__Sigma__sub,axiom,
! [A: $tType,A5: set @ A,R: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) )
@ ( rel_restrict @ A
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
@ R )
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ A5 @ R )
@ ^ [Uu3: A] : ( minus_minus @ ( set @ A ) @ A5 @ R ) ) ) ) ).
% rel_restrict_Sigma_sub
thf(fact_4343_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_4344_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( set2 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F ) @ Xs @ ( top_top @ A ) ) ) ) ).
% INF_set_fold
thf(fact_4345_same__fst__trancl,axiom,
! [B: $tType,A: $tType,P: A > $o,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( transitive_trancl @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P @ R ) )
= ( same_fst @ A @ B @ P
@ ^ [X3: A] : ( transitive_trancl @ B @ ( R @ X3 ) ) ) ) ).
% same_fst_trancl
thf(fact_4346_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( set2 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F ) @ Xs @ ( bot_bot @ A ) ) ) ) ).
% SUP_set_fold
thf(fact_4347_fold__append,axiom,
! [A: $tType,B: $tType,F: B > A > A,Xs: list @ B,Ys: list @ B] :
( ( fold @ B @ A @ F @ ( append @ B @ Xs @ Ys ) )
= ( comp @ A @ A @ A @ ( fold @ B @ A @ F @ Ys ) @ ( fold @ B @ A @ F @ Xs ) ) ) ).
% fold_append
thf(fact_4348_fold__replicate,axiom,
! [A: $tType,B: $tType,F: B > A > A,N: nat,X: B] :
( ( fold @ B @ A @ F @ ( replicate @ B @ N @ X ) )
= ( compow @ ( A > A ) @ N @ ( F @ X ) ) ) ).
% fold_replicate
thf(fact_4349_fold__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > A > A,S4: A] :
( ( fold @ B @ A @ F @ ( nil @ B ) @ S4 )
= S4 ) ).
% fold_simps(1)
thf(fact_4350_fold__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > A > A,X: B,Xs: list @ B,S4: A] :
( ( fold @ B @ A @ F @ ( cons @ B @ X @ Xs ) @ S4 )
= ( fold @ B @ A @ F @ Xs @ ( F @ X @ S4 ) ) ) ).
% fold_simps(2)
thf(fact_4351_fold__invariant,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Q: A > $o,P: B > $o,S4: B,F: A > B > B] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( Q @ X2 ) )
=> ( ( P @ S4 )
=> ( ! [X2: A,S8: B] :
( ( Q @ X2 )
=> ( ( P @ S8 )
=> ( P @ ( F @ X2 @ S8 ) ) ) )
=> ( P @ ( fold @ A @ B @ F @ Xs @ S4 ) ) ) ) ) ).
% fold_invariant
thf(fact_4352_List_Ofold__cong,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,Xs: list @ B,Ys: list @ B,F: B > A > A,G: B > A > A] :
( ( A3 = B2 )
=> ( ( Xs = Ys )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( set2 @ B @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( fold @ B @ A @ F @ Xs @ A3 )
= ( fold @ B @ A @ G @ Ys @ B2 ) ) ) ) ) ).
% List.fold_cong
thf(fact_4353_foldl__conv__fold,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F3: A > B > A,S6: A,Xs3: list @ B] :
( fold @ B @ A
@ ^ [X3: B,T2: A] : ( F3 @ T2 @ X3 )
@ Xs3
@ S6 ) ) ) ).
% foldl_conv_fold
thf(fact_4354_fold__Cons,axiom,
! [B: $tType,A: $tType,F: A > B > B,X: A,Xs: list @ A] :
( ( fold @ A @ B @ F @ ( cons @ A @ X @ Xs ) )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ Xs ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_4355_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H: B > C,G: A > B > B,F: A > C > C] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H @ ( G @ X2 ) )
= ( comp @ C @ C @ B @ ( F @ X2 ) @ H ) ) )
=> ( ( comp @ B @ C @ B @ H @ ( fold @ A @ B @ G @ Xs ) )
= ( comp @ C @ C @ B @ ( fold @ A @ C @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_4356_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H: B > C,G: A > B > B,F: A > C > C,S4: B] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H @ ( G @ X2 ) )
= ( comp @ C @ C @ B @ ( F @ X2 ) @ H ) ) )
=> ( ( H @ ( fold @ A @ B @ G @ Xs @ S4 ) )
= ( fold @ A @ C @ F @ Xs @ ( H @ S4 ) ) ) ) ).
% fold_commute_apply
thf(fact_4357_fold__Nil,axiom,
! [A: $tType,B: $tType,F: A > B > B] :
( ( fold @ A @ B @ F @ ( nil @ A ) )
= ( id @ B ) ) ).
% fold_Nil
thf(fact_4358_fold__id,axiom,
! [A: $tType,B: $tType,Xs: list @ A,F: A > B > B] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( F @ X2 )
= ( id @ B ) ) )
=> ( ( fold @ A @ B @ F @ Xs )
= ( id @ B ) ) ) ).
% fold_id
thf(fact_4359_fold__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: B > A > A,F: C > B,Xs: list @ C] :
( ( fold @ B @ A @ G @ ( map @ C @ B @ F @ Xs ) )
= ( fold @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F ) @ Xs ) ) ).
% fold_map
thf(fact_4360_foldr__conv__fold,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F3: B > A > A,Xs3: list @ B] : ( fold @ B @ A @ F3 @ ( rev @ B @ Xs3 ) ) ) ) ).
% foldr_conv_fold
thf(fact_4361_fold__filter,axiom,
! [A: $tType,B: $tType,F: B > A > A,P: B > $o,Xs: list @ B] :
( ( fold @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) )
= ( fold @ B @ A
@ ^ [X3: B] : ( if @ ( A > A ) @ ( P @ X3 ) @ ( F @ X3 ) @ ( id @ A ) )
@ Xs ) ) ).
% fold_filter
thf(fact_4362_union__set__fold,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
= ( fold @ A @ ( set @ A ) @ ( insert2 @ A ) @ Xs @ A5 ) ) ).
% union_set_fold
thf(fact_4363_rev__conv__fold,axiom,
! [A: $tType] :
( ( rev @ A )
= ( ^ [Xs3: list @ A] : ( fold @ A @ ( list @ A ) @ ( cons @ A ) @ Xs3 @ ( nil @ A ) ) ) ) ).
% rev_conv_fold
thf(fact_4364_fold__Cons__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( fold @ A @ ( list @ A ) @ ( cons @ A ) @ Xs )
= ( append @ A @ ( rev @ A @ Xs ) ) ) ).
% fold_Cons_rev
thf(fact_4365_fold__rev,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B] :
( ! [X2: A,Y2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member2 @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
= ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) )
=> ( ( fold @ A @ B @ F @ ( rev @ A @ Xs ) )
= ( fold @ A @ B @ F @ Xs ) ) ) ).
% fold_rev
thf(fact_4366_List_Ounion__def,axiom,
! [A: $tType] :
( ( union @ A )
= ( fold @ A @ ( list @ A ) @ ( insert @ A ) ) ) ).
% List.union_def
thf(fact_4367_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B] :
( ! [X2: A,Y2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member2 @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
= ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) )
=> ( ( foldr @ A @ B @ F @ Xs )
= ( fold @ A @ B @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_4368_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B,X: A] :
( ! [X2: A,Y2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member2 @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) )
= ( comp @ B @ B @ B @ ( F @ Y2 ) @ ( F @ X2 ) ) ) ) )
=> ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( fold @ A @ B @ F @ Xs )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ ( remove1 @ A @ X @ Xs ) ) @ ( F @ X ) ) ) ) ) ).
% fold_remove1_split
thf(fact_4369_minus__set__fold,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( set2 @ A @ Xs ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs @ A5 ) ) ).
% minus_set_fold
thf(fact_4370_sort__conv__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3
@ Xs )
= ( fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3 )
@ Xs
@ ( nil @ A ) ) ) ) ).
% sort_conv_fold
thf(fact_4371_inter__coset__fold,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( coset @ A @ Xs ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs @ A5 ) ) ).
% inter_coset_fold
thf(fact_4372_fold__append__concat__rev,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( fold @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss2 )
= ( append @ A @ ( concat @ A @ ( rev @ ( list @ A ) @ Xss2 ) ) ) ) ).
% fold_append_concat_rev
thf(fact_4373_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_4374_Inf__set__fold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Xs: list @ A] :
( ( complete_Inf_Inf @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs @ ( top_top @ A ) ) ) ) ).
% Inf_set_fold
thf(fact_4375_Inf__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic7752659483105999362nf_fin @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs @ X ) ) ) ).
% Inf_fin.set_eq_fold
thf(fact_4376_Sup__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic5882676163264333800up_fin @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs @ X ) ) ) ).
% Sup_fin.set_eq_fold
thf(fact_4377_Max_Oset__eq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic643756798349783984er_Max @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( ord_max @ A ) @ Xs @ X ) ) ) ).
% Max.set_eq_fold
thf(fact_4378_Min_Oset__eq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic643756798350308766er_Min @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( ord_min @ A ) @ Xs @ X ) ) ) ).
% Min.set_eq_fold
thf(fact_4379_semilattice__set_Oset__eq__fold,axiom,
! [A: $tType,F: A > A > A,X: A,Xs: list @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ F @ Xs @ X ) ) ) ).
% semilattice_set.set_eq_fold
thf(fact_4380_sort__key__conv__fold,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( inj_on @ B @ A @ F @ ( set2 @ B @ Xs ) )
=> ( ( linorder_sort_key @ B @ A @ F @ Xs )
= ( fold @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F ) @ Xs @ ( nil @ B ) ) ) ) ) ).
% sort_key_conv_fold
thf(fact_4381_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
! [A: $tType,B: $tType,S: set @ A,F: A > B > B,Xs: list @ A,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ S )
=> ( ( finite_fold @ A @ B @ F @ Y @ ( set2 @ A @ Xs ) )
= ( fold @ A @ B @ F @ ( remdups @ A @ Xs ) @ Y ) ) ) ) ).
% comp_fun_commute_on.fold_set_fold_remdups
thf(fact_4382_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_4383_comp__fun__idem__on_Ofold__set__fold,axiom,
! [A: $tType,B: $tType,S: set @ A,F: A > B > B,Xs: list @ A,Y: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ S )
=> ( ( finite_fold @ A @ B @ F @ Y @ ( set2 @ A @ Xs ) )
= ( fold @ A @ B @ F @ Xs @ Y ) ) ) ) ).
% comp_fun_idem_on.fold_set_fold
thf(fact_4384_list_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R6: A > B > $o,A4: list @ A,B3: list @ B] :
? [Z3: list @ ( product_prod @ A @ B )] :
( ( member2 @ ( list @ ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X3: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z3 )
= A4 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z3 )
= B3 ) ) ) ) ).
% list.in_rel
thf(fact_4385_properties__for__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Ys: list @ B,Xs: list @ B,F: B > A] :
( ( ( mset @ B @ Ys )
= ( mset @ B @ Xs ) )
=> ( ! [K3: B] :
( ( member2 @ B @ K3 @ ( set2 @ B @ Ys ) )
=> ( ( filter2 @ B
@ ^ [X3: B] :
( ( F @ K3 )
= ( F @ X3 ) )
@ Ys )
= ( filter2 @ B
@ ^ [X3: B] :
( ( F @ K3 )
= ( F @ X3 ) )
@ Xs ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Ys ) )
=> ( ( linorder_sort_key @ B @ A @ F @ Xs )
= Ys ) ) ) ) ) ).
% properties_for_sort_key
thf(fact_4386_list__all2__Nil2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: list @ A] :
( ( list_all2 @ A @ B @ P @ Xs @ ( nil @ B ) )
= ( Xs
= ( nil @ A ) ) ) ).
% list_all2_Nil2
thf(fact_4387_list__all2__Nil,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( nil @ A ) @ Ys )
= ( Ys
= ( nil @ B ) ) ) ).
% list_all2_Nil
thf(fact_4388_mset__zero__iff,axiom,
! [A: $tType,X: list @ A] :
( ( ( mset @ A @ X )
= ( zero_zero @ ( multiset @ A ) ) )
= ( X
= ( nil @ A ) ) ) ).
% mset_zero_iff
thf(fact_4389_mset__zero__iff__right,axiom,
! [A: $tType,X: list @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( mset @ A @ X ) )
= ( X
= ( nil @ A ) ) ) ).
% mset_zero_iff_right
thf(fact_4390_list__all2__rev,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( rev @ A @ Xs ) @ ( rev @ B @ Ys ) )
= ( list_all2 @ A @ B @ P @ Xs @ Ys ) ) ).
% list_all2_rev
thf(fact_4391_mergesort__by__rel__permutes,axiom,
! [A: $tType,R: A > A > $o,Xs: list @ A] :
( ( mset @ A @ ( mergesort_by_rel @ A @ R @ Xs ) )
= ( mset @ A @ Xs ) ) ).
% mergesort_by_rel_permutes
thf(fact_4392_mset__mergesort__by__rel__merge,axiom,
! [A: $tType,R: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( mset @ A @ ( merges9089515139780605204_merge @ A @ R @ Xs @ Ys ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys ) ) ) ).
% mset_mergesort_by_rel_merge
thf(fact_4393_quicksort__permutes,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( mset @ A @ ( linorder_quicksort @ A @ Xs ) )
= ( mset @ A @ Xs ) ) ) ).
% quicksort_permutes
thf(fact_4394_quicksort__by__rel__permutes,axiom,
! [A: $tType,R: A > A > $o,Sl2: list @ A,Xs: list @ A] :
( ( mset @ A @ ( quicksort_by_rel @ A @ R @ Sl2 @ Xs ) )
= ( mset @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% quicksort_by_rel_permutes
thf(fact_4395_mset__mergesort__by__rel__split,axiom,
! [A: $tType,Xs1: list @ A,Xs23: 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 @ Xs23 ) @ Xs ) ) ) @ ( mset @ A @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs23 ) @ Xs ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Xs1 ) ) @ ( mset @ A @ Xs23 ) ) ) ).
% mset_mergesort_by_rel_split
thf(fact_4396_list_Orel__cong,axiom,
! [A: $tType,B: $tType,X: list @ A,Ya: list @ A,Y: list @ B,Xa: list @ B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: A,Yb: B] :
( ( member2 @ A @ Z4 @ ( set2 @ A @ Ya ) )
=> ( ( member2 @ B @ Yb @ ( set2 @ B @ Xa ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( list_all2 @ A @ B @ R @ X @ Y )
= ( list_all2 @ A @ B @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_4397_list_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: list @ A,Y: list @ B,Ra: A > B > $o] :
( ( list_all2 @ A @ B @ R @ X @ Y )
=> ( ! [Z4: A,Yb: B] :
( ( member2 @ A @ Z4 @ ( set2 @ A @ X ) )
=> ( ( member2 @ B @ Yb @ ( set2 @ B @ Y ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( list_all2 @ A @ B @ Ra @ X @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_4398_list_Orel__refl__strong,axiom,
! [A: $tType,X: list @ A,Ra: A > A > $o] :
( ! [Z4: A] :
( ( member2 @ A @ Z4 @ ( set2 @ A @ X ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( list_all2 @ A @ A @ Ra @ X @ X ) ) ).
% list.rel_refl_strong
thf(fact_4399_list__all2__same,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( list_all2 @ A @ A @ P @ Xs @ Xs )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 @ X3 ) ) ) ) ).
% list_all2_same
thf(fact_4400_list_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,X22: list @ A,Y21: B,Y22: list @ B] :
( ( list_all2 @ A @ B @ R @ ( cons @ A @ X21 @ X22 ) @ ( cons @ B @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( list_all2 @ A @ B @ R @ X22 @ Y22 ) ) ) ).
% list.rel_inject(2)
thf(fact_4401_list_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X22: list @ A,Y22: list @ B] :
( ( R @ X21 @ Y21 )
=> ( ( list_all2 @ A @ B @ R @ X22 @ Y22 )
=> ( list_all2 @ A @ B @ R @ ( cons @ A @ X21 @ X22 ) @ ( cons @ B @ Y21 @ Y22 ) ) ) ) ).
% list.rel_intros(2)
thf(fact_4402_list__all2__Cons,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( ( P @ X @ Y )
& ( list_all2 @ A @ B @ P @ Xs @ Ys ) ) ) ).
% list_all2_Cons
thf(fact_4403_list__all2__Cons1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( cons @ A @ X @ Xs ) @ Ys )
= ( ? [Z3: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z3 @ Zs2 ) )
& ( P @ X @ Z3 )
& ( list_all2 @ A @ B @ P @ Xs @ Zs2 ) ) ) ) ).
% list_all2_Cons1
thf(fact_4404_list__all2__Cons2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: list @ A,Y: B,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs @ ( cons @ B @ Y @ Ys ) )
= ( ? [Z3: A,Zs2: list @ A] :
( ( Xs
= ( cons @ A @ Z3 @ Zs2 ) )
& ( P @ Z3 @ Y )
& ( list_all2 @ A @ B @ P @ Zs2 @ Ys ) ) ) ) ).
% list_all2_Cons2
thf(fact_4405_list_Orel__mono,axiom,
! [B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
=> ( ord_less_eq @ ( ( list @ A ) > ( list @ B ) > $o ) @ ( list_all2 @ A @ B @ R ) @ ( list_all2 @ A @ B @ Ra ) ) ) ).
% list.rel_mono
thf(fact_4406_list__all2__appendI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,A3: list @ A,B2: list @ B,C3: list @ A,D3: list @ B] :
( ( list_all2 @ A @ B @ P @ A3 @ B2 )
=> ( ( list_all2 @ A @ B @ P @ C3 @ D3 )
=> ( list_all2 @ A @ B @ P @ ( append @ A @ A3 @ C3 ) @ ( append @ B @ B2 @ D3 ) ) ) ) ).
% list_all2_appendI
thf(fact_4407_list__all2__dropI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,N: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( list_all2 @ A @ B @ P @ ( drop @ A @ N @ Xs ) @ ( drop @ B @ N @ Ys ) ) ) ).
% list_all2_dropI
thf(fact_4408_list__all2__update__cong,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,X: A,Y: B,I: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( P @ X @ Y )
=> ( list_all2 @ A @ B @ P @ ( list_update @ A @ Xs @ I @ X ) @ ( list_update @ B @ Ys @ I @ Y ) ) ) ) ).
% list_all2_update_cong
thf(fact_4409_list__all2__lengthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_4410_list__all2__antisym,axiom,
! [A: $tType,P: A > A > $o,Q: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ! [X2: A,Y2: A] :
( ( P @ X2 @ Y2 )
=> ( ( Q @ Y2 @ X2 )
=> ( X2 = Y2 ) ) )
=> ( ( list_all2 @ A @ A @ P @ Xs @ Ys )
=> ( ( list_all2 @ A @ A @ Q @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% list_all2_antisym
thf(fact_4411_list__all2__trans,axiom,
! [B: $tType,A: $tType,C: $tType,P1: A > B > $o,P22: B > C > $o,P32: A > C > $o,As2: list @ A,Bs: list @ B,Cs2: list @ C] :
( ! [A6: A,B6: B,C2: C] :
( ( P1 @ A6 @ B6 )
=> ( ( P22 @ B6 @ C2 )
=> ( P32 @ A6 @ C2 ) ) )
=> ( ( list_all2 @ A @ B @ P1 @ As2 @ Bs )
=> ( ( list_all2 @ B @ C @ P22 @ Bs @ Cs2 )
=> ( list_all2 @ A @ C @ P32 @ As2 @ Cs2 ) ) ) ) ).
% list_all2_trans
thf(fact_4412_list__all2__refl,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ! [X2: A] : ( P @ X2 @ X2 )
=> ( list_all2 @ A @ A @ P @ Xs @ Xs ) ) ).
% list_all2_refl
thf(fact_4413_list__all2__mono,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,Q: A > B > $o] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ! [Xs2: A,Ys2: B] :
( ( P @ Xs2 @ Ys2 )
=> ( Q @ Xs2 @ Ys2 ) )
=> ( list_all2 @ A @ B @ Q @ Xs @ Ys ) ) ) ).
% list_all2_mono
thf(fact_4414_list__all2__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z5: list @ A] : Y5 = Z5 )
= ( list_all2 @ A @ A
@ ^ [Y5: A,Z5: A] : Y5 = Z5 ) ) ).
% list_all2_eq
thf(fact_4415_list_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: list @ B] :
( ! [X2: B] : ( Ra @ X2 @ X2 )
=> ( list_all2 @ B @ B @ Ra @ X @ X ) ) ).
% list.rel_refl
thf(fact_4416_list_Orel__eq,axiom,
! [A: $tType] :
( ( list_all2 @ A @ A
@ ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [Y5: list @ A,Z5: list @ A] : Y5 = Z5 ) ) ).
% list.rel_eq
thf(fact_4417_list__all2__rev1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( rev @ A @ Xs ) @ Ys )
= ( list_all2 @ A @ B @ P @ Xs @ ( rev @ B @ Ys ) ) ) ).
% list_all2_rev1
thf(fact_4418_list__all2__takeI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,N: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( list_all2 @ A @ B @ P @ ( take @ A @ N @ Xs ) @ ( take @ B @ N @ Ys ) ) ) ).
% list_all2_takeI
thf(fact_4419_mset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( mset @ A @ ( nil @ A ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% mset.simps(1)
thf(fact_4420_list_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( list_all2 @ A @ B @ R @ ( nil @ A ) @ ( nil @ B ) ) ).
% list.ctr_transfer(1)
thf(fact_4421_list__all2__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As2: list @ A,F: C > B,Bs: list @ C] :
( ( list_all2 @ A @ B @ P @ As2 @ ( map @ C @ B @ F @ Bs ) )
= ( list_all2 @ A @ C
@ ^ [X3: A,Y3: C] : ( P @ X3 @ ( F @ Y3 ) )
@ As2
@ Bs ) ) ).
% list_all2_map2
thf(fact_4422_list__all2__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F: C > A,As2: list @ C,Bs: list @ B] :
( ( list_all2 @ A @ B @ P @ ( map @ C @ A @ F @ As2 ) @ Bs )
= ( list_all2 @ C @ B
@ ^ [X3: C] : ( P @ ( F @ X3 ) )
@ As2
@ Bs ) ) ).
% list_all2_map1
thf(fact_4423_list_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: list @ A,Y: list @ B] :
( ( list_all2 @ C @ B @ Sb @ ( map @ A @ C @ I @ X ) @ Y )
= ( list_all2 @ A @ B
@ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
@ X
@ Y ) ) ).
% list.rel_map(1)
thf(fact_4424_list_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: list @ A,G: B > C,Y: list @ B] :
( ( list_all2 @ A @ C @ Sa @ X @ ( map @ B @ C @ G @ Y ) )
= ( list_all2 @ A @ B
@ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
@ X
@ Y ) ) ).
% list.rel_map(2)
thf(fact_4425_list_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: A,Y22: list @ A] :
~ ( list_all2 @ A @ B @ R @ ( cons @ A @ Y21 @ Y22 ) @ ( nil @ B ) ) ).
% list.rel_distinct(2)
thf(fact_4426_list_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: B,Y22: list @ B] :
~ ( list_all2 @ A @ B @ R @ ( nil @ A ) @ ( cons @ B @ Y21 @ Y22 ) ) ).
% list.rel_distinct(1)
thf(fact_4427_list_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A3: list @ A,B2: list @ B] :
( ( list_all2 @ A @ B @ R @ A3 @ B2 )
=> ( ( ( A3
= ( nil @ A ) )
=> ( B2
!= ( nil @ B ) ) )
=> ~ ! [X1: A,X23: list @ A] :
( ( A3
= ( cons @ A @ X1 @ X23 ) )
=> ! [Y1: B,Y23: list @ B] :
( ( B2
= ( cons @ B @ Y1 @ Y23 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( list_all2 @ A @ B @ R @ X23 @ Y23 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_4428_list_Orel__induct,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: list @ A,Y: list @ B,Q: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ R @ X @ Y )
=> ( ( Q @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [A21: A,A222: list @ A,B21: B,B222: list @ B] :
( ( R @ A21 @ B21 )
=> ( ( Q @ A222 @ B222 )
=> ( Q @ ( cons @ A @ A21 @ A222 ) @ ( cons @ B @ B21 @ B222 ) ) ) )
=> ( Q @ X @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_4429_List_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,R: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( R @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( P @ X2 @ Y2 )
=> ( ( list_all2 @ A @ B @ P @ Xs2 @ Ys2 )
=> ( ( R @ Xs2 @ Ys2 )
=> ( R @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) ) )
=> ( R @ Xs @ Ys ) ) ) ) ).
% List.list_all2_induct
thf(fact_4430_Misc_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: A > B > $o,L: list @ A,L3: list @ B,Q: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ P @ L @ L3 )
=> ( ( Q @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,X10: B,Ls2: list @ A,Ls4: list @ B] :
( ( P @ X2 @ X10 )
=> ( ( list_all2 @ A @ B @ P @ Ls2 @ Ls4 )
=> ( ( Q @ Ls2 @ Ls4 )
=> ( Q @ ( cons @ A @ X2 @ Ls2 ) @ ( cons @ B @ X10 @ Ls4 ) ) ) ) )
=> ( Q @ L @ L3 ) ) ) ) ).
% Misc.list_all2_induct
thf(fact_4431_list__all2__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: A > B > $o,Us2: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( list_all2 @ A @ B @ P @ ( append @ A @ Xs @ Us2 ) @ ( append @ B @ Ys @ Vs ) )
= ( ( list_all2 @ A @ B @ P @ Xs @ Ys )
& ( list_all2 @ A @ B @ P @ Us2 @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_4432_list__all2__append1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ A,Zs: list @ B] :
( ( list_all2 @ A @ B @ P @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( ? [Us: list @ B,Vs2: list @ B] :
( ( Zs
= ( append @ B @ Us @ Vs2 ) )
& ( ( size_size @ ( list @ B ) @ Us )
= ( size_size @ ( list @ A ) @ Xs ) )
& ( ( size_size @ ( list @ B ) @ Vs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
& ( list_all2 @ A @ B @ P @ Xs @ Us )
& ( list_all2 @ A @ B @ P @ Ys @ Vs2 ) ) ) ) ).
% list_all2_append1
thf(fact_4433_list__all2__append2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,Zs: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs @ ( append @ B @ Ys @ Zs ) )
= ( ? [Us: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us @ Vs2 ) )
& ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ B ) @ Ys ) )
& ( ( size_size @ ( list @ A ) @ Vs2 )
= ( size_size @ ( list @ B ) @ Zs ) )
& ( list_all2 @ A @ B @ P @ Us @ Ys )
& ( list_all2 @ A @ B @ P @ Vs2 @ Zs ) ) ) ) ).
% list_all2_append2
thf(fact_4434_list__all2__conv__all__nth,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P3: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ ( nth @ A @ Xs3 @ I2 ) @ ( nth @ B @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_4435_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ! [N7: nat] :
( ( ord_less @ nat @ N7 @ ( size_size @ ( list @ A ) @ A3 ) )
=> ( P @ ( nth @ A @ A3 @ N7 ) @ ( nth @ B @ B2 @ N7 ) ) )
=> ( list_all2 @ A @ B @ P @ A3 @ B2 ) ) ) ).
% list_all2_all_nthI
thf(fact_4436_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,P4: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ord_less @ nat @ P4 @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( P @ ( nth @ A @ Xs @ P4 ) @ ( nth @ B @ Ys @ P4 ) ) ) ) ).
% list_all2_nthD2
thf(fact_4437_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,P4: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ord_less @ nat @ P4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ P4 ) @ ( nth @ B @ Ys @ P4 ) ) ) ) ).
% list_all2_nthD
thf(fact_4438_list_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R6: A > B > $o,A4: list @ A,B3: list @ B] :
( ( ( A4
= ( nil @ A ) )
= ( B3
= ( nil @ B ) ) )
& ( ( A4
!= ( nil @ A ) )
=> ( ( B3
!= ( nil @ B ) )
=> ( ( R6 @ ( hd @ A @ A4 ) @ ( hd @ B @ B3 ) )
& ( list_all2 @ A @ B @ R6 @ ( tl @ A @ A4 ) @ ( tl @ B @ B3 ) ) ) ) ) ) ) ) ).
% list.rel_sel
thf(fact_4439_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_4440_product__lists__set,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) )
= ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( list_all2 @ A @ ( list @ A )
@ ^ [X3: A,Ys3: list @ A] : ( member2 @ A @ X3 @ ( set2 @ A @ Ys3 ) )
@ Xs3
@ Xss2 ) ) ) ).
% product_lists_set
thf(fact_4441_list__all2I,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
( ! [X2: product_prod @ A @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ X2 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ A3 @ B2 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P @ X2 ) )
=> ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( list_all2 @ A @ B @ P @ A3 @ B2 ) ) ) ).
% list_all2I
thf(fact_4442_Linear__order__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_679001287576687338der_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) ) ) ).
% Linear_order_Restr
thf(fact_4443_list__all2__iff,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P3: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [X3: product_prod @ A @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ X3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P3 @ X3 ) ) ) ) ) ).
% list_all2_iff
thf(fact_4444_sort__key__inj__key__eq,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,Ys: list @ B,F: B > A] :
( ( ( mset @ B @ Xs )
= ( mset @ B @ Ys ) )
=> ( ( inj_on @ B @ A @ F @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Ys ) )
=> ( ( linorder_sort_key @ B @ A @ F @ Xs )
= Ys ) ) ) ) ) ).
% sort_key_inj_key_eq
thf(fact_4445_linear__order__on__Restr,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( order_679001287576687338der_on @ A @ A5 @ R3 )
=> ( order_679001287576687338der_on @ A @ ( inf_inf @ ( set @ A ) @ A5 @ ( order_above @ A @ R3 @ X ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_above @ A @ R3 @ X )
@ ^ [Uu3: A] : ( order_above @ A @ R3 @ X ) ) ) ) ) ).
% linear_order_on_Restr
thf(fact_4446_mset__zip__take__Cons__drop__twice,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,J: nat,X: A,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( append @ A @ ( take @ A @ J @ Xs ) @ ( cons @ A @ X @ ( drop @ A @ J @ Xs ) ) ) @ ( append @ B @ ( take @ B @ J @ Ys ) @ ( cons @ B @ Y @ ( drop @ B @ J @ Ys ) ) ) ) )
= ( add_mset @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ) ).
% mset_zip_take_Cons_drop_twice
thf(fact_4447_takeWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( takeWhile @ A
@ ^ [Y3: A] : Y3 != X
@ ( rev @ A @ Xs ) )
= ( rev @ A
@ ( tl @ A
@ ( dropWhile @ A
@ ^ [Y3: A] : Y3 != X
@ Xs ) ) ) ) ) ) ).
% takeWhile_neq_rev
thf(fact_4448_stable__sort__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord3483353639454293061rt_key @ B @ A )
= ( ^ [Sk: ( B > A ) > ( list @ B ) > ( list @ B )] :
! [F3: B > A,Xs3: list @ B,K2: A] :
( ( filter2 @ B
@ ^ [Y3: B] :
( ( F3 @ Y3 )
= K2 )
@ ( Sk @ F3 @ Xs3 ) )
= ( filter2 @ B
@ ^ [Y3: B] :
( ( F3 @ Y3 )
= K2 )
@ Xs3 ) ) ) ) ) ).
% stable_sort_key_def
thf(fact_4449_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X2: A,Y2: B] :
( ( P @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_4450_dropWhile__idem,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( dropWhile @ A @ P @ ( dropWhile @ A @ P @ Xs ) )
= ( dropWhile @ A @ P @ Xs ) ) ).
% dropWhile_idem
thf(fact_4451_dropWhile__eq__Nil__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
= ( nil @ A ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_4452_dropWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_4453_dropWhile__append1,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_4454_dropWhile__replicate,axiom,
! [A: $tType,P: A > $o,X: A,N: nat] :
( ( ( P @ X )
=> ( ( dropWhile @ A @ P @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) )
& ( ~ ( P @ X )
=> ( ( dropWhile @ A @ P @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) ) ) ).
% dropWhile_replicate
thf(fact_4455_takeWhile__dropWhile__id,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( append @ A @ ( takeWhile @ A @ P @ Xs ) @ ( dropWhile @ A @ P @ Xs ) )
= Xs ) ).
% takeWhile_dropWhile_id
thf(fact_4456_mset__single__iff__right,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) )
= ( mset @ A @ Xs ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% mset_single_iff_right
thf(fact_4457_mset__single__iff,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( mset @ A @ Xs )
= ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% mset_single_iff
thf(fact_4458_union__diff__assoc,axiom,
! [A: $tType,C4: multiset @ A,B4: multiset @ A,A5: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C4 @ B4 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) @ C4 )
= ( plus_plus @ ( multiset @ A ) @ A5 @ ( minus_minus @ ( multiset @ A ) @ B4 @ C4 ) ) ) ) ).
% union_diff_assoc
thf(fact_4459_mset__single__cases,axiom,
! [A: $tType,S4: A,C3: multiset @ A,R8: A,C9: multiset @ A] :
( ( ( add_mset @ A @ S4 @ C3 )
= ( add_mset @ A @ R8 @ C9 ) )
=> ( ( ( S4 = R8 )
=> ( C3 != C9 ) )
=> ~ ( ( C9
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( C3
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R8 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R8 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R8 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases
thf(fact_4460_mset__single__cases_H,axiom,
! [A: $tType,S4: A,C3: multiset @ A,R8: A,C9: multiset @ A] :
( ( ( add_mset @ A @ S4 @ C3 )
= ( add_mset @ A @ R8 @ C9 ) )
=> ( ( ( S4 = R8 )
=> ( C3 != C9 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C9
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( C3
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R8 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R8 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases'
thf(fact_4461_mset__single__cases2,axiom,
! [A: $tType,S4: A,C3: multiset @ A,R8: A,C9: multiset @ A] :
( ( ( add_mset @ A @ S4 @ C3 )
= ( add_mset @ A @ R8 @ C9 ) )
=> ( ( ( S4 = R8 )
=> ( C3 != C9 ) )
=> ~ ( ( C9
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C3
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R8 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ R8 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R8 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases2
thf(fact_4462_mset__single__cases2_H,axiom,
! [A: $tType,S4: A,C3: multiset @ A,R8: A,C9: multiset @ A] :
( ( ( add_mset @ A @ S4 @ C3 )
= ( add_mset @ A @ R8 @ C9 ) )
=> ( ( ( S4 = R8 )
=> ( C3 != C9 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C9
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C3
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ R8 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ R8 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases2'
thf(fact_4463_mset__unplusm__dist__cases,axiom,
! [A: $tType,S4: A,A5: multiset @ A,B4: multiset @ A,C4: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) @ A5 )
= ( plus_plus @ ( multiset @ A ) @ B4 @ C4 ) )
=> ( ( ( B4
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C4 ) ) )
=> ~ ( ( C4
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ B4 @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases
thf(fact_4464_mset__unplusm__dist__cases2,axiom,
! [A: $tType,B4: multiset @ A,C4: multiset @ A,S4: A,A5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ B4 @ C4 )
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) @ A5 ) )
=> ( ( ( B4
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C4 ) ) )
=> ~ ( ( C4
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ B4 @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases2
thf(fact_4465_dropWhile_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( dropWhile @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% dropWhile.simps(1)
thf(fact_4466_mset__union__2__elem,axiom,
! [A: $tType,A3: A,B2: A,C3: A,M: multiset @ A] :
( ( ( add_mset @ A @ A3 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( add_mset @ A @ C3 @ M ) )
=> ( ( ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
= M )
& ( B2 = C3 ) )
| ( ( A3 = C3 )
& ( ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
= M ) ) ) ) ).
% mset_union_2_elem
thf(fact_4467_mset__distrib,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,M: multiset @ A,N4: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A5 @ B4 )
= ( plus_plus @ ( multiset @ A ) @ M @ N4 ) )
=> ~ ! [Am: multiset @ A,An: multiset @ A] :
( ( A5
= ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
=> ! [Bm: multiset @ A,Bn: multiset @ A] :
( ( B4
= ( plus_plus @ ( multiset @ A ) @ Bm @ Bn ) )
=> ( ( M
= ( plus_plus @ ( multiset @ A ) @ Am @ Bm ) )
=> ( N4
!= ( plus_plus @ ( multiset @ A ) @ An @ Bn ) ) ) ) ) ) ).
% mset_distrib
thf(fact_4468_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_4469_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_4470_distinct__dropWhile,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( dropWhile @ A @ P @ Xs ) ) ) ).
% distinct_dropWhile
thf(fact_4471_dropWhile_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( dropWhile @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( dropWhile @ A @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( dropWhile @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_4472_set__dropWhileD,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_4473_dropWhile__cong,axiom,
! [A: $tType,L: list @ A,K: list @ A,P: A > $o,Q: A > $o] :
( ( L = K )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ L ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( dropWhile @ A @ P @ L )
= ( dropWhile @ A @ Q @ K ) ) ) ) ).
% dropWhile_cong
thf(fact_4474_mset_Osimps_I2_J,axiom,
! [A: $tType,A3: A,X: list @ A] :
( ( mset @ A @ ( cons @ A @ A3 @ X ) )
= ( add_mset @ A @ A3 @ ( mset @ A @ X ) ) ) ).
% mset.simps(2)
thf(fact_4475_dropWhile__append3,axiom,
! [A: $tType,P: A > $o,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( P @ Y )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_4476_length__dropWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_dropWhile_le
thf(fact_4477_sorted__dropWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P @ Xs ) ) ) ) ).
% sorted_dropWhile
thf(fact_4478_hd__dropWhile,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
!= ( nil @ A ) )
=> ~ ( P @ ( hd @ A @ ( dropWhile @ A @ P @ Xs ) ) ) ) ).
% hd_dropWhile
thf(fact_4479_dropWhile__eq__self__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
= Xs )
= ( ( Xs
= ( nil @ A ) )
| ~ ( P @ ( hd @ A @ Xs ) ) ) ) ).
% dropWhile_eq_self_iff
thf(fact_4480_dropWhile__last,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( last @ A @ ( dropWhile @ A @ P @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% dropWhile_last
thf(fact_4481_dropWhile__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs: list @ B] :
( ( dropWhile @ A @ P @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( dropWhile @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs ) ) ) ).
% dropWhile_map
thf(fact_4482_remdups__adj__Cons_H,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y3: A] : Y3 = X
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_4483_dropWhile__eq__Cons__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ( Xs
= ( append @ A @ ( takeWhile @ A @ P @ Xs ) @ ( cons @ A @ Y @ Ys ) ) )
& ~ ( P @ Y ) ) ) ).
% dropWhile_eq_Cons_conv
thf(fact_4484_takeWhile__eq__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs ) ) )
=> ~ ( P @ X2 ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% takeWhile_eq_filter
thf(fact_4485_dropWhile__eq__drop,axiom,
! [A: $tType] :
( ( dropWhile @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] : ( drop @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs3 ) ) @ Xs3 ) ) ) ).
% dropWhile_eq_drop
thf(fact_4486_dropWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) )
& ( ~ ! [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
=> ( P @ X6 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append
thf(fact_4487_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_4488_remdups__adj__append__dropWhile,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X3: A] : X3 = Y
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_4489_remdups__adj__append_H_H,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( remdups_adj @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remdups_adj @ A @ Xs )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y3: A] :
( Y3
= ( last @ A @ Xs ) )
@ Ys ) ) ) ) ) ).
% remdups_adj_append''
thf(fact_4490_tl__remdups__adj,axiom,
! [A: $tType,Ys: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( tl @ A @ ( remdups_adj @ A @ Ys ) )
= ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X3: A] :
( X3
= ( hd @ A @ Ys ) )
@ ( tl @ A @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_4491_stable__sort__key__sort__key,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( linord3483353639454293061rt_key @ A @ B @ ( linorder_sort_key @ A @ B ) ) ) ).
% stable_sort_key_sort_key
thf(fact_4492_dropWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P @ Xs ) @ J )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_4493_dropWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( dropWhile @ A
@ ^ [Y3: A] : Y3 != X
@ ( rev @ A @ Xs ) )
= ( cons @ A @ X
@ ( rev @ A
@ ( takeWhile @ A
@ ^ [Y3: A] : Y3 != X
@ Xs ) ) ) ) ) ) ).
% dropWhile_neq_rev
thf(fact_4494_sorted__list__of__multiset__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( linord6283353356039996273ltiset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted_list_of_multiset_singleton
thf(fact_4495_shuffles_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( insert2 @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( insert2 @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ( Y
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y2 @ Ys2 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y2 ) @ ( shuffles @ A @ ( cons @ A @ X2 @ Xs2 ) @ Ys2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_4496_map__rec,axiom,
! [A: $tType,B: $tType] :
( ( map @ B @ A )
= ( ^ [F3: B > A] :
( rec_list @ ( list @ A ) @ B @ ( nil @ A )
@ ^ [X3: B,Uu3: list @ B] : ( cons @ A @ ( F3 @ X3 ) ) ) ) ) ).
% map_rec
thf(fact_4497_set__rec,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( rec_list @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X3: A,Uu3: list @ A] : ( insert2 @ A @ X3 ) ) ) ).
% set_rec
thf(fact_4498_sorted__list__of__multiset__empty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord6283353356039996273ltiset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( nil @ A ) ) ) ).
% sorted_list_of_multiset_empty
thf(fact_4499_shuffles_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ( P @ ( nil @ A ) @ Ys2 ) )
=> ( ! [Xs2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ( P @ Xs2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ( P @ Xs2 @ ( cons @ A @ Y2 @ Ys2 ) )
=> ( ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_4500_list_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F22: A > ( list @ A ) > C > C,X21: A,X22: list @ A] :
( ( rec_list @ C @ A @ F1 @ F22 @ ( cons @ A @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 @ ( rec_list @ C @ A @ F1 @ F22 @ X22 ) ) ) ).
% list.simps(7)
thf(fact_4501_list_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F22: A > ( list @ A ) > C > C] :
( ( rec_list @ C @ A @ F1 @ F22 @ ( nil @ A ) )
= F1 ) ).
% list.simps(6)
thf(fact_4502_rec__list__Cons__imp,axiom,
! [B: $tType,A: $tType,F: ( list @ A ) > B,F1: B,F22: A > ( list @ A ) > B > B,X: A,Xs: list @ A] :
( ( F
= ( rec_list @ B @ A @ F1 @ F22 ) )
=> ( ( F @ ( cons @ A @ X @ Xs ) )
= ( F22 @ X @ Xs @ ( F @ Xs ) ) ) ) ).
% rec_list_Cons_imp
thf(fact_4503_rec__list__Nil__imp,axiom,
! [A: $tType,B: $tType,F: ( list @ A ) > B,F1: B,F22: A > ( list @ A ) > B > B] :
( ( F
= ( rec_list @ B @ A @ F1 @ F22 ) )
=> ( ( F @ ( nil @ A ) )
= F1 ) ) ).
% rec_list_Nil_imp
thf(fact_4504_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_4505_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_4506_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) )
=> ( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ) ).
% shuffles.psimps(3)
thf(fact_4507_list_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: C,Ga: B > ( list @ B ) > C > C,F: A > B] :
( ( comp @ ( list @ B ) @ C @ ( list @ A ) @ ( rec_list @ C @ B @ G @ Ga ) @ ( map @ A @ B @ F ) )
= ( rec_list @ C @ A @ G
@ ^ [X3: A,Xa4: list @ A] : ( Ga @ ( F @ X3 ) @ ( map @ A @ B @ F @ Xa4 ) ) ) ) ).
% list.rec_o_map
thf(fact_4508_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,As: list @ A] :
( ( X
= ( cons @ A @ A6 @ As ) )
=> ( ( Y
= ( revg @ A @ As @ ( cons @ A @ A6 @ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A6 @ As ) @ Xa ) ) ) ) ) ) ) ).
% revg.pelims
thf(fact_4509_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 ) ) ) )
=> ( ! [V2: A,Va2: list @ A] :
( ( X
= ( cons @ A @ V2 @ Va2 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( cons @ A @ V2 @ Va2 ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X1: A,L1: list @ A] :
( ( X
= ( cons @ A @ X1 @ L1 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ( ( ( ( ord_less @ A @ X1 @ X23 )
=> ( Y
= ( cons @ A @ X1 @ ( merge @ A @ L1 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X23 )
=> ( ( ( X1 = X23 )
=> ( Y
= ( cons @ A @ X1 @ ( merge @ A @ L1 @ L22 ) ) ) )
& ( ( X1 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ L22 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ) ).
% merge.pelims
thf(fact_4510_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 ) ) )
=> ! [L2: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L2 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y = L2 )
=> ~ ( 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 ) @ L2 @ ( 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 ) )
=> ! [L2: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L2 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L2 @ ( 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 ) @ L2 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ~ ! [L1: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) )
=> ( ( Y
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ X ) @ Ls2 ) )
=> ~ ( 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 @ Ls2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
thf(fact_4511_splice_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ( P @ ( nil @ A ) @ Ys2 ) )
=> ( ! [X2: A,Xs2: list @ A,Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Ys2 ) )
=> ( ( P @ Ys2 @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_4512_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_4513_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_4514_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_4515_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_4516_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 ) ) ) )
=> ( ! [L2: 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 ) @ L2 @ ( nil @ ( list @ A ) ) ) ) )
=> ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L2 @ ( 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 ),L2: 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 ) @ L2 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L2 @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L2 @ ( nil @ ( list @ A ) ) ) ) ) )
=> ( ! [Acc2: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc2 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) )
=> ( ( P @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ Acc2 ) @ Ls2 )
=> ( P @ Acc2 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ).
% merge_list.pinduct
thf(fact_4517_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc22: list @ ( list @ A ),L12: list @ A,L23: list @ A,Ls: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) ) )
=> ( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ Acc22 ) @ Ls ) ) ) ) ).
% merge_list.psimps(5)
thf(fact_4518_splice_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( cons @ A @ X2 @ ( splice @ A @ Xa @ Xs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_4519_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ) ).
% splice.psimps(1)
thf(fact_4520_splice_Opsimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) )
=> ( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ) ).
% splice.psimps(2)
thf(fact_4521_map__of__rev__distinct,axiom,
! [B: $tType,A: $tType,M2: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ M2 ) )
=> ( ( map_of @ A @ B @ ( rev @ ( product_prod @ A @ B ) @ M2 ) )
= ( map_of @ A @ B @ M2 ) ) ) ).
% map_of_rev_distinct
thf(fact_4522_split__Nil__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( splice @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% split_Nil_iff
thf(fact_4523_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_4524_splice__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( member2 @ ( list @ A ) @ ( splice @ A @ Xs @ Ys ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% splice_in_shuffles
thf(fact_4525_length__splice,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_splice
thf(fact_4526_splice__replicate,axiom,
! [A: $tType,M2: nat,X: A,N: nat] :
( ( splice @ A @ ( replicate @ A @ M2 @ X ) @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ ( plus_plus @ nat @ M2 @ N ) @ X ) ) ).
% splice_replicate
thf(fact_4527_splice_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_4528_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_4529_splice_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( Y
!= ( cons @ A @ X2 @ ( splice @ A @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_4530_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_4531_map__of__map__to__set,axiom,
! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B ),M2: A > ( option @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
=> ( ( ( map_of @ A @ B @ L )
= M2 )
= ( ( set2 @ ( product_prod @ A @ B ) @ L )
= ( map_to_set @ A @ B @ M2 ) ) ) ) ).
% map_of_map_to_set
thf(fact_4532_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_4533_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] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member2 @ 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_4534_map__to__set__empty,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B
@ ^ [X3: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% map_to_set_empty
thf(fact_4535_map__to__set__ran,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M3: B > ( option @ A )] : ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map_to_set @ B @ A @ M3 ) ) ) ) ).
% map_to_set_ran
thf(fact_4536_map__to__set__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( ( map_to_set @ A @ B @ M2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( M2
= ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(1)
thf(fact_4537_map__to__set__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
= ( map_to_set @ A @ B @ M2 ) )
= ( M2
= ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(2)
thf(fact_4538_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_4539_inj__on__fst__map__to__set,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) ) ).
% inj_on_fst_map_to_set
thf(fact_4540_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_4541_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X3: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_4542_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
( ( rel_of @ A @ B
@ ^ [X3: A] : ( none @ B )
@ P )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% rel_of_empty
thf(fact_4543_inf__None__2,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [X: option @ A] :
( ( inf_inf @ ( option @ A ) @ X @ ( none @ A ) )
= ( none @ A ) ) ) ).
% inf_None_2
thf(fact_4544_inf__None__1,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [Y: option @ A] :
( ( inf_inf @ ( option @ A ) @ ( none @ A ) @ Y )
= ( none @ A ) ) ) ).
% inf_None_1
thf(fact_4545_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] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member2 @ 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_4546_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] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member2 @ 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_4547_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A,V: B] :
( ( map_to_set @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( some @ B @ V ) ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V )
@ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( map_to_set @ A @ B @ M2 )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [V5: B] :
( Uu3
= ( product_Pair @ A @ B @ K @ V5 ) ) ) ) ) ) ).
% map_to_set_upd
thf(fact_4548_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_4549_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
( ( ! [X3: A,Y3: B] :
( V
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) )
= ( V
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_4550_set__to__map__empty,axiom,
! [B: $tType,A: $tType] :
( ( set_to_map @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% set_to_map_empty
thf(fact_4551_inf__Some,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( some @ A @ ( inf_inf @ A @ X @ Y ) ) ) ) ).
% inf_Some
thf(fact_4552_map__update__eta__repair_I2_J,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A,V: B] :
( ( ( M2 @ K )
= ( none @ B ) )
=> ( ( ran @ A @ B
@ ^ [X3: A] : ( if @ ( option @ B ) @ ( X3 = K ) @ ( some @ B @ V ) @ ( M2 @ X3 ) ) )
= ( insert2 @ B @ V @ ( ran @ A @ B @ M2 ) ) ) ) ).
% map_update_eta_repair(2)
thf(fact_4553_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ran @ nat @ A
@ ^ [I2: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I2 ) ) @ ( none @ A ) ) )
= ( set2 @ A @ L ) ) ).
% ran_nth_set_encoding_conv
thf(fact_4554_le__some__optE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M2: A,X: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M2 ) @ X )
=> ~ ! [M5: A] :
( ( X
= ( some @ A @ M5 ) )
=> ~ ( ord_less_eq @ A @ M2 @ M5 ) ) ) ) ).
% le_some_optE
thf(fact_4555_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 ) )
= ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ S ) ) ) ).
% set_to_map_simp
thf(fact_4556_Some__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ A @ ( complete_Sup_Sup @ A @ A5 ) )
= ( complete_Sup_Sup @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) ) ) ) ) ).
% Some_Sup
thf(fact_4557_map__to__set__inverse,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( set_to_map @ A @ B @ ( map_to_set @ A @ B @ M2 ) )
= M2 ) ).
% map_to_set_inverse
thf(fact_4558_map__to__set__def,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K2: A,V4: B] :
( ( M3 @ K2 )
= ( some @ B @ V4 ) ) ) ) ) ) ).
% map_to_set_def
thf(fact_4559_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ B )
=> ! [A5: set @ A,F: A > B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F @ A5 ) ) )
= ( complete_Sup_Sup @ ( option @ B )
@ ( image2 @ A @ ( option @ B )
@ ^ [X3: A] : ( some @ B @ ( F @ X3 ) )
@ A5 ) ) ) ) ) ).
% Some_SUP
thf(fact_4560_set__to__map__insert,axiom,
! [B: $tType,A: $tType,Kv: product_prod @ A @ B,S: set @ ( product_prod @ A @ B )] :
( ~ ( member2 @ 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_4561_set__to__map__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B )] :
( ( ( set_to_map @ A @ B @ S )
= ( ^ [X3: A] : ( none @ B ) ) )
= ( S
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(1)
thf(fact_4562_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M3: A > ( option @ B ),P3: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K2: A,V4: B] :
( ( ( M3 @ K2 )
= ( some @ B @ V4 ) )
& ( P3 @ ( product_Pair @ A @ B @ K2 @ V4 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_4563_set__to__map__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A] : ( none @ B ) )
= ( set_to_map @ A @ B @ S ) )
= ( S
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(2)
thf(fact_4564_map__of__zip__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ ( map @ A @ B @ F @ Xs ) ) )
= ( ^ [X3: A] : ( if @ ( option @ B ) @ ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) ) @ ( some @ B @ ( F @ X3 ) ) @ ( none @ B ) ) ) ) ).
% map_of_zip_map
thf(fact_4565_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 ),Zs3: list @ ( product_prod @ B @ A )] :
( ( Xs
= ( append @ ( product_prod @ B @ A ) @ Ys2 @ ( cons @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ Zs3 ) ) )
& ( ( map_of @ B @ A @ Ys2 @ K )
= ( none @ A ) ) ) ) ).
% map_of_Some_split
thf(fact_4566_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_4567_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_4568_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] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member2 @ 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_4569_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,M2: A > ( option @ B ),X: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ Xs @ Ys ) @ X @ ( some @ B @ ( nth @ B @ Ys @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_4570_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y3: A,Ys3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Ys3 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs3 ) ) ) ) ).
% extract_def
thf(fact_4571_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F: B > A,P: B > $o,Xs: list @ B] :
( ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) )
= ( map_filter @ B @ A
@ ^ [X3: B] : ( if @ ( option @ A ) @ ( P @ X3 ) @ ( some @ A @ ( F @ X3 ) ) @ ( none @ A ) )
@ Xs ) ) ).
% map_filter_map_filter
thf(fact_4572_extract__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
= ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Ys ) )
& ( P @ X3 ) ) ) ) ).
% extract_Some_iff
thf(fact_4573_map__upds__Nil1,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),Bs: list @ B] :
( ( map_upds @ A @ B @ M2 @ ( nil @ A ) @ Bs )
= M2 ) ).
% map_upds_Nil1
thf(fact_4574_map__upds__Nil2,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),As2: list @ A] :
( ( map_upds @ A @ B @ M2 @ As2 @ ( nil @ B ) )
= M2 ) ).
% map_upds_Nil2
thf(fact_4575_map__upds__Cons,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),A3: A,As2: list @ A,B2: B,Bs: list @ B] :
( ( map_upds @ A @ B @ M2 @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs ) )
= ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ B2 ) ) @ As2 @ Bs ) ) ).
% map_upds_Cons
thf(fact_4576_extract__Nil__code,axiom,
! [A: $tType,P: A > $o] :
( ( extract @ A @ P @ ( nil @ A ) )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
% extract_Nil_code
thf(fact_4577_extract__None__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) )
= ( ~ ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% extract_None_iff
thf(fact_4578_map__filter__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > ( option @ A )] :
( ( map_filter @ B @ A @ F @ ( nil @ B ) )
= ( nil @ A ) ) ).
% map_filter_simps(2)
thf(fact_4579_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] :
? [X3: B] :
( ( member2 @ B @ X3 @ ( set2 @ B @ Xs ) )
& ( ( G @ X3 )
= ( some @ A @ Y3 ) ) ) ) ) ).
% set_map_filter
thf(fact_4580_extract__SomeE,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
=> ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Ys ) )
& ( P @ X6 ) ) ) ) ).
% extract_SomeE
thf(fact_4581_extract__Cons__code,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X @ Xs ) ) ) ) )
& ( ~ ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Ys3: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y3: A,Zs2: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X @ Ys3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Zs2 ) ) ) ) )
@ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_4582_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X3: A,Xa4: list @ A] : ( some @ A @ X3 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs3 ) ) ) ) ).
% find_dropWhile
thf(fact_4583_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
& ( P @ X6 ) )
=> ( ( find @ A @ P @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_4584_find__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( find @ A @ P @ Xs )
= ( find @ A @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_4585_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_4586_find_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( some @ A @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( find @ A @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_4587_find_Osimps_I1_J,axiom,
! [A: $tType,Uu: A > $o] :
( ( find @ A @ Uu @ ( nil @ A ) )
= ( none @ A ) ) ).
% find.simps(1)
thf(fact_4588_find__SomeD_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% find_SomeD(2)
thf(fact_4589_find__None__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( find @ A @ P @ Xs )
= ( none @ A ) )
= ( ~ ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff
thf(fact_4590_find__None__iff2,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( none @ A )
= ( find @ A @ P @ Xs ) )
= ( ~ ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff2
thf(fact_4591_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I2 ) )
& ( X
= ( nth @ A @ Xs @ I2 ) )
& ! [J2: nat] :
( ( ord_less @ nat @ J2 @ I2 )
=> ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_4592_find__Some__iff2,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I2 ) )
& ( X
= ( nth @ A @ Xs @ I2 ) )
& ! [J2: nat] :
( ( ord_less @ nat @ J2 @ I2 )
=> ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_4593_inf__option__def,axiom,
! [A: $tType] :
( ( inf @ A )
=> ( ( inf_inf @ ( option @ A ) )
= ( ^ [X3: option @ A,Y3: option @ A] :
( case_option @ ( option @ A ) @ A @ ( none @ A )
@ ^ [Z3: A] :
( case_option @ ( option @ A ) @ A @ ( none @ A )
@ ^ [Aa2: A] : ( some @ A @ ( inf_inf @ A @ Z3 @ Aa2 ) )
@ Y3 )
@ X3 ) ) ) ) ).
% inf_option_def
thf(fact_4594_map__of__distinct__upd,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_add @ A @ B
@ ( fun_upd @ A @ ( option @ B )
@ ^ [X3: A] : ( none @ B )
@ X
@ ( some @ B @ Y ) )
@ ( map_of @ A @ B @ Xs ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Xs ) @ X @ ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_upd
thf(fact_4595_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_4596_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D2: set @ A,M2: A > ( option @ B )] :
( ( ( member2 @ A @ X @ D2 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D2 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member2 @ A @ X @ D2 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D2 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ D2 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_4597_restrict__map__UNIV,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= F ) ).
% restrict_map_UNIV
thf(fact_4598_restrict__restrict,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),A5: set @ A,B4: set @ A] :
( ( restrict_map @ A @ B @ ( restrict_map @ A @ B @ M2 @ A5 ) @ B4 )
= ( restrict_map @ A @ B @ M2 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% restrict_restrict
thf(fact_4599_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_4600_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D2: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( member2 @ A @ X @ D2 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D2 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_4601_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X: A,D2: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( ( member2 @ A @ X @ D2 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ Y ) @ D2 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) )
& ( ~ ( member2 @ A @ X @ D2 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ Y ) @ D2 )
= ( restrict_map @ A @ B @ M2 @ D2 ) ) ) ) ).
% restrict_fun_upd
thf(fact_4602_le__map__restrict,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M2: A > ( option @ B ),X5: set @ A] : ( ord_less_eq @ ( A > ( option @ B ) ) @ ( restrict_map @ A @ B @ M2 @ X5 ) @ M2 ) ) ).
% le_map_restrict
thf(fact_4603_map__add__first__le,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M2: A > ( option @ B ),M6: A > ( option @ B ),N: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M6 )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M2 @ N ) @ ( map_add @ A @ B @ M6 @ N ) ) ) ) ).
% map_add_first_le
thf(fact_4604_restrict__map__subset__eq,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),R: set @ A,M6: A > ( option @ B ),R9: set @ A] :
( ( ( restrict_map @ A @ B @ M2 @ R )
= M6 )
=> ( ( ord_less_eq @ ( set @ A ) @ R9 @ R )
=> ( ( restrict_map @ A @ B @ M2 @ R9 )
= ( restrict_map @ A @ B @ M6 @ R9 ) ) ) ) ).
% restrict_map_subset_eq
thf(fact_4605_map__add__left__None,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),K: B,G: B > ( option @ A )] :
( ( ( F @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F @ G @ K )
= ( G @ K ) ) ) ).
% map_add_left_None
thf(fact_4606_map__add__find__left,axiom,
! [A: $tType,B: $tType,G: B > ( option @ A ),K: B,F: B > ( option @ A )] :
( ( ( G @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F @ G @ K )
= ( F @ K ) ) ) ).
% map_add_find_left
thf(fact_4607_restrict__map__eq_I2_J,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A5: set @ B,K: B,V: A] :
( ( ( restrict_map @ B @ A @ M2 @ A5 @ K )
= ( some @ A @ V ) )
= ( ( ( M2 @ K )
= ( some @ A @ V ) )
& ( member2 @ B @ K @ A5 ) ) ) ).
% restrict_map_eq(2)
thf(fact_4608_map__restrict__insert__none__simp,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),X: B,S4: set @ B] :
( ( ( M2 @ X )
= ( none @ A ) )
=> ( ( restrict_map @ B @ A @ M2 @ ( uminus_uminus @ ( set @ B ) @ ( insert2 @ B @ X @ S4 ) ) )
= ( restrict_map @ B @ A @ M2 @ ( uminus_uminus @ ( set @ B ) @ S4 ) ) ) ) ).
% map_restrict_insert_none_simp
thf(fact_4609_restrict__map__upd,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),S: set @ A,K: A,V: B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F @ S ) @ K @ ( some @ B @ V ) )
= ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ K @ ( some @ B @ V ) ) @ ( insert2 @ A @ K @ S ) ) ) ).
% restrict_map_upd
thf(fact_4610_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),D2: set @ A,X: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D2 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ).
% fun_upd_restrict
thf(fact_4611_map__of__concat,axiom,
! [B: $tType,A: $tType,Xss2: list @ ( list @ ( product_prod @ A @ B ) )] :
( ( map_of @ A @ B @ ( concat @ ( product_prod @ A @ B ) @ Xss2 ) )
= ( foldr @ ( list @ ( product_prod @ A @ B ) ) @ ( A > ( option @ B ) )
@ ^ [Xs3: list @ ( product_prod @ A @ B ),F3: A > ( option @ B )] : ( map_add @ A @ B @ F3 @ ( map_of @ A @ B @ Xs3 ) )
@ Xss2
@ ^ [X3: A] : ( none @ B ) ) ) ).
% map_of_concat
thf(fact_4612_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_4613_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A] :
( ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% map_upd_eq_restrict
thf(fact_4614_subset__eq__mset__impl_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
( ( ( subset_eq_mset_impl @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( some @ $o
@ ( Xa
!= ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( Y
!= ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
@ ^ [Ys1: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
@ ^ [Y3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z5: A] : Y5 = Z5
@ X2 )
@ Xa ) ) ) ) ) ) ).
% subset_eq_mset_impl.elims
thf(fact_4615_subset__eq__mset__impl_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
( ( ( subset_eq_mset_impl @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( some @ $o
@ ( Xa
!= ( nil @ A ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
@ ^ [Ys1: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
@ ^ [Y3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z5: A] : Y5 = Z5
@ X2 )
@ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% subset_eq_mset_impl.pelims
thf(fact_4616_subset__eq__mset__impl_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( subset_eq_mset_impl @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
@ ^ [Ys1: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
@ ^ [X3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z5: A] : Y5 = Z5
@ X )
@ Ys ) ) ) ).
% subset_eq_mset_impl.simps(2)
thf(fact_4617_subset__eq__mset__impl_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( subset_eq_mset_impl @ A @ ( nil @ A ) @ Ys )
= ( some @ $o
@ ( Ys
!= ( nil @ A ) ) ) ) ).
% subset_eq_mset_impl.simps(1)
thf(fact_4618_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( inf_inf @ A @ X3 ) @ Y3 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_4619_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),X: B,Y: A,Z2: A] :
( ( ( M2 @ X )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M2 @ ( dom @ B @ A @ M2 ) )
=> ( ~ ( member2 @ A @ Z2 @ ( ran @ B @ A @ M2 ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ X @ ( some @ A @ Z2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M2 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_4620_Eps__Opt__eq__None,axiom,
! [A: $tType,P: A > $o] :
( ( ( eps_Opt @ A @ P )
= ( none @ A ) )
= ( ~ ? [X8: A] : ( P @ X8 ) ) ) ).
% Eps_Opt_eq_None
thf(fact_4621_restrict__map__self,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M2 @ ( dom @ A @ B @ M2 ) )
= M2 ) ).
% restrict_map_self
thf(fact_4622_some__opt__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( eps_Opt @ A
@ ( ^ [Y5: A,Z5: A] : Y5 = Z5
@ X ) )
= ( some @ A @ X ) ) ).
% some_opt_sym_eq_trivial
thf(fact_4623_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_4624_some__opt__false__trivial,axiom,
! [A: $tType] :
( ( eps_Opt @ A
@ ^ [Uu3: A] : $false )
= ( none @ A ) ) ).
% some_opt_false_trivial
thf(fact_4625_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B )] :
( ( ( dom @ A @ B @ F )
= ( bot_bot @ ( set @ A ) ) )
= ( F
= ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_4626_dom__restrict,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),A5: set @ A] :
( ( dom @ A @ B @ ( restrict_map @ A @ B @ M2 @ A5 ) )
= ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M2 ) @ A5 ) ) ).
% dom_restrict
thf(fact_4627_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X3: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_4628_map__update__eta__repair_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M2: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X3: A] : ( if @ ( option @ B ) @ ( X3 = K ) @ ( some @ B @ V ) @ ( M2 @ X3 ) ) )
= ( insert2 @ A @ K @ ( dom @ A @ B @ M2 ) ) ) ).
% map_update_eta_repair(1)
thf(fact_4629_dom__const_H,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( dom @ A @ B
@ ^ [X3: A] : ( some @ B @ ( F @ X3 ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const'
thf(fact_4630_restrict__map__inv,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( dom @ A @ B @ F ) ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% restrict_map_inv
thf(fact_4631_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option @ B,F: A > ( option @ B ),X: A] :
( ( ( Y
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ Y ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ Y ) )
= ( insert2 @ A @ X @ ( dom @ A @ B @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_4632_ran__map__add,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ M1 @ M22 ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ M1 ) @ ( ran @ A @ B @ M22 ) ) ) ) ).
% ran_map_add
thf(fact_4633_ran__add,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( dom @ A @ B @ G ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ F @ G ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ F ) @ ( ran @ A @ B @ G ) ) ) ) ).
% ran_add
thf(fact_4634_nempty__dom,axiom,
! [B: $tType,A: $tType,E3: A > ( option @ B )] :
( ( E3
!= ( ^ [X3: A] : ( none @ B ) ) )
=> ~ ! [M4: A] :
~ ( member2 @ A @ M4 @ ( dom @ A @ B @ E3 ) ) ) ).
% nempty_dom
thf(fact_4635_inj__on__map__the,axiom,
! [B: $tType,A: $tType,D2: set @ A,M2: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ D2 @ ( dom @ A @ B @ M2 ) )
=> ( ( inj_on @ A @ ( option @ B ) @ M2 @ D2 )
=> ( inj_on @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ M2 ) @ D2 ) ) ) ).
% inj_on_map_the
thf(fact_4636_ran__is__image,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M7: B > ( option @ A )] : ( image2 @ B @ A @ ( comp @ ( option @ A ) @ A @ B @ ( the2 @ A ) @ M7 ) @ ( dom @ B @ A @ M7 ) ) ) ) ).
% ran_is_image
thf(fact_4637_map__dom__ran__finite,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M ) )
=> ( finite_finite2 @ B @ ( ran @ A @ B @ M ) ) ) ).
% map_dom_ran_finite
thf(fact_4638_dom__if,axiom,
! [B: $tType,A: $tType,P: A > $o,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X3: A] : ( if @ ( option @ B ) @ ( P @ X3 ) @ ( F @ X3 ) @ ( G @ X3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( collect @ A @ P ) )
@ ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ G )
@ ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) ) ) ) ).
% dom_if
thf(fact_4639_le__map__dom__mono,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M2: A > ( option @ B ),M6: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M6 )
=> ( ord_less_eq @ ( set @ A ) @ ( dom @ A @ B @ M2 ) @ ( dom @ A @ B @ M6 ) ) ) ) ).
% le_map_dom_mono
thf(fact_4640_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_4641_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_4642_map__add__comm,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ M1 @ M22 )
= ( map_add @ A @ B @ M22 @ M1 ) ) ) ).
% map_add_comm
thf(fact_4643_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A5: A > ( option @ B ),B4: A > ( option @ B ),C4: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ A5 ) @ ( dom @ A @ B @ B4 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ A5 @ ( map_add @ A @ B @ B4 @ C4 ) )
= ( map_add @ A @ B @ B4 @ ( map_add @ A @ B @ A5 @ C4 ) ) ) ) ).
% map_add_left_comm
thf(fact_4644_restrict__map__eq_I1_J,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A5: set @ B,K: B] :
( ( ( restrict_map @ B @ A @ M2 @ A5 @ K )
= ( none @ A ) )
= ( ~ ( member2 @ B @ K @ ( inf_inf @ ( set @ B ) @ ( dom @ B @ A @ M2 ) @ A5 ) ) ) ) ).
% restrict_map_eq(1)
thf(fact_4645_map__card__eq__iff,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X: A,Y: A] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M ) )
=> ( ( ( finite_card @ A @ ( dom @ A @ B @ M ) )
= ( finite_card @ B @ ( ran @ A @ B @ M ) ) )
=> ( ( member2 @ A @ X @ ( dom @ A @ B @ M ) )
=> ( ( ( M @ X )
= ( M @ Y ) )
= ( X = Y ) ) ) ) ) ).
% map_card_eq_iff
thf(fact_4646_finite__map__to__set,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) )
= ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) ) ) ).
% finite_map_to_set
thf(fact_4647_map__to__set__dom,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M3: A > ( option @ B )] : ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M3 ) ) ) ) ).
% map_to_set_dom
thf(fact_4648_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_4649_card__map__to__set,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( finite_card @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) )
= ( finite_card @ A @ ( dom @ A @ B @ M2 ) ) ) ).
% card_map_to_set
thf(fact_4650_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,Xs: list @ B] :
( ( map_filter @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F @ Xs )
@ ^ [Y3: A] : ( cons @ A @ Y3 @ ( map_filter @ B @ A @ F @ Xs ) )
@ ( F @ X ) ) ) ).
% map_filter_simps(1)
thf(fact_4651_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M2: A > ( option @ B ),M6: A > ( option @ B ),N: A > ( option @ B ),N9: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M6 )
=> ( ( ord_less_eq @ ( A > ( option @ B ) ) @ N @ N9 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M6 ) @ ( dom @ A @ B @ N9 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M2 @ N ) @ ( map_add @ A @ B @ M6 @ N9 ) ) ) ) ) ) ).
% map_add_distinct_le
thf(fact_4652_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
( ( ( dom @ A @ B @ F )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V4: B] :
( F
= ( fun_upd @ A @ ( option @ B )
@ ^ [X3: A] : ( none @ B )
@ X
@ ( some @ B @ V4 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_4653_set__to__map__def,axiom,
! [A: $tType,B: $tType] :
( ( set_to_map @ B @ A )
= ( ^ [S7: set @ ( product_prod @ B @ A ),K2: B] :
( eps_Opt @ A
@ ^ [V4: A] : ( member2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K2 @ V4 ) @ S7 ) ) ) ) ).
% set_to_map_def
thf(fact_4654_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F3: A > ( option @ B ),Xs3: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F3 )
@ ( filter2 @ A
@ ^ [X3: A] :
( ( F3 @ X3 )
!= ( none @ B ) )
@ Xs3 ) ) ) ) ).
% map_filter_def
thf(fact_4655_dom__override__on,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B ),A5: set @ A] :
( ( dom @ A @ B @ ( override_on @ A @ ( option @ B ) @ F @ G @ A5 ) )
= ( sup_sup @ ( set @ A )
@ ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F )
@ ( collect @ A
@ ^ [A4: A] : ( member2 @ A @ A4 @ ( minus_minus @ ( set @ A ) @ A5 @ ( dom @ A @ B @ G ) ) ) ) )
@ ( collect @ A
@ ^ [A4: A] : ( member2 @ A @ A4 @ ( inf_inf @ ( set @ A ) @ A5 @ ( dom @ A @ B @ G ) ) ) ) ) ) ).
% dom_override_on
thf(fact_4656_graph__map__add,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M1 ) @ ( dom @ A @ B @ M22 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( graph @ A @ B @ ( map_add @ A @ B @ M1 @ M22 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( graph @ A @ B @ M1 ) @ ( graph @ A @ B @ M22 ) ) ) ) ).
% graph_map_add
thf(fact_4657_eq__f__restr__conv,axiom,
! [B: $tType,A: $tType,S4: set @ A,F: ( A > ( option @ B ) ) > A > ( option @ B ),A5: A > ( option @ B )] :
( ( ( ord_less_eq @ ( set @ A ) @ S4 @ ( dom @ A @ B @ ( F @ A5 ) ) )
& ( A5
= ( restrict_map @ A @ B @ ( F @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ S4 ) ) ) )
= ( ( map_le @ A @ B @ A5 @ ( F @ A5 ) )
& ( S4
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F @ A5 ) ) @ ( dom @ A @ B @ A5 ) ) ) ) ) ).
% eq_f_restr_conv
thf(fact_4658_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
= F ) ).
% override_on_emptyset
thf(fact_4659_map__leI,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ! [X2: A,V2: B] :
( ( ( M1 @ X2 )
= ( some @ B @ V2 ) )
=> ( ( M22 @ X2 )
= ( some @ B @ V2 ) ) )
=> ( map_le @ A @ B @ M1 @ M22 ) ) ).
% map_leI
thf(fact_4660_map__leD,axiom,
! [A: $tType,B: $tType,M1: A > ( option @ B ),M22: A > ( option @ B ),K: A,V: B] :
( ( map_le @ A @ B @ M1 @ M22 )
=> ( ( ( M1 @ K )
= ( some @ B @ V ) )
=> ( ( M22 @ K )
= ( some @ B @ V ) ) ) ) ).
% map_leD
thf(fact_4661_eq__f__restr__ss__eq,axiom,
! [B: $tType,A: $tType,S4: set @ A,F: ( A > ( option @ B ) ) > A > ( option @ B ),A5: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ S4 @ ( dom @ A @ B @ ( F @ A5 ) ) )
=> ( ( A5
= ( restrict_map @ A @ B @ ( F @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ S4 ) ) )
= ( ( map_le @ A @ B @ A5 @ ( F @ A5 ) )
& ( S4
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F @ A5 ) ) @ ( dom @ A @ B @ A5 ) ) ) ) ) ) ).
% eq_f_restr_ss_eq
thf(fact_4662_le__map__mmupd__not__dom,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),K5: set @ A,V: B] : ( map_le @ A @ B @ M2 @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K5 @ ( dom @ A @ B @ M2 ) ) @ V ) ) ).
% le_map_mmupd_not_dom
thf(fact_4663_map__mmupd__update__less,axiom,
! [A: $tType,B: $tType,K5: set @ A,K6: set @ A,M2: A > ( option @ B ),V: B] :
( ( ord_less_eq @ ( set @ A ) @ K5 @ K6 )
=> ( map_le @ A @ B @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K5 @ ( dom @ A @ B @ M2 ) ) @ V ) @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K6 @ ( dom @ A @ B @ M2 ) ) @ V ) ) ) ).
% map_mmupd_update_less
thf(fact_4664_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_4665_mmupd__notin__upd,axiom,
! [B: $tType,A: $tType,K: A,K5: set @ A,M2: A > ( option @ B ),V: B] :
( ~ ( member2 @ A @ K @ K5 )
=> ( ( map_mmupd @ A @ B @ M2 @ K5 @ V @ K )
= ( M2 @ K ) ) ) ).
% mmupd_notin_upd
thf(fact_4666_map__mmupd__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),V: B] :
( ( map_mmupd @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) @ V )
= M2 ) ).
% map_mmupd_empty
thf(fact_4667_mmupd__in__upd,axiom,
! [A: $tType,B: $tType,K: A,K5: set @ A,M2: A > ( option @ B ),V: B] :
( ( member2 @ A @ K @ K5 )
=> ( ( map_mmupd @ A @ B @ M2 @ K5 @ V @ K )
= ( some @ B @ V ) ) ) ).
% mmupd_in_upd
thf(fact_4668_the__dflt__None__empty,axiom,
! [A: $tType] :
( ( dflt_None_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( none @ ( set @ A ) ) ) ).
% the_dflt_None_empty
thf(fact_4669_dom__mmupd,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K5: set @ A,V: B] :
( ( dom @ A @ B @ ( map_mmupd @ A @ B @ M2 @ K5 @ V ) )
= ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ M2 ) @ K5 ) ) ).
% dom_mmupd
thf(fact_4670_map__mmupd__def,axiom,
! [A: $tType,B: $tType] :
( ( map_mmupd @ B @ A )
= ( ^ [M3: B > ( option @ A ),K7: set @ B,V4: A,K2: B] : ( if @ ( option @ A ) @ ( member2 @ B @ K2 @ K7 ) @ ( some @ A @ V4 ) @ ( M3 @ K2 ) ) ) ) ).
% map_mmupd_def
thf(fact_4671_map__mmupdE,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),K5: set @ B,V: A,K: B,X: A] :
( ( ( map_mmupd @ B @ A @ M2 @ K5 @ V @ K )
= ( some @ A @ X ) )
=> ( ( ~ ( member2 @ B @ K @ K5 )
=> ( ( M2 @ K )
!= ( some @ A @ X ) ) )
=> ~ ( ( member2 @ B @ K @ K5 )
=> ( X != V ) ) ) ) ).
% map_mmupdE
thf(fact_4672_dflt__None__set__def,axiom,
! [A: $tType] :
( ( dflt_None_set @ A )
= ( ^ [S7: set @ A] :
( if @ ( option @ ( set @ A ) )
@ ( S7
= ( bot_bot @ ( set @ A ) ) )
@ ( none @ ( set @ A ) )
@ ( some @ ( set @ A ) @ S7 ) ) ) ) ).
% dflt_None_set_def
thf(fact_4673_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_4674_subset__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member2 @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member2 @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_4675_at__most__one__mset__mset__diff,axiom,
! [A: $tType,A3: A,M: multiset @ A] :
( ~ ( member2 @ A @ A3 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( set_mset @ A @ M ) @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_4676_Image__empty2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
( ( image @ B @ A @ R @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty2
thf(fact_4677_set__mset__eq__empty__iff,axiom,
! [A: $tType,M: multiset @ A] :
( ( ( set_mset @ A @ M )
= ( bot_bot @ ( set @ A ) ) )
= ( M
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% set_mset_eq_empty_iff
thf(fact_4678_set__mset__empty,axiom,
! [A: $tType] :
( ( set_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% set_mset_empty
thf(fact_4679_Image__empty1,axiom,
! [B: $tType,A: $tType,X5: set @ B] :
( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty1
thf(fact_4680_Image__Id__on,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( image @ A @ A @ ( id_on @ A @ A5 ) @ B4 )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ).
% Image_Id_on
thf(fact_4681_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A3: B] :
( ( member2 @ A @ B2 @ ( image @ B @ A @ R3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_4682_mset__diff__cancel1elem,axiom,
! [A: $tType,A3: A,B4: multiset @ A] :
( ~ ( member2 @ A @ A3 @ ( set_mset @ A @ B4 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B4 )
= ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% mset_diff_cancel1elem
thf(fact_4683_listrel__Nil,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ A )] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R3 ) @ ( insert2 @ ( list @ B ) @ ( nil @ B ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listrel_Nil
thf(fact_4684_pair__vimage__is__Image,axiom,
! [A: $tType,B: $tType,U: B,E5: set @ ( product_prod @ B @ A )] :
( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ U ) @ E5 )
= ( image @ B @ A @ E5 @ ( insert2 @ B @ U @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% pair_vimage_is_Image
thf(fact_4685_mset__right__cancel__union,axiom,
! [A: $tType,A3: A,A5: multiset @ A,B4: multiset @ A] :
( ( member2 @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) ) )
=> ( ~ ( member2 @ A @ A3 @ ( set_mset @ A @ B4 ) )
=> ( member2 @ A @ A3 @ ( set_mset @ A @ A5 ) ) ) ) ).
% mset_right_cancel_union
thf(fact_4686_mset__left__cancel__union,axiom,
! [A: $tType,A3: A,A5: multiset @ A,B4: multiset @ A] :
( ( member2 @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) ) )
=> ( ~ ( member2 @ A @ A3 @ ( set_mset @ A @ A5 ) )
=> ( member2 @ A @ A3 @ ( set_mset @ A @ B4 ) ) ) ) ).
% mset_left_cancel_union
thf(fact_4687_mset__un__cases,axiom,
! [A: $tType,A3: A,A5: multiset @ A,B4: multiset @ A] :
( ( member2 @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) ) )
=> ( ~ ( member2 @ A @ A3 @ ( set_mset @ A @ A5 ) )
=> ( member2 @ A @ A3 @ ( set_mset @ A @ B4 ) ) ) ) ).
% mset_un_cases
thf(fact_4688_ex__Melem__conv,axiom,
! [A: $tType,A5: multiset @ A] :
( ( ? [X3: A] : ( member2 @ A @ X3 @ ( set_mset @ A @ A5 ) ) )
= ( A5
!= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% ex_Melem_conv
thf(fact_4689_Image__Int__subset,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A5: set @ B,B4: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R @ A5 ) @ ( image @ B @ A @ R @ B4 ) ) ) ).
% Image_Int_subset
thf(fact_4690_rtrancl__image__advance,axiom,
! [A: $tType,Q3: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
( ( member2 @ A @ Q3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ X ) @ R )
=> ( member2 @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_4691_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q3: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
( ( member2 @ A @ Q3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ X ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member2 @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_4692_finite__Image__subset,axiom,
! [A: $tType,B: $tType,A5: set @ ( product_prod @ B @ A ),B4: set @ B,C4: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ ( image @ B @ A @ A5 @ B4 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ C4 @ A5 )
=> ( finite_finite2 @ A @ ( image @ B @ A @ C4 @ B4 ) ) ) ) ).
% finite_Image_subset
thf(fact_4693_rtrancl__image__unfold__right,axiom,
! [A: $tType,E5: set @ ( product_prod @ A @ A ),V3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E5 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ V3 ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ V3 ) ) ).
% rtrancl_image_unfold_right
thf(fact_4694_rtrancl__reachable__induct,axiom,
! [A: $tType,I5: set @ A,INV: set @ A,E5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ I5 @ INV )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E5 @ INV ) @ INV )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ I5 ) @ INV ) ) ) ).
% rtrancl_reachable_induct
thf(fact_4695_trancl__Image__unfold__right,axiom,
! [A: $tType,E5: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E5 ) @ S )
= ( image @ A @ A @ E5 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ S ) ) ) ).
% trancl_Image_unfold_right
thf(fact_4696_trancl__Image__unfold__left,axiom,
! [A: $tType,E5: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E5 ) @ S )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ ( image @ A @ A @ E5 @ S ) ) ) ).
% trancl_Image_unfold_left
thf(fact_4697_Image__empty__rtrancl__Image__id,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),V: A] :
( ( ( image @ A @ A @ R @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_4698_Image__empty__trancl__Image__empty,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),V: A] :
( ( ( image @ A @ A @ R @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_trancl @ A @ R ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_4699_reachable__mono,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A @ A ),X5: set @ A,X11: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ R9 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ X11 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ X5 ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R9 ) @ X11 ) ) ) ) ).
% reachable_mono
thf(fact_4700_quotientI,axiom,
! [A: $tType,X: A,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ A @ X @ A5 )
=> ( member2 @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( equiv_quotient @ A @ A5 @ R3 ) ) ) ).
% quotientI
thf(fact_4701_quotientE,axiom,
! [A: $tType,X5: set @ A,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ~ ! [X2: A] :
( ( X5
= ( image @ A @ A @ R3 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ~ ( member2 @ A @ X2 @ A5 ) ) ) ).
% quotientE
thf(fact_4702_Image__subset__snd__image,axiom,
! [A: $tType,B: $tType,A5: set @ ( product_prod @ B @ A ),B4: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ A5 @ B4 ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A5 ) ) ).
% Image_subset_snd_image
thf(fact_4703_trancl__image__by__rtrancl,axiom,
! [A: $tType,E5: set @ ( product_prod @ A @ A ),Vi: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E5 ) @ Vi ) @ Vi )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ Vi ) ) ).
% trancl_image_by_rtrancl
thf(fact_4704_mset__left__cancel__elem,axiom,
! [A: $tType,A3: A,B2: A,A5: multiset @ A] :
( ( member2 @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A5 ) ) )
=> ( ( A3 != B2 )
=> ( member2 @ A @ A3 @ ( set_mset @ A @ A5 ) ) ) ) ).
% mset_left_cancel_elem
thf(fact_4705_mset__right__cancel__elem,axiom,
! [A: $tType,A3: A,A5: multiset @ A,B2: A] :
( ( member2 @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( A3 != B2 )
=> ( member2 @ A @ A3 @ ( set_mset @ A @ A5 ) ) ) ) ).
% mset_right_cancel_elem
thf(fact_4706_Image__singleton,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ A ),A3: B] :
( ( image @ B @ A @ R3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B3: A] : ( member2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B3 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_4707_Image__fold,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( image @ A @ B @ R @ S )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X3: A,Y3: B,A7: set @ B] : ( if @ ( set @ B ) @ ( member2 @ A @ X3 @ S ) @ ( insert2 @ B @ Y3 @ A7 ) @ A7 ) )
@ ( bot_bot @ ( set @ B ) )
@ R ) ) ) ).
% Image_fold
thf(fact_4708_listrel__Cons,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A ),X: B,Xs: list @ B] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R3 ) @ ( insert2 @ ( list @ B ) @ ( cons @ B @ X @ Xs ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( set_Cons @ A @ ( image @ B @ A @ R3 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) @ ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R3 ) @ ( insert2 @ ( list @ B ) @ Xs @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) ) ) ) ).
% listrel_Cons
thf(fact_4709_rtrancl__apply__insert,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,S: set @ A] :
( ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ ( insert2 @ A @ X @ S ) )
= ( insert2 @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ ( sup_sup @ ( set @ A ) @ S @ ( image @ A @ A @ R @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_4710_rtrancl__Image__in__Field,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),V3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ V3 ) @ ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R ) @ V3 ) ) ).
% rtrancl_Image_in_Field
thf(fact_4711_set__mset__single,axiom,
! [A: $tType,B2: A] :
( ( set_mset @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_mset_single
thf(fact_4712_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ E5 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E5 @ R ) @ R )
=> ( ~ ( member2 @ A @ V @ R )
=> ~ ( ~ ( member2 @ A @ U @ R )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E5 @ R ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_4713_rel__restrict__tranclI,axiom,
! [A: $tType,X: A,Y: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ E5 ) )
=> ( ~ ( member2 @ A @ X @ R )
=> ( ~ ( member2 @ A @ Y @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E5 @ R ) @ R )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E5 @ R ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_4714_mset__contains__eq,axiom,
! [A: $tType,M2: A,M: multiset @ A] :
( ( member2 @ A @ M2 @ ( set_mset @ A @ M ) )
= ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ M2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ M @ ( add_mset @ A @ M2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= M ) ) ).
% mset_contains_eq
thf(fact_4715_mset__union__diff__comm,axiom,
! [A: $tType,T4: A,S: multiset @ A,T: multiset @ A] :
( ( member2 @ A @ T4 @ ( set_mset @ A @ S ) )
=> ( ( plus_plus @ ( multiset @ A ) @ T @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ T @ S ) @ ( add_mset @ A @ T4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_union_diff_comm
thf(fact_4716_diff__union__single__conv2,axiom,
! [A: $tType,A3: A,J4: multiset @ A,I5: multiset @ A] :
( ( member2 @ A @ A3 @ ( set_mset @ A @ J4 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ J4 @ I5 ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ J4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ I5 ) ) ) ).
% diff_union_single_conv2
thf(fact_4717_mset__un__single__un__cases,axiom,
! [A: $tType,A3: A,A5: multiset @ A,B4: multiset @ A,C4: multiset @ A] :
( ( ( add_mset @ A @ A3 @ A5 )
= ( plus_plus @ ( multiset @ A ) @ B4 @ C4 ) )
=> ( ( ( member2 @ A @ A3 @ ( set_mset @ A @ B4 ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C4 ) ) )
=> ~ ( ( member2 @ A @ A3 @ ( set_mset @ A @ C4 ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ B4 @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_un_single_un_cases
thf(fact_4718_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A ),B5: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y3: B] : ( image @ B @ A @ R4 @ ( insert2 @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) )
@ B5 ) ) ) ) ).
% Image_eq_UN
thf(fact_4719_Sigma__Image,axiom,
! [A: $tType,B: $tType,A5: set @ B,B4: B > ( set @ A ),X5: set @ B] :
( ( image @ B @ A @ ( product_Sigma @ B @ A @ A5 @ B4 ) @ X5 )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ B ) @ X5 @ A5 ) ) ) ) ).
% Sigma_Image
thf(fact_4720_finite__reachable__advance,axiom,
! [A: $tType,E5: set @ ( product_prod @ A @ A ),V0: A,V: A] :
( ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ ( insert2 @ A @ V0 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V0 @ V ) @ ( transitive_rtrancl @ A @ E5 ) )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_4721_rtrancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E5: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E5 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_4722_trancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E5: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E5 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E5 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E5 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_4723_the__default_Osimps_I1_J,axiom,
! [A: $tType,Uu: A,X: A] :
( ( the_default @ A @ Uu @ ( some @ A @ X ) )
= X ) ).
% the_default.simps(1)
thf(fact_4724_the__default_Osimps_I2_J,axiom,
! [A: $tType,X: A] :
( ( the_default @ A @ X @ ( none @ A ) )
= X ) ).
% the_default.simps(2)
thf(fact_4725_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V: A,E5: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_trancl @ A @ E5 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E5 @ S ) @ S )
=> ( ( member2 @ A @ U @ S )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E5
@ ( product_Sigma @ A @ A @ S
@ ^ [Uu3: A] : S ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_4726_singleton__quotient,axiom,
! [A: $tType,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
= ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% singleton_quotient
thf(fact_4727_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A7: set @ A,R4: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X3: A] : ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R4 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A7 ) ) ) ) ).
% quotient_def
thf(fact_4728_sum__wcount__Int,axiom,
! [A: $tType,A5: set @ A,F: A > nat,N4: multiset @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( wcount @ A @ F @ N4 ) @ ( inf_inf @ ( set @ A ) @ A5 @ ( set_mset @ A @ N4 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ ( wcount @ A @ F @ N4 ) @ A5 ) ) ) ).
% sum_wcount_Int
thf(fact_4729_set__mset__replicate__mset__subset,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_4730_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( antisym @ A @ R3 )
=> ( ( member2 @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member2 @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B2 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_4731_antisym__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( antisym @ A @ R3 )
=> ( antisym @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) ) ) ).
% antisym_Restr
thf(fact_4732_size__diff__se,axiom,
! [A: $tType,T4: A,S: multiset @ A] :
( ( member2 @ A @ T4 @ ( set_mset @ A @ S ) )
=> ( ( size_size @ ( multiset @ A ) @ S )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% size_diff_se
thf(fact_4733_sorted__list__of__multiset__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord6283353356039996273ltiset @ A )
= ( fold_mset @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3 )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_multiset_def
thf(fact_4734_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B3: A,A7: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3
@ ( the @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_4735_mset__size__le1__cases,axiom,
! [A: $tType,M: multiset @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ M ) @ ( suc @ ( zero_zero @ nat ) ) )
=> ( ( M
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [M4: A] :
( M
!= ( add_mset @ A @ M4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size_le1_cases
thf(fact_4736_mset__size1elem,axiom,
! [A: $tType,P: multiset @ A,Q3: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( one_one @ nat ) )
=> ( ( member2 @ A @ Q3 @ ( set_mset @ A @ P ) )
=> ( P
= ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size1elem
thf(fact_4737_mset__size2elem,axiom,
! [A: $tType,P: multiset @ A,Q3: A,Q6: 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 @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q6 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ P )
=> ( P
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q6 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% mset_size2elem
thf(fact_4738_mset__set__Union,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( mset_set @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset_set @ A @ A5 ) @ ( mset_set @ A @ B4 ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_4739_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member2 @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member2 @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B2 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_4740_mset__set_Oempty,axiom,
! [A: $tType] :
( ( mset_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% mset_set.empty
thf(fact_4741_mset__le__addE,axiom,
! [A: $tType,Xs: multiset @ A,Ys: multiset @ A] :
( ( subseteq_mset @ A @ Xs @ Ys )
=> ~ ! [Zs3: multiset @ A] :
( Ys
!= ( plus_plus @ ( multiset @ A ) @ Xs @ Zs3 ) ) ) ).
% mset_le_addE
thf(fact_4742_mset__le__distrib,axiom,
! [A: $tType,X5: multiset @ A,A5: multiset @ A,B4: multiset @ A] :
( ( subseteq_mset @ A @ X5 @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) )
=> ~ ! [Xa5: multiset @ A,Xb3: multiset @ A] :
( ( X5
= ( plus_plus @ ( multiset @ A ) @ Xa5 @ Xb3 ) )
=> ( ( subseteq_mset @ A @ Xa5 @ A5 )
=> ~ ( subseteq_mset @ A @ Xb3 @ B4 ) ) ) ) ).
% mset_le_distrib
thf(fact_4743_mset__union__subset,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,C4: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) @ C4 )
=> ( ( subseteq_mset @ A @ A5 @ C4 )
& ( subseteq_mset @ A @ B4 @ C4 ) ) ) ).
% mset_union_subset
thf(fact_4744_mset__le__decr__left1,axiom,
! [A: $tType,A3: multiset @ A,C3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C3 ) @ B2 )
=> ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% mset_le_decr_left1
thf(fact_4745_mset__le__decr__left2,axiom,
! [A: $tType,C3: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C3 @ A3 ) @ B2 )
=> ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% mset_le_decr_left2
thf(fact_4746_mset__le__incr__right1,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C3: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ B2 @ C3 ) ) ) ).
% mset_le_incr_right1
thf(fact_4747_mset__le__incr__right2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C3: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ C3 @ B2 ) ) ) ).
% mset_le_incr_right2
thf(fact_4748_mset__le__subtract,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,C4: multiset @ A] :
( ( subseteq_mset @ A @ A5 @ B4 )
=> ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A5 @ C4 ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ C4 ) ) ) ).
% mset_le_subtract
thf(fact_4749_mset__le__add__mset__decr__left1,axiom,
! [A: $tType,C3: A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C3 @ A3 ) @ B2 )
=> ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% mset_le_add_mset_decr_left1
thf(fact_4750_mset__le__add__mset,axiom,
! [A: $tType,X: A,B4: multiset @ A,C4: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B4 ) @ C4 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ C4 )
& ( subseteq_mset @ A @ B4 @ C4 ) ) ) ).
% mset_le_add_mset
thf(fact_4751_mset__le__single__cases,axiom,
! [A: $tType,M: multiset @ A,A3: A] :
( ( subseteq_mset @ A @ M @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( ( M
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( M
= ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_le_single_cases
thf(fact_4752_mset__le__add__mset__decr__left2,axiom,
! [A: $tType,C3: A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C3 @ A3 ) @ B2 )
=> ( subseteq_mset @ A @ ( add_mset @ A @ C3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B2 ) ) ).
% mset_le_add_mset_decr_left2
thf(fact_4753_mset__le__subtract__left,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,X5: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) @ X5 )
=> ( ( subseteq_mset @ A @ B4 @ ( minus_minus @ ( multiset @ A ) @ X5 @ A5 ) )
& ( subseteq_mset @ A @ A5 @ X5 ) ) ) ).
% mset_le_subtract_left
thf(fact_4754_mset__le__subtract__right,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,X5: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) @ X5 )
=> ( ( subseteq_mset @ A @ A5 @ ( minus_minus @ ( multiset @ A ) @ X5 @ B4 ) )
& ( subseteq_mset @ A @ B4 @ X5 ) ) ) ).
% mset_le_subtract_right
thf(fact_4755_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_4756_mset__le__subtract__add__mset__right,axiom,
! [A: $tType,X: A,B4: multiset @ A,X5: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B4 ) @ X5 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ X5 @ B4 ) )
& ( subseteq_mset @ A @ B4 @ X5 ) ) ) ).
% mset_le_subtract_add_mset_right
thf(fact_4757_mset__le__subtract__add__mset__left,axiom,
! [A: $tType,X: A,B4: multiset @ A,X5: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B4 ) @ X5 )
=> ( ( subseteq_mset @ A @ B4 @ ( minus_minus @ ( multiset @ A ) @ X5 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
& ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ X5 ) ) ) ).
% mset_le_subtract_add_mset_left
thf(fact_4758_subset__mset_OcINF__greatest,axiom,
! [A: $tType,B: $tType,A5: set @ B,M2: multiset @ A,F: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ A5 )
=> ( subseteq_mset @ A @ M2 @ ( F @ X2 ) ) )
=> ( subseteq_mset @ A @ M2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) ) ) ) ).
% subset_mset.cINF_greatest
thf(fact_4759_subset__mset_OcSUP__least,axiom,
! [B: $tType,A: $tType,A5: set @ B,F: B > ( multiset @ A ),M: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ A5 )
=> ( subseteq_mset @ A @ ( F @ X2 ) @ M ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) @ M ) ) ) ).
% subset_mset.cSUP_least
thf(fact_4760_mset__set__empty__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( mset_set @ A @ A5 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% mset_set_empty_iff
thf(fact_4761_infinite__set__mset__mset__set,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( set_mset @ A @ ( mset_set @ A @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_set_mset_mset_set
thf(fact_4762_mset__le__mono__add__single,axiom,
! [A: $tType,A3: A,Ys: multiset @ A,B2: A,Ws2: multiset @ A] :
( ( member2 @ A @ A3 @ ( set_mset @ A @ Ys ) )
=> ( ( member2 @ A @ B2 @ ( set_mset @ A @ Ws2 ) )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ Ys @ Ws2 ) ) ) ) ).
% mset_le_mono_add_single
thf(fact_4763_mset__union__subset__s,axiom,
! [A: $tType,A3: A,B4: multiset @ A,C4: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B4 ) @ C4 )
=> ( ( member2 @ A @ A3 @ ( set_mset @ A @ C4 ) )
& ( subseteq_mset @ A @ B4 @ C4 ) ) ) ).
% mset_union_subset_s
thf(fact_4764_mset__2dist2__cases,axiom,
! [A: $tType,A3: A,B2: A,A5: multiset @ A,B4: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ A5 )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ B4 )
=> ( ( ( member2 @ A @ A3 @ ( set_mset @ A @ A5 ) )
=> ~ ( member2 @ A @ B2 @ ( set_mset @ A @ B4 ) ) )
=> ~ ( ( member2 @ A @ A3 @ ( set_mset @ A @ B4 ) )
=> ~ ( member2 @ A @ B2 @ ( set_mset @ A @ A5 ) ) ) ) ) ) ) ).
% mset_2dist2_cases
thf(fact_4765_Partial__order__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_7125193373082350890der_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) ) ) ).
% Partial_order_Restr
thf(fact_4766_mset__set_Oinsert__remove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( mset_set @ A @ ( insert2 @ A @ X @ A5 ) )
= ( add_mset @ A @ X @ ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% mset_set.insert_remove
thf(fact_4767_mset__set_Oremove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( mset_set @ A @ A5 )
= ( add_mset @ A @ X @ ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% mset_set.remove
thf(fact_4768_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_4769_prod__mset_Oremove,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [X: A,A5: multiset @ A] :
( ( member2 @ A @ X @ ( set_mset @ A @ A5 ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ A5 )
= ( times_times @ A @ X @ ( comm_m9189036328036947845d_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% prod_mset.remove
thf(fact_4770_proj__def,axiom,
! [A: $tType,B: $tType] :
( ( equiv_proj @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A ),X3: B] : ( image @ B @ A @ R4 @ ( insert2 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% proj_def
thf(fact_4771_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_4772_prod__mset_Oadd__mset,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [X: A,N4: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( add_mset @ A @ X @ N4 ) )
= ( times_times @ A @ X @ ( comm_m9189036328036947845d_mset @ A @ N4 ) ) ) ) ).
% prod_mset.add_mset
thf(fact_4773_prod__mset_Ounion,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: multiset @ A,N4: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M @ N4 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ M ) @ ( comm_m9189036328036947845d_mset @ A @ N4 ) ) ) ) ).
% prod_mset.union
thf(fact_4774_prod__mset__Un,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: multiset @ A,B4: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ A5 ) @ ( comm_m9189036328036947845d_mset @ A @ B4 ) ) ) ) ).
% prod_mset_Un
thf(fact_4775_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( Less_eq @ A3 @ Top ) ) ).
% ordering_top.extremum
thf(fact_4776_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A3 ) ) ).
% ordering_top.extremum_strict
thf(fact_4777_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
= ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_4778_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( A3 != Top )
= ( Less @ A3 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_4779_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_4780_prod__mset_Oneutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: multiset @ A] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set_mset @ A @ A5 ) )
=> ( X2
= ( one_one @ A ) ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ A5 )
= ( one_one @ A ) ) ) ) ).
% prod_mset.neutral
thf(fact_4781_is__unit__prod__mset__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A5: multiset @ A] :
( ( dvd_dvd @ A @ ( comm_m9189036328036947845d_mset @ A @ A5 ) @ ( one_one @ A ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set_mset @ A @ A5 ) )
=> ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% is_unit_prod_mset_iff
thf(fact_4782_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_4783_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_4784_proj__iff,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 )
=> ( ( ( equiv_proj @ A @ A @ R3 @ X )
= ( equiv_proj @ A @ A @ R3 @ Y ) )
= ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% proj_iff
thf(fact_4785_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A5: set @ C,B4: C > ( set @ D ),F: D > B] :
( ( A5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X2: C] :
( ( member2 @ C @ X2 @ A5 )
=> ( ( B4 @ X2 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X3: C] : ( image2 @ D @ B @ F @ ( B4 @ X3 ) )
@ A5 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B4 @ A5 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F @ ( B4 @ X3 ) ) )
@ A5 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_4786_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( condit941137186595557371_above @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F @ A5 ) ) @ ( F @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_4787_bdd__above__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_above_empty
thf(fact_4788_equiv__listrel,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( equiv_equiv @ ( list @ A ) @ ( lists @ A @ A5 ) @ ( listrel @ A @ A @ R3 ) ) ) ).
% equiv_listrel
thf(fact_4789_bdd__above__Int1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( condit941137186595557371_above @ A @ A5 )
=> ( condit941137186595557371_above @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% bdd_above_Int1
thf(fact_4790_bdd__above__Int2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B4: set @ A,A5: set @ A] :
( ( condit941137186595557371_above @ A @ B4 )
=> ( condit941137186595557371_above @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% bdd_above_Int2
thf(fact_4791_in__quotient__imp__non__empty,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X5: set @ A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member2 @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( X5
!= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_4792_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S ) @ A3 )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ S )
=> ( ord_less_eq @ A @ X3 @ A3 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_4793_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B4: set @ A,A5: set @ A] :
( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A5 )
=> ( ! [B6: A] :
( ( member2 @ A @ B6 @ B4 )
=> ? [X6: A] :
( ( member2 @ A @ X6 @ A5 )
& ( ord_less_eq @ A @ B6 @ X6 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B4 ) @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_4794_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,Y: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X5 )
=> ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X5 ) )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ X5 )
& ( ord_less @ A @ Y @ X3 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_4795_equiv__class__self,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member2 @ A @ A3 @ A5 )
=> ( member2 @ A @ A3 @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_self
thf(fact_4796_quotient__disj,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X5: set @ A,Y6: set @ A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member2 @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( member2 @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( X5 = Y6 )
| ( ( inf_inf @ ( set @ A ) @ X5 @ Y6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% quotient_disj
thf(fact_4797_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,U: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ U )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_4798_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,G: C > A,B4: set @ C,F: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G @ B4 ) )
=> ( ! [N7: B] :
( ( member2 @ B @ N7 @ A5 )
=> ? [X6: C] :
( ( member2 @ C @ X6 @ B4 )
& ( ord_less_eq @ A @ ( F @ N7 ) @ ( G @ X6 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B4 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_4799_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B4 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_4800_cSup__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A3: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X5 )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X5 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ X5 ) ) ) ) ) ) ).
% cSup_insert
thf(fact_4801_cSup__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A3: A] :
( ( condit941137186595557371_above @ A @ X5 )
=> ( ( ( X5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X5 ) )
= A3 ) )
& ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert2 @ A @ A3 @ X5 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ X5 ) ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_4802_cSup__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B4 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B4 ) ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_4803_equiv__class__eq__iff,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
= ( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member2 @ A @ X @ A5 )
& ( member2 @ A @ Y @ A5 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_4804_eq__equiv__class__iff,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( member2 @ A @ Y @ A5 )
=> ( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_4805_equiv__class__eq,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_4806_eq__equiv__class,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,A5: set @ A] :
( ( ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member2 @ A @ B2 @ A5 )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% eq_equiv_class
thf(fact_4807_eq__equiv__class__iff2,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( member2 @ A @ Y @ A5 )
=> ( ( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
= ( equiv_quotient @ A @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R3 ) )
= ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_4808_refines__equiv__class__eq2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A5: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_equiv @ A @ A5 @ S )
=> ( ( image @ A @ A @ S @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_4809_refines__equiv__class__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A5: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_equiv @ A @ A5 @ S )
=> ( ( image @ A @ A @ R @ ( image @ A @ A @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_4810_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A5: set @ B,F: B > A,A3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( ? [X3: B] :
( ( member2 @ B @ X3 @ A5 )
& ( ord_less @ A @ A3 @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_4811_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,G: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ A5 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ A5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A4: B] : ( sup_sup @ A @ ( F @ A4 ) @ ( G @ A4 ) )
@ A5 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_4812_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,G: B > A,B4: set @ B,F: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ B4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B4 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_4813_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( condit941137186595557371_above @ A @ A5 )
=> ( ( condit941137186595557371_above @ A @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B4 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_4814_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,A3: B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( insert2 @ B @ A3 @ A5 ) ) )
= ( sup_sup @ A @ ( F @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_4815_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ B4 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ B4 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_4816_equiv__class__subset,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_4817_subset__equiv__class,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member2 @ A @ B2 @ A5 )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% subset_equiv_class
thf(fact_4818_equiv__class__nondisjoint,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member2 @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_4819_in__quotient__imp__in__rel,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X5: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member2 @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X5 )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_4820_cSup__cInf,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S )
=> ( ( complete_Sup_Sup @ A @ S )
= ( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [X3: A] :
! [Y3: A] :
( ( member2 @ A @ Y3 @ S )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_4821_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A5: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ A5 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( F @ ( A5 @ X3 ) )
@ I5 ) )
@ ( F @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A5 @ I5 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_4822_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A16: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R2: set @ ( product_prod @ B @ B ),F: A > B > ( set @ C ),A1: A,A22: B] :
( ( equiv_equiv @ A @ A16 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R2 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R2 @ F )
=> ( ( member2 @ A @ A1 @ A16 )
=> ( ( member2 @ B @ A22 @ A25 )
=> ( ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X13: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F @ X13 ) @ ( image @ B @ B @ R2 @ ( insert2 @ B @ A22 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R1 @ ( insert2 @ A @ A1 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F @ A1 @ A22 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_4823_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),A3: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R3 @ F )
=> ( ( member2 @ A @ A3 @ A5 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F @ A3 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_4824_disjnt__equiv__class,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( disjnt @ A @ ( image @ A @ A @ R3 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_4825_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_4826_disjnt__insert2,axiom,
! [A: $tType,Y6: set @ A,A3: A,X5: set @ A] :
( ( disjnt @ A @ Y6 @ ( insert2 @ A @ A3 @ X5 ) )
= ( ~ ( member2 @ A @ A3 @ Y6 )
& ( disjnt @ A @ Y6 @ X5 ) ) ) ).
% disjnt_insert2
thf(fact_4827_disjnt__insert1,axiom,
! [A: $tType,A3: A,X5: set @ A,Y6: set @ A] :
( ( disjnt @ A @ ( insert2 @ A @ A3 @ X5 ) @ Y6 )
= ( ~ ( member2 @ A @ A3 @ Y6 )
& ( disjnt @ A @ X5 @ Y6 ) ) ) ).
% disjnt_insert1
thf(fact_4828_disjnt__Un2,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( disjnt @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ( disjnt @ A @ C4 @ A5 )
& ( disjnt @ A @ C4 @ B4 ) ) ) ).
% disjnt_Un2
thf(fact_4829_disjnt__Un1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( disjnt @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( ( disjnt @ A @ A5 @ C4 )
& ( disjnt @ A @ B4 @ C4 ) ) ) ).
% disjnt_Un1
thf(fact_4830_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C4: set @ A,A5: set @ B,B4: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C4
@ ^ [Uu3: A] : A5 )
@ ( product_Sigma @ A @ B @ C4
@ ^ [Uu3: A] : B4 ) )
= ( ( C4
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A5 @ B4 ) ) ) ).
% disjnt_Times1_iff
thf(fact_4831_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A5: set @ A,C4: set @ B,B4: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : C4 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu3: A] : C4 ) )
= ( ( C4
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A5 @ B4 ) ) ) ).
% disjnt_Times2_iff
thf(fact_4832_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A5: set @ A,C4: A > ( set @ B ),B4: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ C4 ) @ ( product_Sigma @ A @ B @ B4 @ C4 ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
=> ( ( C4 @ X3 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A5 @ B4 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_4833_disjnt__insert,axiom,
! [A: $tType,X: A,N4: set @ A,M: set @ A] :
( ~ ( member2 @ A @ X @ N4 )
=> ( ( disjnt @ A @ M @ N4 )
=> ( disjnt @ A @ ( insert2 @ A @ X @ M ) @ N4 ) ) ) ).
% disjnt_insert
thf(fact_4834_disjnt__iff,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A7: set @ A,B5: set @ A] :
! [X3: A] :
~ ( ( member2 @ A @ X3 @ A7 )
& ( member2 @ A @ X3 @ B5 ) ) ) ) ).
% disjnt_iff
thf(fact_4835_disjnt__sym,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( disjnt @ A @ A5 @ B4 )
=> ( disjnt @ A @ B4 @ A5 ) ) ).
% disjnt_sym
thf(fact_4836_disjnt__empty2,axiom,
! [A: $tType,A5: set @ A] : ( disjnt @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ).
% disjnt_empty2
thf(fact_4837_disjnt__empty1,axiom,
! [A: $tType,A5: set @ A] : ( disjnt @ A @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% disjnt_empty1
thf(fact_4838_disjnt__subset1,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A,Z8: set @ A] :
( ( disjnt @ A @ X5 @ Y6 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z8 @ X5 )
=> ( disjnt @ A @ Z8 @ Y6 ) ) ) ).
% disjnt_subset1
thf(fact_4839_disjnt__subset2,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A,Z8: set @ A] :
( ( disjnt @ A @ X5 @ Y6 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z8 @ Y6 )
=> ( disjnt @ A @ X5 @ Z8 ) ) ) ).
% disjnt_subset2
thf(fact_4840_disjnt__def,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A7: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A7 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjnt_def
thf(fact_4841_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A16: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R2: set @ ( product_prod @ B @ B ),F: A > B > ( set @ C ),A3: B] :
( ( equiv_equiv @ A @ A16 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R2 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R2 @ F )
=> ( ( member2 @ B @ A3 @ A25 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R1
@ ^ [X13: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F @ X13 ) @ ( image @ B @ B @ R2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_4842_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A5: set @ C,B4: C > ( set @ D ),F: D > B] :
( ( A5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X2: C] :
( ( member2 @ C @ X2 @ A5 )
=> ( ( B4 @ X2 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X3: C] : ( image2 @ D @ B @ F @ ( B4 @ X3 ) )
@ A5 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B4 @ A5 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F @ ( B4 @ X3 ) ) )
@ A5 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_4843_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( condit1013018076250108175_below @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ A5 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_4844_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A5: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ A5 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A5 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( F @ ( A5 @ X3 ) )
@ I5 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_4845_bdd__below__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_below_empty
thf(fact_4846_bdd__below__image__inf,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F: B > A,G: B > A,A5: set @ B] :
( ( condit1013018076250108175_below @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( inf_inf @ A @ ( F @ X3 ) @ ( G @ X3 ) )
@ A5 ) )
= ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
& ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ A5 ) ) ) ) ) ).
% bdd_below_image_inf
thf(fact_4847_bdd__below__Int2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B4: set @ A,A5: set @ A] :
( ( condit1013018076250108175_below @ A @ B4 )
=> ( condit1013018076250108175_below @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% bdd_below_Int2
thf(fact_4848_bdd__below__Int1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( condit1013018076250108175_below @ A @ A5 )
=> ( condit1013018076250108175_below @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% bdd_below_Int1
thf(fact_4849_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B4: set @ A,A5: set @ A] :
( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A5 )
=> ( ! [B6: A] :
( ( member2 @ A @ B6 @ B4 )
=> ? [X6: A] :
( ( member2 @ A @ X6 @ A5 )
& ( ord_less_eq @ A @ X6 @ B6 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B4 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_4850_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S )
=> ( ( ord_less_eq @ A @ A3 @ ( complete_Inf_Inf @ A @ S ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ S )
=> ( ord_less_eq @ A @ A3 @ X3 ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_4851_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,Y: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X5 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X5 ) @ Y )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ X5 )
& ( ord_less @ A @ X3 @ Y ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_4852_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B4: set @ B,F: C > A,A5: set @ C,G: B > A] :
( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F @ A5 ) )
=> ( ! [M4: B] :
( ( member2 @ B @ M4 @ B4 )
=> ? [X6: C] :
( ( member2 @ C @ X6 @ A5 )
& ( ord_less_eq @ A @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B4 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_4853_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,U: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ U @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_4854_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B4 ) @ ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_4855_cInf__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A3: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X5 )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X5 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ X5 ) ) ) ) ) ) ).
% cInf_insert
thf(fact_4856_cInf__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A3: A] :
( ( condit1013018076250108175_below @ A @ X5 )
=> ( ( ( X5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X5 ) )
= A3 ) )
& ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert2 @ A @ A3 @ X5 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ X5 ) ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_4857_cInf__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B4 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B4 ) ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_4858_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A5: set @ B,F: B > A,A3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ A3 )
= ( ? [X3: B] :
( ( member2 @ B @ X3 @ A5 )
& ( ord_less @ A @ ( F @ X3 ) @ A3 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_4859_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,G: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ A5 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A4: B] : ( inf_inf @ A @ ( F @ A4 ) @ ( G @ A4 ) )
@ A5 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_4860_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,G: B > A,B4: set @ B,F: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ B4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ B4 )
=> ( ord_less_eq @ A @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B4 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_4861_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( condit1013018076250108175_below @ A @ A5 )
=> ( ( condit1013018076250108175_below @ A @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B4 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_4862_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,A3: B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( insert2 @ B @ A3 @ A5 ) ) )
= ( inf_inf @ A @ ( F @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ) ) ).
% cINF_insert
thf(fact_4863_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ B4 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ B4 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_4864_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A5 )
=> ( ( condit1013018076250108175_below @ A @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_4865_cInf__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S )
=> ( ( complete_Inf_Inf @ A @ S )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [X3: A] :
! [Y3: A] :
( ( member2 @ A @ Y3 @ S )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_4866_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% numeral_num_of_nat_unfold
thf(fact_4867_subset__singleton__iff__Uniq,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [A4: A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( uniq @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ A5 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_4868_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B2: C] :
( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( F @ A3 @ B2 ) ) ).
% uncurry_apply
thf(fact_4869_uncurry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( uncurry @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% uncurry_def
thf(fact_4870_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( uniq @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs3 )
& ( ( set2 @ A @ Xs3 )
= A5 ) ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_4871_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] :
( lattic501386751177426532rg_min @ A @ B @ F
@ ^ [X3: A] : ( member2 @ A @ X3 @ S )
@ X_1 ) ) ) ) ).
% ex_is_arg_min_if_finite
thf(fact_4872_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A] :
( ! [A6: A,B6: A,X2: A] :
( ( member2 @ A @ A6 @ S )
=> ( ( member2 @ A @ B6 @ S )
=> ( ( ord_less_eq @ A @ A6 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ B6 )
=> ( member2 @ A @ X2 @ S ) ) ) ) )
=> ? [A6: A,B6: A] :
( ( S
= ( bot_bot @ ( set @ A ) ) )
| ( S
= ( top_top @ ( set @ A ) ) )
| ( S
= ( set_ord_lessThan @ A @ B6 ) )
| ( S
= ( set_ord_atMost @ A @ B6 ) )
| ( S
= ( set_ord_greaterThan @ A @ A6 ) )
| ( S
= ( set_ord_atLeast @ A @ A6 ) )
| ( S
= ( set_or5935395276787703475ssThan @ A @ A6 @ B6 ) )
| ( S
= ( set_or3652927894154168847AtMost @ A @ A6 @ B6 ) )
| ( S
= ( set_or7035219750837199246ssThan @ A @ A6 @ B6 ) )
| ( S
= ( set_or1337092689740270186AtMost @ A @ A6 @ B6 ) ) ) ) ) ).
% interval_cases
thf(fact_4873_take__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( nil @ A )
@ ^ [M3: nat] : ( cons @ A @ X @ ( take @ A @ M3 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_4874_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_4875_Int__atLeastAtMostR2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C3 ) @ D3 ) ) ) ).
% Int_atLeastAtMostR2
thf(fact_4876_Int__atLeastAtMostL2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atLeast @ A @ C3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C3 ) @ B2 ) ) ) ).
% Int_atLeastAtMostL2
thf(fact_4877_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_4878_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_4879_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_4880_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_4881_lessThan__Int__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( set_ord_greaterThan @ A @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% lessThan_Int_lessThan
thf(fact_4882_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_4883_atLeastAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or1337092689740270186AtMost @ A )
= ( ^ [L4: A,U3: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L4 ) @ ( set_ord_atMost @ A @ U3 ) ) ) ) ) ).
% atLeastAtMost_def
thf(fact_4884_atLeastLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [L4: A,U3: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L4 ) @ ( set_ord_lessThan @ A @ U3 ) ) ) ) ) ).
% atLeastLessThan_def
thf(fact_4885_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_4886_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_4887_drop__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ X @ Xs )
@ ^ [M3: nat] : ( drop @ A @ M3 @ Xs )
@ N ) ) ).
% drop_Cons
thf(fact_4888_list__update_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I: nat,V: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ I @ V )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V @ Xs )
@ ^ [J2: nat] : ( cons @ A @ X @ ( list_update @ A @ Xs @ J2 @ V ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_4889_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_4890_greaterThanAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [L4: A,U3: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L4 ) @ ( set_ord_atMost @ A @ U3 ) ) ) ) ) ).
% greaterThanAtMost_def
thf(fact_4891_nth__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( case_nat @ A @ X @ ( nth @ A @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_4892_greaterThanLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [L4: A,U3: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L4 ) @ ( set_ord_lessThan @ A @ U3 ) ) ) ) ) ).
% greaterThanLessThan_def
thf(fact_4893_greaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [A4: A,B3: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A4 ) @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ).
% greaterThanLessThan_eq
thf(fact_4894_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N ) @ ( set_ord_atLeast @ A @ N ) )
= ( insert2 @ A @ N @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_4895_subset__mset_OcINF__superset__mono,axiom,
! [A: $tType,B: $tType,A5: set @ B,G: B > ( multiset @ A ),B4: set @ B,F: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ B4 )
=> ( subseteq_mset @ A @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B4 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) ) ) ) ) ) ).
% subset_mset.cINF_superset_mono
thf(fact_4896_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ Xs )
= ( ! [I2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ ( suc @ I2 ) ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
thf(fact_4897_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_4898_subset__mset_Obdd__below__Int1,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) ) ).
% subset_mset.bdd_below_Int1
thf(fact_4899_subset__mset_Obdd__below__Int2,axiom,
! [A: $tType,B4: set @ ( multiset @ A ),A5: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) ) ).
% subset_mset.bdd_below_Int2
thf(fact_4900_transp__inf,axiom,
! [A: $tType,R3: A > A > $o,S4: A > A > $o] :
( ( transp @ A @ R3 )
=> ( ( transp @ A @ S4 )
=> ( transp @ A @ ( inf_inf @ ( A > A > $o ) @ R3 @ S4 ) ) ) ) ).
% transp_inf
thf(fact_4901_list_Orel__transp,axiom,
! [A: $tType,R: A > A > $o] :
( ( transp @ A @ R )
=> ( transp @ ( list @ A ) @ ( list_all2 @ A @ A @ R ) ) ) ).
% list.rel_transp
thf(fact_4902_sorted__wrt2,axiom,
! [A: $tType,P: A > A > $o,X: A,Y: A,Zs: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( ( P @ X @ Y )
& ( sorted_wrt @ A @ P @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted_wrt2
thf(fact_4903_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I3: int,J3: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J3 ) )
=> ( ( ( ord_less_eq @ int @ I3 @ J3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J3 ) )
=> ( P @ I3 @ J3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_4904_subset__mset_OcINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType,B4: set @ B,F: C > ( multiset @ A ),A5: set @ C,G: B > ( multiset @ A )] :
( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F @ A5 ) )
=> ( ! [M4: B] :
( ( member2 @ B @ M4 @ B4 )
=> ? [X6: C] :
( ( member2 @ C @ X6 @ A5 )
& ( subseteq_mset @ A @ ( F @ X6 ) @ ( G @ M4 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F @ A5 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B4 ) ) ) ) ) ) ).
% subset_mset.cINF_mono
thf(fact_4905_subset__mset_Ole__cINF__iff,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A ),U: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( subseteq_mset @ A @ U @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ A5 )
=> ( subseteq_mset @ A @ U @ ( F @ X3 ) ) ) ) ) ) ) ).
% subset_mset.le_cINF_iff
thf(fact_4906_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
=> ( ( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
& ( ~ ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_4907_subset__mset_Omono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F: ( multiset @ A ) > B,A5: C > ( multiset @ A ),I5: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( F @ ( A5 @ X3 ) )
@ I5 ) ) ) ) ) ) ) ).
% subset_mset.mono_cINF
thf(fact_4908_subset__mset_OcINF__union,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A ),B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ B4 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ B4 ) ) ) ) ) ) ) ) ).
% subset_mset.cINF_union
thf(fact_4909_subset__mset_OcINF__insert,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A ),A3: B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ ( insert2 @ B @ A3 @ A5 ) ) )
= ( inter_mset @ A @ ( F @ A3 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) ) ) ) ) ).
% subset_mset.cINF_insert
thf(fact_4910_set__mset__inter,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A] :
( ( set_mset @ A @ ( inter_mset @ A @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ ( set_mset @ A @ A5 ) @ ( set_mset @ A @ B4 ) ) ) ).
% set_mset_inter
thf(fact_4911_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( mono @ A @ B )
= ( mono @ A @ B ) ) ) ).
% order.mono.cong
thf(fact_4912_subset__mset_Omono__inf,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ! [F: ( multiset @ A ) > B,A5: multiset @ A,B4: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
=> ( ord_less_eq @ B @ ( F @ ( inter_mset @ A @ A5 @ B4 ) ) @ ( inf_inf @ B @ ( F @ A5 ) @ ( F @ B4 ) ) ) ) ) ).
% subset_mset.mono_inf
thf(fact_4913_subset__mset_Oless__eq__cInf__inter,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B4 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) ) ) ) ) ).
% subset_mset.less_eq_cInf_inter
thf(fact_4914_subset__mset_OcINF__inf__distrib,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A5 ) )
=> ( ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A5 ) ) )
= ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A4: B] : ( inter_mset @ A @ ( F @ A4 ) @ ( G @ A4 ) )
@ A5 ) ) ) ) ) ) ).
% subset_mset.cINF_inf_distrib
thf(fact_4915_subset__mset_Omono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F: ( multiset @ A ) > B,A5: C > ( multiset @ A ),I5: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X3: C] : ( F @ ( A5 @ X3 ) )
@ I5 ) )
@ ( F @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) ) ) ) ) ) ) ) ).
% subset_mset.mono_cSUP
thf(fact_4916_subset__mset_OcSUP__subset__mono,axiom,
! [A: $tType,B: $tType,A5: set @ B,G: B > ( multiset @ A ),B4: set @ B,F: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ A5 )
=> ( subseteq_mset @ A @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B4 ) ) ) ) ) ) ) ).
% subset_mset.cSUP_subset_mono
thf(fact_4917_Eps__Opt__def,axiom,
! [A: $tType] :
( ( eps_Opt @ A )
= ( ^ [P3: A > $o] :
( if @ ( option @ A )
@ ? [X8: A] : ( P3 @ X8 )
@ ( some @ A @ ( fChoice @ A @ P3 ) )
@ ( none @ A ) ) ) ) ).
% Eps_Opt_def
thf(fact_4918_some__insert__self,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( insert2 @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ S ) )
@ S )
= S ) ) ).
% some_insert_self
thf(fact_4919_some__elem,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ S ) )
@ S ) ) ).
% some_elem
thf(fact_4920_some__in__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( member2 @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ A5 ) )
@ A5 )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_4921_inv__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inv_on @ A @ B )
= ( ^ [F3: A > B,A7: set @ A,X3: B] :
( fChoice @ A
@ ^ [Y3: A] :
( ( member2 @ A @ Y3 @ A7 )
& ( ( F3 @ Y3 )
= X3 ) ) ) ) ) ).
% inv_on_def
thf(fact_4922_subset__mset_Obdd__above__Int1,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) ) ).
% subset_mset.bdd_above_Int1
thf(fact_4923_subset__mset_Obdd__above__Int2,axiom,
! [A: $tType,B4: set @ ( multiset @ A ),A5: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) ) ).
% subset_mset.bdd_above_Int2
thf(fact_4924_some__theI,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ? [A11: A,X_12: B] : ( P @ A11 @ X_12 )
=> ( ! [B13: B,B23: B] :
( ? [A6: A] : ( P @ A6 @ B13 )
=> ( ? [A6: A] : ( P @ A6 @ B23 )
=> ( B13 = B23 ) ) )
=> ( P
@ ( fChoice @ A
@ ^ [A4: A] :
? [X8: B] : ( P @ A4 @ X8 ) )
@ ( the @ B
@ ^ [B3: B] :
? [A4: A] : ( P @ A4 @ B3 ) ) ) ) ) ).
% some_theI
thf(fact_4925_subset__mset_OcSUP__le__iff,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A ),U: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) @ U )
= ( ! [X3: B] :
( ( member2 @ B @ X3 @ A5 )
=> ( subseteq_mset @ A @ ( F @ X3 ) @ U ) ) ) ) ) ) ).
% subset_mset.cSUP_le_iff
thf(fact_4926_subset__mset_OcSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType,A5: set @ B,G: C > ( multiset @ A ),B4: set @ C,F: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G @ B4 ) )
=> ( ! [N7: B] :
( ( member2 @ B @ N7 @ A5 )
=> ? [X6: C] :
( ( member2 @ C @ X6 @ B4 )
& ( subseteq_mset @ A @ ( F @ N7 ) @ ( G @ X6 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G @ B4 ) ) ) ) ) ) ).
% subset_mset.cSUP_mono
thf(fact_4927_fun__of__rel__def,axiom,
! [A: $tType,B: $tType] :
( ( fun_of_rel @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ A ),X3: B] :
( fChoice @ A
@ ^ [Y3: A] : ( member2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ).
% fun_of_rel_def
thf(fact_4928_subset__mset_OcSUP__union,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A ),B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ B4 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ B4 ) ) ) ) ) ) ) ) ).
% subset_mset.cSUP_union
thf(fact_4929_subset__mset_OcSUP__insert,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A ),A3: B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ ( insert2 @ B @ A3 @ A5 ) ) )
= ( union_mset @ A @ ( F @ A3 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) ) ) ) ) ).
% subset_mset.cSUP_insert
thf(fact_4930_subset__mset_OcSup__inter__less__eq,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) @ ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B4 ) ) ) ) ) ) ).
% subset_mset.cSup_inter_less_eq
thf(fact_4931_subset__mset_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A5 ) )
=> ( ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A5 ) ) )
= ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A4: B] : ( union_mset @ A @ ( F @ A4 ) @ ( G @ A4 ) )
@ A5 ) ) ) ) ) ) ).
% subset_mset.SUP_sup_distrib
thf(fact_4932_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_4933_irrefl__tranclI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R3 ) @ ( transitive_rtrancl @ A @ R3 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R3 ) ) ) ).
% irrefl_tranclI
thf(fact_4934_su__rel__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( su_rel_fun @ A @ B )
= ( ^ [F7: set @ ( product_prod @ A @ B ),F3: A > B] :
( ! [A7: A,B5: B,B14: B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ F7 )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B14 ) @ F7 )
=> ( B5 = B14 ) ) )
& ! [A7: A,P3: $o] :
( ! [B5: B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ F7 )
=> P3 )
=> P3 )
& ! [A7: A] :
( ( F3 @ A7 )
= ( the @ B
@ ^ [B5: B] : ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ F7 ) ) ) ) ) ) ).
% su_rel_fun_def
thf(fact_4935_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_4936_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
@ ^ [B3: A,A4: B] : ( P @ A4 @ B3 ) ) ) ) ).
% pair_set_inverse
thf(fact_4937_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ A3 @ ( sgn_sgn @ A @ B2 ) )
= ( times_times @ A @ A3 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% divide_sgn
thf(fact_4938_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_4939_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_4940_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_4941_converse__Int,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A ),S4: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ R3 @ S4 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R3 ) @ ( converse @ B @ A @ S4 ) ) ) ).
% converse_Int
thf(fact_4942_listrel1__converse,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( listrel1 @ A @ ( converse @ A @ A @ R3 ) )
= ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel1_converse
thf(fact_4943_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_mult
thf(fact_4944_in__listrel1__converse,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( listrel1 @ A @ ( converse @ A @ A @ R3 ) ) )
= ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ).
% in_listrel1_converse
thf(fact_4945_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_4946_su__rel__fun_Orepr,axiom,
! [B: $tType,A: $tType,F2: set @ ( product_prod @ A @ B ),F: A > B,A5: A,B4: B] :
( ( su_rel_fun @ A @ B @ F2 @ F )
=> ( ( ( F @ A5 )
= B4 )
= ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ F2 ) ) ) ).
% su_rel_fun.repr
thf(fact_4947_su__rel__fun_Orepr1,axiom,
! [B: $tType,A: $tType,F2: set @ ( product_prod @ A @ B ),F: A > B,A5: A] :
( ( su_rel_fun @ A @ B @ F2 @ F )
=> ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ ( F @ A5 ) ) @ F2 ) ) ).
% su_rel_fun.repr1
thf(fact_4948_su__rel__fun_Orepr2,axiom,
! [B: $tType,A: $tType,F2: set @ ( product_prod @ A @ B ),F: A > B,A5: A,B4: B] :
( ( su_rel_fun @ A @ B @ F2 @ F )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ F2 )
=> ( B4
= ( F @ A5 ) ) ) ) ).
% su_rel_fun.repr2
thf(fact_4949_su__rel__fun_Ounique,axiom,
! [A: $tType,B: $tType,F2: set @ ( product_prod @ A @ B ),F: A > B,A5: A,B4: B,B15: B] :
( ( su_rel_fun @ A @ B @ F2 @ F )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ F2 )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B15 ) @ F2 )
=> ( B4 = B15 ) ) ) ) ).
% su_rel_fun.unique
thf(fact_4950_su__rel__fun_Osurjective,axiom,
! [B: $tType,A: $tType,F2: set @ ( product_prod @ A @ B ),F: A > B,A5: A] :
( ( su_rel_fun @ A @ B @ F2 @ F )
=> ~ ! [B7: B] :
~ ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B7 ) @ F2 ) ) ).
% su_rel_fun.surjective
thf(fact_4951_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_4952_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= A3 ) ) ).
% sgn_mult_abs
thf(fact_4953_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= A3 ) ) ).
% abs_mult_sgn
thf(fact_4954_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K2: A] : ( times_times @ A @ K2 @ ( sgn_sgn @ A @ K2 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_4955_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_1_pos
thf(fact_4956_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_4957_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X3: A] :
( if @ A
@ ( X3
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_4958_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_4959_su__rel__fun_Of__def,axiom,
! [A: $tType,B: $tType,F2: set @ ( product_prod @ A @ B ),F: A > B,A5: A] :
( ( su_rel_fun @ A @ B @ F2 @ F )
=> ( ( F @ A5 )
= ( the @ B
@ ^ [B5: B] : ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ F2 ) ) ) ) ).
% su_rel_fun.f_def
thf(fact_4960_su__rel__fun_Ointro,axiom,
! [B: $tType,A: $tType,F2: set @ ( product_prod @ A @ B ),F: A > B] :
( ! [A9: A,B7: B,B8: B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B7 ) @ F2 )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B8 ) @ F2 )
=> ( B7 = B8 ) ) )
=> ( ! [A9: A,P2: $o] :
( ! [B16: B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B16 ) @ F2 )
=> P2 )
=> P2 )
=> ( ! [A9: A] :
( ( F @ A9 )
= ( the @ B
@ ^ [B5: B] : ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B5 ) @ F2 ) ) )
=> ( su_rel_fun @ A @ B @ F2 @ F ) ) ) ) ).
% su_rel_fun.intro
thf(fact_4961_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R3: set @ ( product_prod @ B @ A ),A5: set @ C,B4: C > ( set @ B )] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R3 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( image @ B @ A @ R3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X3: C] : ( image @ B @ A @ R3 @ ( B4 @ X3 ) )
@ A5 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_4962_length__upto,axiom,
! [I: int,J: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_4963_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_4964_single__valued__inter1,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ).
% single_valued_inter1
thf(fact_4965_single__valued__inter2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ S @ R ) ) ) ).
% single_valued_inter2
thf(fact_4966_bijective__alt,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R6 )
& ( single_valued @ B @ A @ ( converse @ A @ B @ R6 ) ) ) ) ) ).
% bijective_alt
thf(fact_4967_Image__Int__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A5: set @ B,B4: set @ B] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R ) )
=> ( ( image @ B @ A @ R @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R @ A5 ) @ ( image @ B @ A @ R @ B4 ) ) ) ) ).
% Image_Int_eq
thf(fact_4968_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A7: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A7 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A7 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_4969_lexn_Osimps_I2_J,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),N: nat] :
( ( lexn @ A @ R3 @ ( suc @ N ) )
= ( inf_inf @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ ( product_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_map_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) ) @ ( lex_prod @ A @ ( list @ A ) @ R3 @ ( lexn @ A @ R3 @ N ) ) )
@ ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= ( suc @ N ) ) ) ) ) ) ) ).
% lexn.simps(2)
thf(fact_4970_Linear__order__Well__order__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R3 ) )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member2 @ A @ X3 @ A7 )
& ! [Y3: A] :
( ( member2 @ A @ Y3 @ A7 )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_4971_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ A3 @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.bottom_right_bottom
thf(fact_4972_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ ( one_one @ A ) @ A3 )
= ( one_one @ A ) ) ) ).
% gcd.bottom_left_bottom
thf(fact_4973_is__unit__gcd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) ) ) ) ).
% is_unit_gcd_iff
thf(fact_4974_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% Gcd_2
thf(fact_4975_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,K: A,N: A] :
( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ ( times_times @ A @ K @ M2 ) @ N ) )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add_mult
thf(fact_4976_gcd__dvd__prod,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ).
% gcd_dvd_prod
thf(fact_4977_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_4978_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A3 ) @ C3 )
= ( gcd_gcd @ A @ B2 @ C3 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_4979_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C3 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C3 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_4980_gcd__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C3 )
= ( gcd_gcd @ A @ B2 @ C3 ) ) ) ) ).
% gcd_div_unit1
thf(fact_4981_gcd__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C3 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C3 ) ) ) ) ).
% gcd_div_unit2
thf(fact_4982_Well__order__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_well_order_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) ) ) ).
% Well_order_Restr
thf(fact_4983_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( member2 @ A @ A3 @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Gcd_fin.remove
thf(fact_4984_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert2 @ A @ A3 @ A5 ) )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Gcd_fin.insert_remove
thf(fact_4985_well__order__on__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R3 ) )
=> ( order_well_order_on @ A @ A5
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) ) ) ) ).
% well_order_on_Restr
thf(fact_4986_ofilter__Restr__Int,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( order_ofilter @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu3: A] : B4 ) )
@ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% ofilter_Restr_Int
thf(fact_4987_ofilter__Restr__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( order_ofilter @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu3: A] : B4 ) )
@ A5 ) ) ) ) ).
% ofilter_Restr_subset
thf(fact_4988_Field__Restr__ofilter,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) )
= A5 ) ) ) ).
% Field_Restr_ofilter
thf(fact_4989_ofilter__subset__embedS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( order_ofilter @ A @ R3 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu3: A] : B4 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embedS
thf(fact_4990_ofilter__subset__embed,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( order_ofilter @ A @ R3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
= ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu3: A] : B4 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embed
thf(fact_4991_ofilter__embed,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R3 ) )
& ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
@ R3
@ ( id @ A ) ) ) ) ) ).
% ofilter_embed
thf(fact_4992_ofilter__subset__embedS__iso,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( order_ofilter @ A @ R3 @ B4 )
=> ( ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu3: A] : B4 ) )
@ ( id @ A ) ) )
& ( ( A5 = B4 )
= ( bNF_Wellorder_iso @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu3: A] : B4 ) )
@ ( id @ A ) ) ) ) ) ) ) ).
% ofilter_subset_embedS_iso
thf(fact_4993_embed__implies__iso__Restr,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R8: set @ ( product_prod @ B @ B ),F: B > A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R8 ) @ R8 )
=> ( ( bNF_Wellorder_embed @ B @ A @ R8 @ R3 @ F )
=> ( bNF_Wellorder_iso @ B @ A @ R8
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F @ ( field2 @ B @ R8 ) )
@ ^ [Uu3: A] : ( image2 @ B @ A @ F @ ( field2 @ B @ R8 ) ) ) )
@ F ) ) ) ) ).
% embed_implies_iso_Restr
thf(fact_4994_ofilter__subset__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( order_ofilter @ A @ R3 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
= ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu3: A] : B4 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_4995_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
~ ! [B6: $o,H4: A,As: set @ B] :
( X
!= ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B6 @ ( product_Pair @ A @ ( set @ B ) @ H4 @ As ) ) ) ).
% pure_assn_raw.cases
thf(fact_4996_ofilter__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ ( field2 @ A @ R3 ) )
= ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
@ R3 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% ofilter_ordLess
thf(fact_4997_underS__Restr__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ A3 )
@ ^ [Uu3: A] : ( order_underS @ A @ R3 @ A3 ) ) )
@ R3 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% underS_Restr_ordLess
thf(fact_4998_ofilter__subset__ordLeq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( order_ofilter @ A @ R3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
= ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu3: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLeq
thf(fact_4999_ofilter__Restr__under,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( member2 @ A @ A3 @ A5 )
=> ( ( order_under @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
@ A3 )
= ( order_under @ A @ R3 @ A3 ) ) ) ) ) ).
% ofilter_Restr_under
thf(fact_5000_underS__empty,axiom,
! [A: $tType,A3: A,R3: set @ ( product_prod @ A @ A )] :
( ~ ( member2 @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( order_underS @ A @ R3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% underS_empty
thf(fact_5001_Refl__under__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member2 @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( order_under @ A @ R3 @ A3 )
= ( sup_sup @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Refl_under_underS
thf(fact_5002_ordLeq__iff__ordLess__Restr,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R8: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R8 ) @ R8 )
=> ( ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R8 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( field2 @ A @ R3 ) )
=> ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ X3 )
@ ^ [Uu3: A] : ( order_underS @ A @ R3 @ X3 ) ) )
@ R8 )
@ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
thf(fact_5003_underS__Field3,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( field2 @ A @ R3 ) ) ) ).
% underS_Field3
thf(fact_5004_ordLess__iff__ordIso__Restr,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R8: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R8 ) @ R8 )
=> ( ( member2 @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R8 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( member2 @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R8
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ X3 )
@ ^ [Uu3: A] : ( order_underS @ A @ R3 @ X3 ) ) ) )
@ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
thf(fact_5005_min__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S ) @ ( insert2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( min_ext @ A @ R ) ) ) ).
% min_ext_compat
thf(fact_5006_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_5007_max__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S ) @ ( insert2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( max_ext @ A @ R ) ) ) ).
% max_ext_compat
thf(fact_5008_bounded__quasi__semilattice__set_Oremove,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( member2 @ A @ A3 @ A5 )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 )
= ( F @ A3 @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.remove
thf(fact_5009_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( insert2 @ A @ A3 @ A5 ) )
= ( F @ A3 @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.insert_remove
thf(fact_5010_bounded__quasi__semilattice__set_Oempty,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( bot_bot @ ( set @ A ) ) )
= Top ) ) ).
% bounded_quasi_semilattice_set.empty
thf(fact_5011_max__ext_Ocases,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X6: A] :
( ( member2 @ A @ X6 @ A1 )
=> ? [Xa3: A] :
( ( member2 @ A @ Xa3 @ A22 )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ Xa3 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_5012_max__ext_Osimps,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
= ( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X3: A] :
( ( member2 @ A @ X3 @ A1 )
=> ? [Y3: A] :
( ( member2 @ A @ Y3 @ A22 )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_5013_max__ext_Omax__extI,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X5 )
=> ( ( finite_finite2 @ A @ Y6 )
=> ( ( Y6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ X5 )
=> ? [Xa2: A] :
( ( member2 @ A @ Xa2 @ Y6 )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R ) ) )
=> ( member2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X5 @ Y6 ) @ ( max_ext @ A @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_5014_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu3: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X8: set @ A,Y9: set @ A] :
( ( Uu3
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X8 @ Y9 ) )
& ( X8
!= ( bot_bot @ ( set @ A ) ) )
& ! [X3: A] :
( ( member2 @ A @ X3 @ Y9 )
=> ? [Y3: A] :
( ( member2 @ A @ Y3 @ X8 )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R4 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_5015_refl__on__reflcl__Image,axiom,
! [A: $tType,B4: set @ A,A5: set @ ( product_prod @ A @ A ),C4: set @ A] :
( ( refl_on @ A @ B4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ B4 )
=> ( ( image @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ ( id2 @ A ) ) @ C4 )
= ( image @ A @ A @ A5 @ C4 ) ) ) ) ).
% refl_on_reflcl_Image
thf(fact_5016_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > A] :
( ( bind2 @ B @ A @ A5
@ ^ [X3: B] : ( insert2 @ A @ ( F @ X3 ) @ ( bot_bot @ ( set @ A ) ) ) )
= ( image2 @ B @ A @ F @ A5 ) ) ).
% bind_singleton_conv_image
thf(fact_5017_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X6: A] :
( ( member2 @ A @ X6 @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X6 ) ) ).
% bex_empty
thf(fact_5018_empty__bind,axiom,
! [B: $tType,A: $tType,F: B > ( set @ A )] :
( ( bind2 @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_bind
thf(fact_5019_bijective__Id,axiom,
! [A: $tType] : ( bijective @ A @ A @ ( id2 @ A ) ) ).
% bijective_Id
thf(fact_5020_bex__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( top_top @ ( set @ A ) ) )
& ( P @ X3 ) ) )
= ( ? [X8: A] : ( P @ X8 ) ) ) ).
% bex_UNIV
thf(fact_5021_below__Id__inv,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R ) @ ( id2 @ A ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ).
% below_Id_inv
thf(fact_5022_Bex__def,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A7: set @ A,P3: A > $o] :
? [X3: A] :
( ( member2 @ A @ X3 @ A7 )
& ( P3 @ X3 ) ) ) ) ).
% Bex_def
thf(fact_5023_Set_Obind__bind,axiom,
! [C: $tType,B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ C ),C4: C > ( set @ B )] :
( ( bind2 @ C @ B @ ( bind2 @ A @ C @ A5 @ B4 ) @ C4 )
= ( bind2 @ A @ B @ A5
@ ^ [X3: A] : ( bind2 @ C @ B @ ( B4 @ X3 ) @ C4 ) ) ) ).
% Set.bind_bind
thf(fact_5024_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ A @ B )
= ( ^ [F3: A > B,A7: set @ A] :
( collect @ B
@ ^ [Y3: B] :
? [X3: A] :
( ( member2 @ A @ X3 @ A7 )
& ( Y3
= ( F3 @ X3 ) ) ) ) ) ) ).
% image_def
thf(fact_5025_Set_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind2 @ A @ B )
= ( ^ [A7: set @ A,F3: A > ( set @ B )] :
( collect @ B
@ ^ [X3: B] :
? [Y3: set @ B] :
( ( member2 @ ( set @ B ) @ Y3 @ ( image2 @ A @ ( set @ B ) @ F3 @ A7 ) )
& ( member2 @ B @ X3 @ Y3 ) ) ) ) ) ).
% Set.bind_def
thf(fact_5026_bind__const,axiom,
! [B: $tType,A: $tType,A5: set @ B,B4: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind2 @ B @ A @ A5
@ ^ [Uu3: B] : B4 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind2 @ B @ A @ A5
@ ^ [Uu3: B] : B4 )
= B4 ) ) ) ).
% bind_const
thf(fact_5027_nonempty__bind__const,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bind2 @ A @ B @ A5
@ ^ [Uu3: A] : B4 )
= B4 ) ) ).
% nonempty_bind_const
thf(fact_5028_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( vimage @ A @ B @ F @ ( image2 @ A @ B @ F @ A5 ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X3: A] :
( ( member2 @ A @ X3 @ A5 )
& ( ( F @ X3 )
= ( F @ Y3 ) ) ) ) ) ).
% vimage_image_eq
thf(fact_5029_single__valued__below__Id,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) )
=> ( single_valued @ A @ A @ R ) ) ).
% single_valued_below_Id
thf(fact_5030_nths__nths,axiom,
! [A: $tType,Xs: list @ A,A5: set @ nat,B4: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs @ A5 ) @ B4 )
= ( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I2: nat] :
( ( member2 @ nat @ I2 @ A5 )
& ( member2 @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I9: nat] :
( ( member2 @ nat @ I9 @ A5 )
& ( ord_less @ nat @ I9 @ I2 ) ) ) )
@ B4 ) ) ) ) ) ).
% nths_nths
thf(fact_5031_rtrancl__Int__subset,axiom,
! [A: $tType,S4: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ S4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ S4 ) @ R3 ) @ S4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ S4 ) ) ) ).
% rtrancl_Int_subset
thf(fact_5032_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F3: A > ( option @ B ),A7: set @ A] :
( collect @ B
@ ^ [B3: B] :
? [X3: A] :
( ( member2 @ A @ X3 @ A7 )
& ( ( F3 @ X3 )
= ( some @ B @ B3 ) ) ) ) ) ) ).
% map_project_def
thf(fact_5033_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
= ( ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R3 ) )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member2 @ A @ X3 @ A7 )
& ! [Y3: A] :
( ( member2 @ A @ Y3 @ A7 )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_5034_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [A12: A,A23: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B17: A,B24: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A12 @ ( insert2 @ A @ A23 @ ( insert2 @ A @ B17 @ ( insert2 @ A @ B24 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R4 ) )
& ( ( ( A12 = B17 )
& ( A23 = B24 ) )
| ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B17 @ B24 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B17 @ B24 ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B17 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B17 @ B24 ) )
& ( A12 = B17 )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B24 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_5035_wf__listrel1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( wf @ ( list @ A ) @ ( listrel1 @ A @ R3 ) )
= ( wf @ A @ R3 ) ) ).
% wf_listrel1_iff
thf(fact_5036_wf__lex,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( wf @ ( list @ A ) @ ( lex @ A @ R3 ) ) ) ).
% wf_lex
thf(fact_5037_wf__lenlex,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( wf @ ( list @ A ) @ ( lenlex @ A @ R3 ) ) ) ).
% wf_lenlex
thf(fact_5038_brk__rel__wf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( wf @ ( product_prod @ $o @ A ) @ ( brk_rel @ A @ A @ R ) ) ) ).
% brk_rel_wf
thf(fact_5039_wf__lexn,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),N: nat] :
( ( wf @ A @ R3 )
=> ( wf @ ( list @ A ) @ ( lexn @ A @ R3 @ N ) ) ) ).
% wf_lexn
thf(fact_5040_wf__measures,axiom,
! [A: $tType,Fs: list @ ( A > nat )] : ( wf @ A @ ( measures @ A @ Fs ) ) ).
% wf_measures
thf(fact_5041_wf__Int1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R8: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( wf @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R8 ) ) ) ).
% wf_Int1
thf(fact_5042_wf__Int2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R8: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( wf @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R8 @ R3 ) ) ) ).
% wf_Int2
thf(fact_5043_wfE__min_H,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Q: set @ A] :
( ( wf @ A @ R )
=> ( ( Q
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z4: A] :
( ( member2 @ A @ Z4 @ Q )
=> ~ ! [Y4: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z4 ) @ R )
=> ~ ( member2 @ A @ Y4 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_5044_finite__wf__eq__wf__converse,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( wf @ A @ ( converse @ A @ A @ R ) )
= ( wf @ A @ R ) ) ) ).
% finite_wf_eq_wf_converse
thf(fact_5045_wfI__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A9 @ ( image @ A @ A @ R @ A9 ) )
=> ( A9
= ( bot_bot @ ( set @ A ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_pf
thf(fact_5046_wfE__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( image @ A @ A @ R @ A5 ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% wfE_pf
thf(fact_5047_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
@ ^ [Q7: set @ A,Q2: set @ A] :
( ( ord_less @ ( set @ A ) @ Q2 @ Q7 )
& ( ord_less_eq @ ( set @ A ) @ Q7 @ S ) ) ) ) ) ) ).
% wf_bounded_supset
thf(fact_5048_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [A7: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R4 ) )
& ( A7
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X3: A] :
( ( member2 @ A @ X3 @ A7 )
& ! [Y3: A] :
( ( member2 @ A @ Y3 @ A7 )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_5049_trans__wf__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( ( wf @ A @ R3 )
= ( ! [A4: A] :
( wf @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( image @ A @ A @ ( converse @ A @ A @ R3 ) @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
@ ^ [Uu3: A] : ( image @ A @ A @ ( converse @ A @ A @ R3 ) @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% trans_wf_iff
thf(fact_5050_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) )
=> ( ( ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
| ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( A3 = B2 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_5051_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X8: set @ A,Y9: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ( finite_finite2 @ A @ Y9 )
& ( Y9
!= ( bot_bot @ ( set @ A ) ) )
& ! [X3: A] :
( ( member2 @ A @ X3 @ X8 )
=> ? [Y3: A] :
( ( member2 @ A @ Y3 @ Y9 )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_5052_lenlex__transI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( trans @ ( list @ A ) @ ( lenlex @ A @ R3 ) ) ) ).
% lenlex_transI
thf(fact_5053_ball__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( top_top @ ( set @ A ) ) )
=> ( P @ X3 ) ) )
= ( ! [X8: A] : ( P @ X8 ) ) ) ).
% ball_UNIV
thf(fact_5054_lexn__transI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),N: nat] :
( ( trans @ A @ R3 )
=> ( trans @ ( list @ A ) @ ( lexn @ A @ R3 @ N ) ) ) ).
% lexn_transI
thf(fact_5055_lexord__transI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( trans @ ( list @ A ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_transI
thf(fact_5056_lex__transI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( trans @ ( list @ A ) @ ( lex @ A @ R3 ) ) ) ).
% lex_transI
thf(fact_5057_listrel__trans,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( trans @ ( list @ A ) @ ( listrel @ A @ A @ R3 ) ) ) ).
% listrel_trans
thf(fact_5058_trans__Int,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( ( trans @ A @ S4 )
=> ( trans @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S4 ) ) ) ) ).
% trans_Int
thf(fact_5059_lexord__trans,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lexord @ A @ R3 ) )
=> ( ( trans @ A @ R3 )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_trans
thf(fact_5060_lenlex__trans,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lenlex @ A @ R3 ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lenlex @ A @ R3 ) )
=> ( ( trans @ A @ R3 )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lenlex @ A @ R3 ) ) ) ) ) ).
% lenlex_trans
thf(fact_5061_trans__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( trans @ A @ R3 )
=> ( trans @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) ) ) ) ).
% trans_Restr
thf(fact_5062_wo__rel_Omax2__among,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member2 @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member2 @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( member2 @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_5063_trans__rtrancl__eq__reflcl,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ A5 )
=> ( ( transitive_rtrancl @ A @ A5 )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ ( id2 @ A ) ) ) ) ).
% trans_rtrancl_eq_reflcl
thf(fact_5064_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X3: A] :
( ( P3 @ X3 )
& ! [Y3: A] :
( ( P3 @ Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_5065_wo__rel_Ocases__Total,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) )
=> ( ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_5066_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member2 @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member2 @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 )
& ( member2 @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_5067_Image__absorb__rtrancl,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B4: set @ A,C4: set @ A] :
( ( trans @ A @ A5 )
=> ( ( refl_on @ A @ B4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ B4 )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ A5 ) @ C4 )
= ( image @ A @ A @ A5 @ C4 ) ) ) ) ) ).
% Image_absorb_rtrancl
thf(fact_5068_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X3: A] :
( ( P3 @ X3 )
& ! [Y3: A] :
( ( P3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_5069_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] : ( bNF_We4791949203932849705sMinim @ A @ R3 @ B4 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_5070_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq2: A > A > $o,P3: A > $o] :
( the @ A
@ ^ [X3: A] :
( ( P3 @ X3 )
& ! [Y3: A] :
( ( P3 @ Y3 )
=> ( Less_eq2 @ X3 @ Y3 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_5071_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_5072_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_5073_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_5074_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( bNF_We4791949203932849705sMinim @ A @ R3 @ B4 @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B4 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_5075_wo__rel_Ominim__inField,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B4 ) @ ( field2 @ A @ R3 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_5076_wo__rel_Ominim__in,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B4 ) @ B4 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_5077_list__ex__length,axiom,
! [A: $tType] :
( ( list_ex @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
? [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P3 @ ( nth @ A @ Xs3 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_5078_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_5079_successively__append__iff,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( successively @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( ( successively @ A @ P @ Xs )
& ( successively @ A @ P @ Ys )
& ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ A ) )
| ( P @ ( last @ A @ Xs ) @ ( hd @ A @ Ys ) ) ) ) ) ).
% successively_append_iff
thf(fact_5080_list__ex__simps_I1_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( list_ex @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( ( P @ X )
| ( list_ex @ A @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_5081_list__ex__simps_I2_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex @ A @ P @ ( nil @ A ) ) ).
% list_ex_simps(2)
thf(fact_5082_list__ex__append,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( list_ex @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( ( list_ex @ A @ P @ Xs )
| ( list_ex @ A @ P @ Ys ) ) ) ).
% list_ex_append
thf(fact_5083_list__ex__rev,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( list_ex @ A @ P @ ( rev @ A @ Xs ) )
= ( list_ex @ A @ P @ Xs ) ) ).
% list_ex_rev
thf(fact_5084_successively__rev,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( successively @ A @ P @ ( rev @ A @ Xs ) )
= ( successively @ A
@ ^ [X3: A,Y3: A] : ( P @ Y3 @ X3 )
@ Xs ) ) ).
% successively_rev
thf(fact_5085_successively__map,axiom,
! [A: $tType,B: $tType,P: A > A > $o,F: B > A,Xs: list @ B] :
( ( successively @ A @ P @ ( map @ B @ A @ F @ Xs ) )
= ( successively @ B
@ ^ [X3: B,Y3: B] : ( P @ ( F @ X3 ) @ ( F @ Y3 ) )
@ Xs ) ) ).
% successively_map
thf(fact_5086_successively__remdups__adjI,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( successively @ A @ P @ Xs )
=> ( successively @ A @ P @ ( remdups_adj @ A @ Xs ) ) ) ).
% successively_remdups_adjI
thf(fact_5087_successively__cong,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o,Q: A > A > $o,Ys: list @ A] :
( ! [X2: A,Y2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member2 @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X2 @ Y2 )
= ( Q @ X2 @ Y2 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( successively @ A @ P @ Xs )
= ( successively @ A @ Q @ Ys ) ) ) ) ).
% successively_cong
thf(fact_5088_successively__mono,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,Q: A > A > $o] :
( ( successively @ A @ P @ Xs )
=> ( ! [X2: A,Y2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member2 @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) ) ) )
=> ( successively @ A @ Q @ Xs ) ) ) ).
% successively_mono
thf(fact_5089_successively__if__sorted__wrt,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( sorted_wrt @ A @ P @ Xs )
=> ( successively @ A @ P @ Xs ) ) ).
% successively_if_sorted_wrt
thf(fact_5090_successively_Oelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( successively @ A @ X @ Xa )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( ( X @ X2 @ Y2 )
& ( successively @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_5091_successively_Osimps_I3_J,axiom,
! [A: $tType,P: A > A > $o,X: A,Y: A,Xs: list @ A] :
( ( successively @ A @ P @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
= ( ( P @ X @ Y )
& ( successively @ A @ P @ ( cons @ A @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_5092_successively_Osimps_I1_J,axiom,
! [A: $tType,P: A > A > $o] : ( successively @ A @ P @ ( nil @ A ) ) ).
% successively.simps(1)
thf(fact_5093_list__ex__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,F: A > $o,G: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( list_ex @ A @ F @ Xs )
= ( list_ex @ A @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_5094_distinct__adj__def,axiom,
! [A: $tType] :
( ( distinct_adj @ A )
= ( successively @ A
@ ^ [X3: A,Y3: A] : X3 != Y3 ) ) ).
% distinct_adj_def
thf(fact_5095_successively_Oelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( successively @ A @ X @ Xa )
=> ( ( Xa
!= ( nil @ A ) )
=> ( ! [X2: A] :
( Xa
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ~ ( ( X @ X2 @ Y2 )
& ( successively @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_5096_successively_Oelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( successively @ A @ X @ Xa )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ Y )
=> ( ( ? [X2: A] :
( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ Y )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( Y
= ( ~ ( ( X @ X2 @ Y2 )
& ( successively @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_5097_successively_Osimps_I2_J,axiom,
! [A: $tType,P: A > A > $o,X: A] : ( successively @ A @ P @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% successively.simps(2)
thf(fact_5098_semilattice__neutr__order_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F @ Z2 @ Less_eq @ Less )
=> ( semilattice_neutr @ A @ F @ Z2 ) ) ).
% semilattice_neutr_order.axioms(1)
thf(fact_5099_successively__iff__sorted__wrt__strong,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ! [X2: A,Y2: A,Z4: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member2 @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( member2 @ A @ Z4 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X2 @ Y2 )
=> ( ( P @ Y2 @ Z4 )
=> ( P @ X2 @ Z4 ) ) ) ) ) )
=> ( ( successively @ A @ P @ Xs )
= ( sorted_wrt @ A @ P @ Xs ) ) ) ).
% successively_iff_sorted_wrt_strong
thf(fact_5100_successively__remdups__adj__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 @ X2 ) )
=> ( ( successively @ A @ P @ ( remdups_adj @ A @ Xs ) )
= ( successively @ A @ P @ Xs ) ) ) ).
% successively_remdups_adj_iff
thf(fact_5101_successively__conv__sorted__wrt,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( transp @ A @ P )
=> ( ( successively @ A @ P @ Xs )
= ( sorted_wrt @ A @ P @ Xs ) ) ) ).
% successively_conv_sorted_wrt
thf(fact_5102_successively__altdef,axiom,
! [A: $tType] :
( ( successively @ A )
= ( ^ [P3: A > A > $o] :
( rec_list @ $o @ A @ $true
@ ^ [X3: A,Xs3: list @ A,B3: $o] :
( case_list @ $o @ A @ $true
@ ^ [Y3: A,Xa4: list @ A] :
( ( P3 @ X3 @ Y3 )
& B3 )
@ Xs3 ) ) ) ) ).
% successively_altdef
thf(fact_5103_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_5104_inf__top_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( semilattice_neutr @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.semilattice_neutr_axioms
thf(fact_5105_Bex__set__list__ex,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( list_ex @ A @ P @ Xs ) ) ).
% Bex_set_list_ex
thf(fact_5106_successively__Cons,axiom,
! [A: $tType,P: A > A > $o,X: A,Xs: list @ A] :
( ( successively @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ( ( P @ X @ ( hd @ A @ Xs ) )
& ( successively @ A @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_5107_successively_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( successively @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ( ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( ( Y
= ( ( X @ X2 @ Y2 )
& ( successively @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% successively.pelims(1)
thf(fact_5108_successively_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( successively @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
=> ( ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) )
=> ~ ( ( X @ X2 @ Y2 )
& ( successively @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.pelims(2)
thf(fact_5109_successively_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( successively @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) )
=> ( ( X @ X2 @ Y2 )
& ( successively @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ).
% successively.pelims(3)
thf(fact_5110_bij__betw__nth,axiom,
! [A: $tType,Xs: list @ A,A5: set @ nat,B4: set @ A] :
( ( distinct @ A @ Xs )
=> ( ( A5
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( ( B4
= ( set2 @ A @ Xs ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs ) @ A5 @ B4 ) ) ) ) ).
% bij_betw_nth
thf(fact_5111_prod__mset_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: multiset @ B,B4: multiset @ B,G: B > A] :
( ( ( inter_mset @ B @ A5 @ B4 )
= ( zero_zero @ ( multiset @ B ) ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ ( union_mset @ B @ A5 @ B4 ) ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ A5 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ B4 ) ) ) ) ) ) ).
% prod_mset.union_disjoint
thf(fact_5112_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ X @ ( power_int @ A @ X @ M2 ) ) ) ) ) ).
% power_int_add_1'
thf(fact_5113_power__int__1__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int] :
( ( power_int @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_int_1_left
thf(fact_5114_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,W: num,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W ) ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_5115_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W: num,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_5116_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_5117_mset__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( mset @ A @ ( map @ B @ A @ F @ Xs ) )
= ( image_mset @ B @ A @ F @ ( mset @ B @ Xs ) ) ) ).
% mset_map
thf(fact_5118_prod__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [Uu3: B] : ( one_one @ A )
@ A5 ) )
= ( one_one @ A ) ) ) ).
% prod_mset.neutral_const
thf(fact_5119_power__int__minus__one__mult__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) )
= ( one_one @ A ) ) ) ).
% power_int_minus_one_mult_self
thf(fact_5120_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int,B2: A] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ B2 ) )
= B2 ) ) ).
% power_int_minus_one_mult_self'
thf(fact_5121_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num,B2: A] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% power_int_add_numeral2
thf(fact_5122_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_5123_prod__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,X: B,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ ( add_mset @ B @ X @ A5 ) ) )
= ( times_times @ A @ ( G @ X ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ A5 ) ) ) ) ) ).
% prod_mset.insert
thf(fact_5124_mset__map__split__orig,axiom,
! [B: $tType,A: $tType,F: B > A,P: multiset @ B,M12: multiset @ A,M23: multiset @ A] :
( ( ( image_mset @ B @ A @ F @ P )
= ( plus_plus @ ( multiset @ A ) @ M12 @ M23 ) )
=> ~ ! [P12: multiset @ B,P23: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P12 @ P23 ) )
=> ( ( ( image_mset @ B @ A @ F @ P12 )
= M12 )
=> ( ( image_mset @ B @ A @ F @ P23 )
!= M23 ) ) ) ) ).
% mset_map_split_orig
thf(fact_5125_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib
thf(fact_5126_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( times_times @ A @ ( power_int @ A @ X @ N ) @ X )
= ( times_times @ A @ X @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_commutes
thf(fact_5127_mset__map__id,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X5: multiset @ A] :
( ! [X2: A] :
( ( F @ ( G @ X2 ) )
= X2 )
=> ( ( image_mset @ B @ A @ F @ ( image_mset @ A @ B @ G @ X5 ) )
= X5 ) ) ).
% mset_map_id
thf(fact_5128_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ B] :
( ( bij_betw @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) @ A5 )
=> ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_5129_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( bij_betw @ A @ B @ F @ A5 @ ( bot_bot @ ( set @ B ) ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_5130_mset__map__split__orig__le,axiom,
! [B: $tType,A: $tType,F: B > A,P: multiset @ B,M12: multiset @ A,M23: multiset @ A] :
( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P ) @ ( plus_plus @ ( multiset @ A ) @ M12 @ M23 ) )
=> ~ ! [P12: multiset @ B,P23: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P12 @ P23 ) )
=> ( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P12 ) @ M12 )
=> ~ ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P23 ) @ M23 ) ) ) ) ).
% mset_map_split_orig_le
thf(fact_5131_power__int__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_one_over
thf(fact_5132_prod__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H: B > A,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( G @ X3 ) @ ( H @ X3 ) )
@ A5 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ A5 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ H @ A5 ) ) ) ) ) ).
% prod_mset.distrib
thf(fact_5133_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A5: set @ A,F: A > B,A13: set @ B] :
( ~ ( member2 @ A @ B2 @ A5 )
=> ( ~ ( member2 @ B @ ( F @ B2 ) @ A13 )
=> ( ( bij_betw @ A @ B @ F @ A5 @ A13 )
= ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A13 @ ( insert2 @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_5134_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A5: set @ A,F: A > B,A13: set @ B] :
( ~ ( member2 @ A @ B2 @ A5 )
=> ( ~ ( member2 @ B @ ( F @ B2 ) @ A13 )
=> ( ( bij_betw @ A @ B @ F @ A5 @ A13 )
=> ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A13 @ ( insert2 @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_5135_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B4: set @ B,C4: set @ A,D2: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ( ( bij_betw @ A @ B @ F @ C4 @ D2 )
=> ( ( ( inf_inf @ ( set @ B ) @ B4 @ D2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ C4 ) @ ( sup_sup @ ( set @ B ) @ B4 @ D2 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_5136_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,C4: set @ A,B4: set @ B,D2: set @ B] :
( ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ C4 ) @ ( sup_sup @ ( set @ B ) @ B4 @ D2 ) )
=> ( ( bij_betw @ A @ B @ F @ C4 @ D2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ C4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B4 @ D2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F @ A5 @ B4 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_5137_power__int__0__left__If,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( ( M2
= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( one_one @ A ) ) )
& ( ( M2
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( zero_zero @ A ) ) ) ) ) ).
% power_int_0_left_If
thf(fact_5138_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A3: A] :
( ( ord_less_eq @ int @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N4 ) ) ) ) ) ).
% power_int_increasing
thf(fact_5139_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A3: A] :
( ( ord_less @ int @ N @ N4 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N4 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_5140_power__int__minus__one__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ int @ N ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) ) ) ).
% power_int_minus_one_minus
thf(fact_5141_power__int__minus__one__diff__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: int,B2: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ A3 @ B2 ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ B2 @ A3 ) ) ) ) ).
% power_int_minus_one_diff_commute
thf(fact_5142_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,C4: set @ B,G: A > B,B4: set @ A,D2: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ C4 )
=> ( ( bij_betw @ A @ B @ G @ B4 @ D2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C4 @ D2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member2 @ A @ X3 @ A5 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup @ ( set @ A ) @ A5 @ B4 )
@ ( sup_sup @ ( set @ B ) @ C4 @ D2 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_5143_infinite__imp__bij__betw2,axiom,
! [A: $tType,A5: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [H4: A > A] : ( bij_betw @ A @ A @ H4 @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_5144_infinite__imp__bij__betw,axiom,
! [A: $tType,A5: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [H4: A > A] : ( bij_betw @ A @ A @ H4 @ A5 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_5145_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A3: A] :
( ( ord_less @ int @ N @ N4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N4 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_5146_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% one_le_power_int
thf(fact_5147_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power_int
thf(fact_5148_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M2 @ N )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ N ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_add
thf(fact_5149_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T5: set @ C,H: B > C,S: set @ B,T: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S @ S3 ) @ ( minus_minus @ ( set @ C ) @ T @ T5 ) )
=> ( ! [A6: B] :
( ( member2 @ B @ A6 @ S3 )
=> ( ( G @ ( H @ A6 ) )
= ( one_one @ A ) ) )
=> ( ! [B6: C] :
( ( member2 @ C @ B6 @ T5 )
=> ( ( G @ B6 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( G @ ( H @ X3 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_5150_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( division_ring @ A )
& ( one @ B )
& ( uminus @ B ) )
=> ! [X: C,A3: A,N: int] :
( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% power_int_minus_left_distrib
thf(fact_5151_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_5152_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N4: int,A3: A] :
( ( ord_less_eq @ int @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( A3
!= ( zero_zero @ A ) )
| ( N4
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N4 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_5153_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_5154_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_5155_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) @ X )
= ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_minus_mult
thf(fact_5156_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ X ) ) ) ) ).
% power_int_add_1
thf(fact_5157_count__image__mset,axiom,
! [A: $tType,B: $tType,F: B > A,A5: multiset @ B,X: A] :
( ( count @ A @ ( image_mset @ B @ A @ F @ A5 ) @ X )
= ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ A5 ) @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( set_mset @ B @ A5 ) ) ) ) ).
% count_image_mset
thf(fact_5158_sum__mset__constant,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [Y: B,A5: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [X3: A] : Y
@ A5 ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( multiset @ A ) @ A5 ) ) @ Y ) ) ) ).
% sum_mset_constant
thf(fact_5159_numeral__BitM,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bitM @ N ) )
= ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_BitM
thf(fact_5160_mset__empty__count,axiom,
! [A: $tType,M: multiset @ A] :
( ( ! [P6: A] :
( ( count @ A @ M @ P6 )
= ( zero_zero @ nat ) ) )
= ( M
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% mset_empty_count
thf(fact_5161_sum__mset__replicate__mset,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,A3: A] :
( ( comm_m7189776963980413722m_mset @ A @ ( replicate_mset @ A @ N @ A3 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ A3 ) ) ) ).
% sum_mset_replicate_mset
thf(fact_5162_sum__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C3: A,A5: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( X3 = Y ) @ C3 @ ( zero_zero @ A ) )
@ A5 ) )
= ( times_times @ A @ C3 @ ( semiring_1_of_nat @ A @ ( count @ B @ A5 @ Y ) ) ) ) ) ).
% sum_mset_delta
thf(fact_5163_sum__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C3: A,A5: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( Y = X3 ) @ C3 @ ( zero_zero @ A ) )
@ A5 ) )
= ( times_times @ A @ C3 @ ( semiring_1_of_nat @ A @ ( count @ B @ A5 @ Y ) ) ) ) ) ).
% sum_mset_delta'
thf(fact_5164_sum__mset__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( times @ A )
& ( semiring_0 @ B ) )
=> ! [F: A > B,A5: multiset @ A,G: C > B,B4: multiset @ C] :
( ( times_times @ B @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ A @ B @ F @ A5 ) ) @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ C @ B @ G @ B4 ) ) )
= ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [I2: A] :
( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ C @ B
@ ^ [J2: C] : ( times_times @ B @ ( F @ I2 ) @ ( G @ J2 ) )
@ B4 ) )
@ A5 ) ) ) ) ).
% sum_mset_product
thf(fact_5165_sum__mset__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F: B > A,M: multiset @ B,C3: A] :
( ( times_times @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F @ M ) ) @ C3 )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( F @ X3 ) @ C3 )
@ M ) ) ) ) ).
% sum_mset_distrib_right
thf(fact_5166_sum__mset__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C3: A,F: B > A,M: multiset @ B] :
( ( times_times @ A @ C3 @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F @ M ) ) )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( times_times @ A @ C3 @ ( F @ X3 ) )
@ M ) ) ) ) ).
% sum_mset_distrib_left
thf(fact_5167_count__ne__remove,axiom,
! [A: $tType,X: A,T4: A,S: multiset @ A] :
( ( X != T4 )
=> ( ( count @ A @ S @ X )
= ( count @ A @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ X ) ) ) ).
% count_ne_remove
thf(fact_5168_count__mset__set__finite__iff,axiom,
! [A: $tType,S: set @ A,A3: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( member2 @ A @ A3 @ S )
=> ( ( count @ A @ ( mset_set @ A @ S ) @ A3 )
= ( one_one @ nat ) ) )
& ( ~ ( member2 @ A @ A3 @ S )
=> ( ( count @ A @ ( mset_set @ A @ S ) @ A3 )
= ( zero_zero @ nat ) ) ) ) ) ).
% count_mset_set_finite_iff
thf(fact_5169_prod__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C3: A,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( Y = X3 ) @ C3 @ ( one_one @ A ) )
@ A5 ) )
= ( power_power @ A @ C3 @ ( count @ B @ A5 @ Y ) ) ) ) ).
% prod_mset_delta'
thf(fact_5170_prod__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C3: A,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( X3 = Y ) @ C3 @ ( one_one @ A ) )
@ A5 ) )
= ( power_power @ A @ C3 @ ( count @ B @ A5 @ Y ) ) ) ) ).
% prod_mset_delta
thf(fact_5171_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M2 ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_5172_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R3: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X3: C] : R3
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R3 ) ) ) ).
% sum_list_triv
thf(fact_5173_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_5174_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_5175_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_5176_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_5177_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_5178_sum__list_OCons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X @ Xs ) )
= ( plus_plus @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% sum_list.Cons
thf(fact_5179_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_5180_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_5181_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_5182_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_5183_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_5184_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_5185_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( Inverse @ ( F @ A3 @ B2 ) )
= ( F @ ( Inverse @ B2 ) @ ( Inverse @ A3 ) ) ) ) ).
% group.inverse_distrib_swap
thf(fact_5186_group_Ogroup__left__neutral,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( F @ Z2 @ A3 )
= A3 ) ) ).
% group.group_left_neutral
thf(fact_5187_group_Oinverse__neutral,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( Inverse @ Z2 )
= Z2 ) ) ).
% group.inverse_neutral
thf(fact_5188_group_Oinverse__inverse,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( Inverse @ ( Inverse @ A3 ) )
= A3 ) ) ).
% group.inverse_inverse
thf(fact_5189_group_Oinverse__unique,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( ( F @ A3 @ B2 )
= Z2 )
=> ( ( Inverse @ A3 )
= B2 ) ) ) ).
% group.inverse_unique
thf(fact_5190_group_Oright__inverse,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( F @ A3 @ ( Inverse @ A3 ) )
= Z2 ) ) ).
% group.right_inverse
thf(fact_5191_group_Oright__cancel,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,B2: A,A3: A,C3: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( ( F @ B2 @ A3 )
= ( F @ C3 @ A3 ) )
= ( B2 = C3 ) ) ) ).
% group.right_cancel
thf(fact_5192_group_Oleft__inverse,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( F @ ( Inverse @ A3 ) @ A3 )
= Z2 ) ) ).
% group.left_inverse
thf(fact_5193_group_Oleft__cancel,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A,C3: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( ( F @ A3 @ B2 )
= ( F @ A3 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% group.left_cancel
thf(fact_5194_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_5195_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_5196_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B3: A] : ( times_times @ A @ ( inverse_inverse @ A @ B3 ) @ A4 ) ) ) ) ).
% divide_inverse_commute
thf(fact_5197_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B3: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% divide_inverse
thf(fact_5198_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A4: A,B3: A] : ( times_times @ A @ A4 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_5199_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_5200_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) @ ( power_power @ A @ X @ M2 ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_5201_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M2 ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_5202_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_5203_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_5204_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ ( G @ X3 ) )
@ Xs ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_addf
thf(fact_5205_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( ( semiring_0 @ A )
=> ! [F: B > A,C3: A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( times_times @ A @ ( F @ X3 ) @ C3 )
@ Xs ) )
= ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ C3 ) ) ) ).
% sum_list_mult_const
thf(fact_5206_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C3: A,F: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( times_times @ A @ C3 @ ( F @ X3 ) )
@ Xs ) )
= ( times_times @ A @ C3 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% sum_list_const_mult
thf(fact_5207_sum__list__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( minus_minus @ A @ ( F @ X3 ) @ ( G @ X3 ) )
@ Xs ) )
= ( minus_minus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_subtractf
thf(fact_5208_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_5209_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_less_inverse
thf(fact_5210_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_5211_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_5212_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_5213_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_5214_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ B2 @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_5215_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_5216_sum__list__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [Xs: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs ) ) ) ) ).
% sum_list_abs
thf(fact_5217_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F ) @ Xs ) ) ) ) ).
% uminus_sum_list_map
thf(fact_5218_sum__list__replicate,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,C3: A] :
( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N @ C3 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ C3 ) ) ) ).
% sum_list_replicate
thf(fact_5219_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F: A > B,G: A > B] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_5220_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F: B > A,P: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( F @ X3 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_5221_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_5222_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_5223_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_le_inverse
thf(fact_5224_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_5225_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_5226_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F: A > B,G: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_5227_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P: B > $o,F: B > A] :
( ! [X2: B] :
( ( member2 @ B @ X2 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ( F @ X2 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_5228_sum__list__distinct__conv__sum__set,axiom,
! [C: $tType,B: $tType] :
( ( comm_monoid_add @ C )
=> ! [Xs: list @ B,F: B > C] :
( ( distinct @ B @ Xs )
=> ( ( groups8242544230860333062m_list @ C @ ( map @ B @ C @ F @ Xs ) )
= ( groups7311177749621191930dd_sum @ B @ C @ F @ ( set2 @ B @ Xs ) ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_5229_sum_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ B,G: B > A] :
( ( distinct @ B @ Xs )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_5230_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X: B,Xs: list @ B,F: B > A] :
( ( member2 @ B @ X @ ( set2 @ B @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) )
= ( plus_plus @ A @ ( F @ X ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_5231_sum__code,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,Xs: list @ B] :
( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ ( remdups @ B @ Xs ) ) ) ) ) ).
% sum_code
thf(fact_5232_uncurry__curry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
= F ) ).
% uncurry_curry_id
thf(fact_5233_curry__uncurry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F ) )
= F ) ).
% curry_uncurry_id
thf(fact_5234_finite__reachable__restrictedI,axiom,
! [A: $tType,Q: set @ A,I5: set @ A,E5: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ Q )
=> ( ( ord_less_eq @ ( set @ A ) @ I5 @ Q )
=> ( ( ord_less_eq @ ( set @ A ) @ ( range @ A @ A @ E5 ) @ Q )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ I5 ) ) ) ) ) ).
% finite_reachable_restrictedI
thf(fact_5235_Range__empty,axiom,
! [B: $tType,A: $tType] :
( ( range @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Range_empty
thf(fact_5236_Range__empty__iff,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A )] :
( ( ( range @ B @ A @ R3 )
= ( bot_bot @ ( set @ A ) ) )
= ( R3
= ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% Range_empty_iff
thf(fact_5237_Range__Int__subset,axiom,
! [A: $tType,B: $tType,A5: set @ ( product_prod @ B @ A ),B4: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( range @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ A5 @ B4 ) ) @ ( inf_inf @ ( set @ A ) @ ( range @ B @ A @ A5 ) @ ( range @ B @ A @ B4 ) ) ) ).
% Range_Int_subset
thf(fact_5238_trancl__Image__in__Range,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),V3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ R ) @ V3 ) @ ( range @ A @ A @ R ) ) ).
% trancl_Image_in_Range
thf(fact_5239_Range__rel__restrict,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( range @ A @ A @ ( rel_restrict @ A @ R @ A5 ) ) @ ( minus_minus @ ( set @ A ) @ ( range @ A @ A @ R ) @ A5 ) ) ).
% Range_rel_restrict
thf(fact_5240_wf__UN,axiom,
! [B: $tType,A: $tType,I5: set @ A,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [I3: A] :
( ( member2 @ A @ I3 @ I5 )
=> ( wf @ B @ ( R3 @ I3 ) ) )
=> ( ! [I3: A,J3: A] :
( ( member2 @ A @ I3 @ I5 )
=> ( ( member2 @ A @ J3 @ I5 )
=> ( ( ( R3 @ I3 )
!= ( R3 @ J3 ) )
=> ( ( inf_inf @ ( set @ B ) @ ( domain @ B @ B @ ( R3 @ I3 ) ) @ ( range @ B @ B @ ( R3 @ J3 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( wf @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ I5 ) ) ) ) ) ).
% wf_UN
thf(fact_5241_dom__ran__disj__comp,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R ) @ ( range @ A @ A @ R ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( relcomp @ A @ A @ A @ R @ R )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% dom_ran_disj_comp
thf(fact_5242_wf__max,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( converse @ A @ A @ R ) )
=> ( ( R
!= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ! [M4: A] :
~ ( member2 @ A @ M4 @ ( minus_minus @ ( set @ A ) @ ( range @ A @ A @ R ) @ ( domain @ A @ A @ R ) ) ) ) ) ).
% wf_max
thf(fact_5243_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_5244_Domain__empty__iff,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( ( domain @ A @ B @ R3 )
= ( bot_bot @ ( set @ A ) ) )
= ( R3
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% Domain_empty_iff
thf(fact_5245_Domain__Int__subset,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B ),B4: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ B4 ) ) @ ( inf_inf @ ( set @ A ) @ ( domain @ A @ B @ A5 ) @ ( domain @ A @ B @ B4 ) ) ) ).
% Domain_Int_subset
thf(fact_5246_for__in__RI,axiom,
! [B: $tType,A: $tType,X: A,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ A @ X @ ( domain @ A @ B @ R ) )
=> ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ ( fun_of_rel @ A @ B @ R @ X ) ) @ R ) ) ).
% for_in_RI
thf(fact_5247_Domain__rel__restrict,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ A @ ( rel_restrict @ A @ R @ A5 ) ) @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R ) @ A5 ) ) ).
% Domain_rel_restrict
thf(fact_5248_wf__no__path,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R ) @ ( range @ A @ A @ R ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ R ) ) ).
% wf_no_path
thf(fact_5249_wf__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( R
!= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ! [M4: A] :
~ ( member2 @ A @ M4 @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R ) @ ( range @ A @ A @ R ) ) ) ) ) ).
% wf_min
thf(fact_5250_wf__Un,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( ( wf @ A @ S4 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R3 ) @ ( range @ A @ A @ S4 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S4 ) ) ) ) ) ).
% wf_Un
thf(fact_5251_wf__Union,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ! [X2: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R )
=> ( wf @ A @ X2 ) )
=> ( ! [X2: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R )
=> ! [Xa3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( set @ ( product_prod @ A @ A ) ) @ Xa3 @ R )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ X2 ) @ ( range @ A @ A @ Xa3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( wf @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R ) ) ) ) ).
% wf_Union
thf(fact_5252_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_5253_folding__insort__key_Oaxioms_I2_J,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F )
=> ( foldin3648464208017769352axioms @ B @ A @ S @ F ) ) ).
% folding_insort_key.axioms(2)
thf(fact_5254_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 )
=> ~ ! [H4: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As ) )
=> ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ).
% pure_assn_raw.elims(3)
thf(fact_5255_monoid_Oleft__neutral,axiom,
! [A: $tType,F: A > A > A,Z2: A,A3: A] :
( ( monoid @ A @ F @ Z2 )
=> ( ( F @ Z2 @ A3 )
= A3 ) ) ).
% monoid.left_neutral
thf(fact_5256_monoid_Oright__neutral,axiom,
! [A: $tType,F: A > A > A,Z2: A,A3: A] :
( ( monoid @ A @ F @ Z2 )
=> ( ( F @ A3 @ Z2 )
= A3 ) ) ).
% monoid.right_neutral
thf(fact_5257_add_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.monoid_axioms
thf(fact_5258_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.monoid_axioms
thf(fact_5259_append_Omonoid__axioms,axiom,
! [A: $tType] : ( monoid @ ( list @ A ) @ ( append @ A ) @ ( nil @ A ) ) ).
% append.monoid_axioms
thf(fact_5260_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.monoid_axioms
thf(fact_5261_inf__top_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.monoid_axioms
thf(fact_5262_folding__insort__key__axioms_Ointro,axiom,
! [A: $tType,B: $tType,F: B > A,S: set @ B] :
( ( inj_on @ B @ A @ F @ S )
=> ( foldin3648464208017769352axioms @ B @ A @ S @ F ) ) ).
% folding_insort_key_axioms.intro
thf(fact_5263_folding__insort__key__axioms__def,axiom,
! [A: $tType,B: $tType] :
( ( foldin3648464208017769352axioms @ B @ A )
= ( ^ [S7: set @ B,F3: B > A] : ( inj_on @ B @ A @ F3 @ S7 ) ) ) ).
% folding_insort_key_axioms_def
thf(fact_5264_pure__assn__raw_Osimps,axiom,
! [A: $tType,B: $tType,B2: $o,H: A,As2: set @ B] :
( ( pure_assn_raw @ A @ B @ B2 @ ( product_Pair @ A @ ( set @ B ) @ H @ As2 ) )
= ( ( As2
= ( bot_bot @ ( set @ B ) ) )
& B2 ) ) ).
% pure_assn_raw.simps
thf(fact_5265_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 )
=> ~ ! [H4: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As ) )
=> ( Y
= ( ~ ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.elims(1)
thf(fact_5266_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 )
=> ~ ! [H4: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As ) )
=> ~ ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ).
% pure_assn_raw.elims(2)
thf(fact_5267_disjoint__image__subset,axiom,
! [A: $tType,A17: set @ ( set @ A ),F: ( set @ A ) > ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A17 )
=> ( ! [X14: set @ A] :
( ( member2 @ ( set @ A ) @ X14 @ A17 )
=> ( ord_less_eq @ ( set @ A ) @ ( F @ X14 ) @ X14 ) )
=> ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ F @ A17 ) ) ) ) ).
% disjoint_image_subset
thf(fact_5268_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 ) )
=> ( ! [Xs12: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs22 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs22 ) )
=> ~ ( 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 ) @ Xs12 @ Xs22 ) @ ( nil @ A ) ) ) ) ) )
=> ( ! [Xs12: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs22 ) )
=> ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs12 ) @ Xs22 ) )
=> ~ ( 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 ) @ Xs12 @ Xs22 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ) )
=> ~ ! [Xs12: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs22 ) )
=> ! [X1: A,X23: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) )
=> ( ( Y
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs12 ) @ ( cons @ A @ X23 @ Xs22 ) ) @ Xs2 ) )
=> ~ ( 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 ) @ Xs12 @ Xs22 ) @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
thf(fact_5269_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_max @ B @ ( F @ X ) @ ( F @ Y ) )
= ( F @ ( ord_min @ A @ X @ Y ) ) ) ) ) ).
% max_of_antimono
thf(fact_5270_pairwise__trivial,axiom,
! [A: $tType,I5: set @ A] :
( pairwise @ A
@ ^ [I2: A,J2: A] : J2 != I2
@ I5 ) ).
% pairwise_trivial
thf(fact_5271_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R6: A > A > $o,S7: set @ A] :
! [X3: A] :
( ( member2 @ A @ X3 @ S7 )
=> ! [Y3: A] :
( ( member2 @ A @ Y3 @ S7 )
=> ( ( X3 != Y3 )
=> ( R6 @ X3 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_5272_pairwiseI,axiom,
! [A: $tType,S: set @ A,R: A > A > $o] :
( ! [X2: A,Y2: A] :
( ( member2 @ A @ X2 @ S )
=> ( ( member2 @ A @ Y2 @ S )
=> ( ( X2 != Y2 )
=> ( R @ X2 @ Y2 ) ) ) )
=> ( pairwise @ A @ R @ S ) ) ).
% pairwiseI
thf(fact_5273_pairwiseD,axiom,
! [A: $tType,R: A > A > $o,S: set @ A,X: A,Y: A] :
( ( pairwise @ A @ R @ S )
=> ( ( member2 @ A @ X @ S )
=> ( ( member2 @ A @ Y @ S )
=> ( ( X != Y )
=> ( R @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_5274_pairwise__insert,axiom,
! [A: $tType,R3: A > A > $o,X: A,S4: set @ A] :
( ( pairwise @ A @ R3 @ ( insert2 @ A @ X @ S4 ) )
= ( ! [Y3: A] :
( ( ( member2 @ A @ Y3 @ S4 )
& ( Y3 != X ) )
=> ( ( R3 @ X @ Y3 )
& ( R3 @ Y3 @ X ) ) )
& ( pairwise @ A @ R3 @ S4 ) ) ) ).
% pairwise_insert
thf(fact_5275_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S: set @ A,T: set @ A] :
( ( pairwise @ A @ P @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T @ S )
=> ( pairwise @ A @ P @ T ) ) ) ).
% pairwise_subset
thf(fact_5276_pairwise__mono,axiom,
! [A: $tType,P: A > A > $o,A5: set @ A,Q: A > A > $o,B4: set @ A] :
( ( pairwise @ A @ P @ A5 )
=> ( ! [X2: A,Y2: A] :
( ( P @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( pairwise @ A @ Q @ B4 ) ) ) ) ).
% pairwise_mono
thf(fact_5277_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_5278_pairwise__imageI,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B,P: B > B > $o] :
( ! [X2: A,Y2: A] :
( ( member2 @ A @ X2 @ A5 )
=> ( ( member2 @ A @ Y2 @ A5 )
=> ( ( X2 != Y2 )
=> ( ( ( F @ X2 )
!= ( F @ Y2 ) )
=> ( P @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) )
=> ( pairwise @ B @ P @ ( image2 @ A @ B @ F @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_5279_pairwise__image,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,F: B > A,S4: set @ B] :
( ( pairwise @ A @ R3 @ ( image2 @ B @ A @ F @ S4 ) )
= ( pairwise @ B
@ ^ [X3: B,Y3: B] :
( ( ( F @ X3 )
!= ( F @ Y3 ) )
=> ( R3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
@ S4 ) ) ).
% pairwise_image
thf(fact_5280_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F3: A > B] :
! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) ) ) ) ) ).
% antimono_def
thf(fact_5281_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X2 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_5282_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% antimonoE
thf(fact_5283_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% antimonoD
thf(fact_5284_pairwise__disjnt__iff,axiom,
! [A: $tType,A17: set @ ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A17 )
= ( ! [X3: A] :
( uniq @ ( set @ A )
@ ^ [X8: set @ A] :
( ( member2 @ ( set @ A ) @ X8 @ A17 )
& ( member2 @ A @ X3 @ X8 ) ) ) ) ) ).
% pairwise_disjnt_iff
thf(fact_5285_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A5: A] : ( pairwise @ A @ P @ ( insert2 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_5286_pairwise__alt,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R6: A > A > $o,S7: set @ A] :
! [X3: A] :
( ( member2 @ A @ X3 @ S7 )
=> ! [Y3: A] :
( ( member2 @ A @ Y3 @ ( minus_minus @ ( set @ A ) @ S7 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( R6 @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_5287_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_min @ B @ ( F @ X ) @ ( F @ Y ) )
= ( F @ ( ord_max @ A @ X @ Y ) ) ) ) ) ).
% min_of_antimono
thf(fact_5288_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A5: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A5 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ A @ ( complete_Sup_Sup @ A @ A5 ) @ A5 ) ) ) ) ) ).
% in_chain_finite
thf(fact_5289_drop__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% drop_bit_of_1
thf(fact_5290_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_5291_bit__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% bit_minus_1_iff
thf(fact_5292_chain__empty,axiom,
! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ).
% chain_empty
thf(fact_5293_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_5294_ccpo__Sup__mono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A5 )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ B4 )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ A5 )
=> ? [Xa2: A] :
( ( member2 @ A @ Xa2 @ B4 )
& ( ord_less_eq @ A @ X2 @ Xa2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B4 ) ) ) ) ) ) ).
% ccpo_Sup_mono
thf(fact_5295_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_5296_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_5297_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( divide_divide @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_5298_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A4: A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A4 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_5299_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_5300_list__ex1__simps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( list_ex1 @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( ( ( P @ X )
=> ( list_all @ A
@ ^ [Y3: A] :
( ~ ( P @ Y3 )
| ( X = Y3 ) )
@ Xs ) )
& ( ~ ( P @ X )
=> ( list_ex1 @ A @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_5301_list__ex__iff__not__all__inverval__int,axiom,
! [P: int > $o,I: int,J: int] :
( ( list_ex @ int @ P @ ( upto @ I @ J ) )
= ( ~ ( all_interval_int @ ( comp @ $o @ $o @ int @ (~) @ P ) @ I @ J ) ) ) ).
% list_ex_iff_not_all_inverval_int
thf(fact_5302_all__interval__int__def,axiom,
( all_interval_int
= ( ^ [P3: int > $o,I2: int,J2: int] :
! [X3: int] :
( ( member2 @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ I2 @ J2 ) )
=> ( P3 @ X3 ) ) ) ) ).
% all_interval_int_def
thf(fact_5303_list_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A3: A,Aa3: list @ A] :
( ( list_all @ A @ P @ ( cons @ A @ A3 @ Aa3 ) )
= ( ( P @ A3 )
& ( list_all @ A @ P @ Aa3 ) ) ) ).
% list.pred_inject(2)
thf(fact_5304_list__all__simps_I1_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( list_all @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( ( P @ X )
& ( list_all @ A @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_5305_list__all__simps_I2_J,axiom,
! [A: $tType,P: A > $o] : ( list_all @ A @ P @ ( nil @ A ) ) ).
% list_all_simps(2)
thf(fact_5306_list__all__append,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( list_all @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( ( list_all @ A @ P @ Xs )
& ( list_all @ A @ P @ Ys ) ) ) ).
% list_all_append
thf(fact_5307_list__all__rev,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( list_all @ A @ P @ ( rev @ A @ Xs ) )
= ( list_all @ A @ P @ Xs ) ) ).
% list_all_rev
thf(fact_5308_list_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: list @ A,Pa: A > $o] :
( ( list_all @ A @ P @ X )
=> ( ! [Z4: A] :
( ( member2 @ A @ Z4 @ ( set2 @ A @ X ) )
=> ( ( P @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( list_all @ A @ Pa @ X ) ) ) ).
% list.pred_mono_strong
thf(fact_5309_list__all__cong,axiom,
! [A: $tType,X: list @ A,Ya: list @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z4: A] :
( ( member2 @ A @ Z4 @ ( set2 @ A @ Ya ) )
=> ( ( P @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( list_all @ A @ P @ X )
= ( list_all @ A @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_5310_list_Opred__True,axiom,
! [A: $tType] :
( ( list_all @ A
@ ^ [Uu3: A] : $true )
= ( ^ [Uu3: list @ A] : $true ) ) ).
% list.pred_True
thf(fact_5311_list__all__iff__all__interval__int,axiom,
! [P: int > $o,I: int,J: int] :
( ( list_all @ int @ P @ ( upto @ I @ J ) )
= ( all_interval_int @ P @ I @ J ) ) ).
% list_all_iff_all_interval_int
thf(fact_5312_list_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( list_all @ A @ P @ ( nil @ A ) ) ).
% list.pred_inject(1)
thf(fact_5313_list_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: list @ A,Ya: list @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( list_all @ A
@ ^ [Z3: A] :
( ( F @ Z3 )
= ( G @ Z3 ) )
@ Ya )
=> ( ( map @ A @ B @ F @ X )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_5314_list_Opred__mono,axiom,
! [A: $tType,P: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
=> ( ord_less_eq @ ( ( list @ A ) > $o ) @ ( list_all @ A @ P ) @ ( list_all @ A @ Pa ) ) ) ).
% list.pred_mono
thf(fact_5315_prop__matchD,axiom,
! [A: $tType,Al: list @ A,E3: A,Bl: list @ A,Al2: list @ A,E2: A,Bl2: list @ A,P: A > $o] :
( ( ( append @ A @ Al @ ( cons @ A @ E3 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E2 @ Bl2 ) ) )
=> ( ( list_all @ A @ P @ Al )
=> ( ~ ( P @ E3 )
=> ( ~ ( P @ E2 )
=> ( ( list_all @ A @ P @ Bl )
=> ( ( Al = Al2 )
& ( E3 = E2 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_matchD
thf(fact_5316_prop__match,axiom,
! [A: $tType,P: A > $o,Al: list @ A,E3: A,E2: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( list_all @ A @ P @ Al )
=> ( ~ ( P @ E3 )
=> ( ~ ( P @ E2 )
=> ( ( list_all @ A @ P @ Bl )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E3 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E2 @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( E3 = E2 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_match
thf(fact_5317_Ball__set__list__all,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) ) )
= ( list_all @ A @ P @ Xs ) ) ).
% Ball_set_list_all
thf(fact_5318_list__all__iff,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P3: A > $o,X3: list @ A] :
! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ X3 ) )
=> ( P3 @ Y3 ) ) ) ) ).
% list_all_iff
thf(fact_5319_list_Opred__set,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P3: A > $o,X3: list @ A] :
! [Y3: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ X3 ) )
=> ( P3 @ Y3 ) ) ) ) ).
% list.pred_set
thf(fact_5320_list_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: list @ A] :
( ( list_all @ B @ Q @ ( map @ A @ B @ F @ X ) )
= ( list_all @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% list.pred_map
thf(fact_5321_list__all__length,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ ( nth @ A @ Xs3 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_5322_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: A > B,C4: set @ A,B4: set @ A,X: A] :
( ( inj_on @ A @ B @ G @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ B4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member2 @ ( B > A )
@ ^ [I2: B] : ( if @ A @ ( member2 @ B @ I2 @ ( image2 @ A @ B @ G @ C4 ) ) @ ( the_inv_into @ A @ B @ C4 @ G @ I2 ) @ X )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_5323_is__num_Osimps,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_is_num @ A )
= ( ^ [A4: A] :
( ( A4
= ( one_one @ A ) )
| ? [X3: A] :
( ( A4
= ( uminus_uminus @ A @ X3 ) )
& ( neg_numeral_is_num @ A @ X3 ) )
| ? [X3: A,Y3: A] :
( ( A4
= ( plus_plus @ A @ X3 @ Y3 ) )
& ( neg_numeral_is_num @ A @ X3 )
& ( neg_numeral_is_num @ A @ Y3 ) ) ) ) ) ) ).
% is_num.simps
thf(fact_5324_Func__is__emp,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A5 @ B4 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_5325_Func__non__emp,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: set @ B] :
( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A5 @ B4 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_5326_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_5327_Func__empty,axiom,
! [B: $tType,A: $tType,B4: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B4 )
= ( insert2 @ ( A > B )
@ ^ [X3: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_5328_is__num_Ocases,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A] :
( ( neg_numeral_is_num @ A @ A3 )
=> ( ( A3
!= ( one_one @ A ) )
=> ( ! [X2: A] :
( ( A3
= ( uminus_uminus @ A @ X2 ) )
=> ~ ( neg_numeral_is_num @ A @ X2 ) )
=> ~ ! [X2: A,Y2: A] :
( ( A3
= ( plus_plus @ A @ X2 @ Y2 ) )
=> ( ( neg_numeral_is_num @ A @ X2 )
=> ~ ( neg_numeral_is_num @ A @ Y2 ) ) ) ) ) ) ) ).
% is_num.cases
thf(fact_5329_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A16: set @ B,B18: set @ A,F22: C > D,B25: set @ C,A25: set @ D] :
( ( ( image2 @ B @ A @ F1 @ A16 )
= B18 )
=> ( ( inj_on @ C @ D @ F22 @ B25 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image2 @ C @ D @ F22 @ B25 ) @ A25 )
=> ( ( ( B25
= ( bot_bot @ ( set @ C ) ) )
=> ( A25
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B25 @ B18 )
= ( image2 @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B25 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A25 @ A16 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_5330_mult__one__div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) )
= ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% mult_one_div_unit_factor
thf(fact_5331_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% unit_factor_mult_unit_left
thf(fact_5332_unit__factor__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% unit_factor_1
thf(fact_5333_inv__unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% inv_unit_factor_eq_0_iff
thf(fact_5334_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B2 @ A3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B2 ) @ A3 ) ) ) ) ).
% unit_factor_mult_unit_right
thf(fact_5335_unit__factor__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A3: A,B2: A] :
( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% unit_factor_mult
thf(fact_5336_is__unit__unit__factor,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ A3 )
= A3 ) ) ) ).
% is_unit_unit_factor
thf(fact_5337_mult__gcd__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( times_times @ A @ C3 @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C3 ) @ ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_gcd_left
thf(fact_5338_mult__gcd__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ C3 )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) @ ( unit_f5069060285200089521factor @ A @ C3 ) ) ) ) ).
% mult_gcd_right
thf(fact_5339_gcd__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K: A,A3: A,B2: A] :
( ( times_times @ A @ K @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ K @ A3 ) @ ( times_times @ A @ K @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% gcd_mult_distrib
thf(fact_5340_unit__factor__is__unit,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_factor_is_unit
thf(fact_5341_unit__factor__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_gcd
thf(fact_5342_unit__factor__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( ( gcd_Gcd @ A @ A5 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A5 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Gcd @ A @ A5 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A5 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Gcd
thf(fact_5343_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_5344_Lcm__no__units,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A )
= ( ^ [A7: set @ A] :
( gcd_Lcm @ A
@ ( minus_minus @ ( set @ A ) @ A7
@ ( collect @ A
@ ^ [A4: A] : ( dvd_dvd @ A @ A4 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% Lcm_no_units
thf(fact_5345_list__all__iff__all__interval__nat,axiom,
! [P: nat > $o,I: nat,J: nat] :
( ( list_all @ nat @ P @ ( upt @ I @ J ) )
= ( all_interval_nat @ P @ I @ J ) ) ).
% list_all_iff_all_interval_nat
thf(fact_5346_normalize__mult__normalize__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% normalize_mult_normalize_left
thf(fact_5347_normalize__mult__normalize__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% normalize_mult_normalize_right
thf(fact_5348_normalize__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% normalize_1
thf(fact_5349_gcd_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.normalize_bottom
thf(fact_5350_Lcm__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_empty
thf(fact_5351_Lcm__1__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( gcd_Lcm @ A @ A5 )
= ( one_one @ A ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% Lcm_1_iff
thf(fact_5352_unit__factor__mult__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ A3 ) )
= A3 ) ) ).
% unit_factor_mult_normalize
thf(fact_5353_normalize__mult__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= A3 ) ) ).
% normalize_mult_unit_factor
thf(fact_5354_normalize__mult__unit__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% normalize_mult_unit_left
thf(fact_5355_normalize__mult__unit__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% normalize_mult_unit_right
thf(fact_5356_unit__factor__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% unit_factor_normalize
thf(fact_5357_normalize__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% normalize_unit_factor
thf(fact_5358_Lcm__singleton,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A] :
( ( gcd_Lcm @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% Lcm_singleton
thf(fact_5359_normalize__div,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ) ).
% normalize_div
thf(fact_5360_Gcd__singleton,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% Gcd_singleton
thf(fact_5361_Lcm__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A,C3: A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( gcd_Lcm @ A @ ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ A5 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_Lcm @ A @ A5 ) ) ) ) ) ) ).
% Lcm_mult
thf(fact_5362_gcd__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% gcd_mult_left
thf(fact_5363_gcd__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_gcd @ A @ B2 @ A3 ) @ C3 ) ) ) ) ).
% gcd_mult_right
thf(fact_5364_gcd__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C3 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ).
% gcd_mult_distrib'
thf(fact_5365_normalize__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% normalize_mult
thf(fact_5366_associated__unit,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% associated_unit
thf(fact_5367_normalize__1__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% normalize_1_iff
thf(fact_5368_is__unit__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% is_unit_normalize
thf(fact_5369_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( A3
= ( one_one @ A ) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
thf(fact_5370_Gcd__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [C3: A,A5: set @ A] :
( ( gcd_Gcd @ A @ ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ A5 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_Gcd @ A @ A5 ) ) ) ) ) ).
% Gcd_mult
thf(fact_5371_unit__factor__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( ( gcd_Lcm @ A @ A5 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A5 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Lcm @ A @ A5 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A5 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Lcm
thf(fact_5372_all__interval__nat__def,axiom,
( all_interval_nat
= ( ^ [P3: nat > $o,I2: nat,J2: nat] :
! [X3: nat] :
( ( member2 @ nat @ X3 @ ( set_or7035219750837199246ssThan @ nat @ I2 @ J2 ) )
=> ( P3 @ X3 ) ) ) ) ).
% all_interval_nat_def
thf(fact_5373_Gcd__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,B2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B2 ) @ A5 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ) ) ).
% Gcd_fin_mult
thf(fact_5374_list__ex__iff__not__all__inverval__nat,axiom,
! [P: nat > $o,I: nat,J: nat] :
( ( list_ex @ nat @ P @ ( upt @ I @ J ) )
= ( ~ ( all_interval_nat @ ( comp @ $o @ $o @ nat @ (~) @ P ) @ I @ J ) ) ) ).
% list_ex_iff_not_all_inverval_nat
thf(fact_5375_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_5376_Lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,B6: A] :
( ( member2 @ A @ A6 @ A5 )
=> ( ( member2 @ A @ B6 @ A5 )
=> ( ( A6 != B6 )
=> ( algebr8660921524188924756oprime @ A @ A6 @ B6 ) ) ) )
=> ( ( gcd_Lcm @ A @ A5 )
= ( normal6383669964737779283malize @ A
@ ( groups7121269368397514597t_prod @ A @ A
@ ^ [X3: A] : X3
@ A5 ) ) ) ) ) ) ) ).
% Lcm_coprime
thf(fact_5377_Lcm__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,B2: A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B2 ) @ A5 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Lcm_fin @ A @ A5 ) ) ) ) ) ) ).
% Lcm_fin_mult
thf(fact_5378_coprime__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C3 @ ( times_times @ A @ A3 @ B2 ) )
= ( ( algebr8660921524188924756oprime @ A @ C3 @ A3 )
& ( algebr8660921524188924756oprime @ A @ C3 @ B2 ) ) ) ) ).
% coprime_mult_right_iff
thf(fact_5379_coprime__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ C3 )
& ( algebr8660921524188924756oprime @ A @ B2 @ C3 ) ) ) ) ).
% coprime_mult_left_iff
thf(fact_5380_coprime__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_self
thf(fact_5381_coprime__imp__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) ) ) ) ).
% coprime_imp_gcd_eq_1
thf(fact_5382_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_5383_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A5 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Lcm_fin @ A @ A5 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Lcm_fin_iff
thf(fact_5384_coprime__0__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( zero_zero @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_right_iff
thf(fact_5385_coprime__0__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_left_iff
thf(fact_5386_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_5387_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_5388_is__unit__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_gcd
thf(fact_5389_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [A3: A,N: A,M2: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ N ) @ M2 )
= ( modulo_modulo @ A @ ( times_times @ A @ B2 @ N ) @ M2 ) )
=> ( ( algebr8660921524188924756oprime @ A @ M2 @ N )
=> ( ( modulo_modulo @ A @ A3 @ M2 )
= ( modulo_modulo @ A @ B2 @ M2 ) ) ) ) ) ).
% mult_mod_cancel_right
thf(fact_5390_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [N: A,A3: A,M2: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ N @ A3 ) @ M2 )
= ( modulo_modulo @ A @ ( times_times @ A @ N @ B2 ) @ M2 ) )
=> ( ( algebr8660921524188924756oprime @ A @ M2 @ N )
=> ( ( modulo_modulo @ A @ A3 @ M2 )
= ( modulo_modulo @ A @ B2 @ M2 ) ) ) ) ) ).
% mult_mod_cancel_left
thf(fact_5391_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_right_imp_coprime
thf(fact_5392_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_left_imp_coprime
thf(fact_5393_coprime__common__divisor,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ).
% coprime_common_divisor
thf(fact_5394_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_5395_coprime__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,D3: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C3 @ D3 )
=> ( ! [E: A] :
( ~ ( dvd_dvd @ A @ E @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E @ A3 )
=> ( ( dvd_dvd @ A @ E @ B2 )
=> ( dvd_dvd @ A @ E @ C3 ) ) ) )
=> ( ! [E: A] :
( ~ ( dvd_dvd @ A @ E @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E @ A3 )
=> ( ( dvd_dvd @ A @ E @ B2 )
=> ( dvd_dvd @ A @ E @ D3 ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% coprime_imp_coprime
thf(fact_5396_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_5397_not__coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ~ ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% not_coprimeI
thf(fact_5398_not__coprimeE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ~ ! [C2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% not_coprimeE
thf(fact_5399_coprime__def,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A4: A,B3: A] :
! [C6: A] :
( ( dvd_dvd @ A @ C6 @ A4 )
=> ( ( dvd_dvd @ A @ C6 @ B3 )
=> ( dvd_dvd @ A @ C6 @ ( one_one @ A ) ) ) ) ) ) ) ).
% coprime_def
thf(fact_5400_coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ! [C2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprimeI
thf(fact_5401_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% coprime_1_left
thf(fact_5402_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_5403_coprime__doff__one__right,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_doff_one_right
thf(fact_5404_coprime__diff__one__left,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% coprime_diff_one_left
thf(fact_5405_coprime__add__one__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_add_one_right
thf(fact_5406_coprime__add__one__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% coprime_add_one_left
thf(fact_5407_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C3 )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% coprime_dvd_mult_right_iff
thf(fact_5408_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C3 )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% coprime_dvd_mult_left_iff
thf(fact_5409_divides__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C3 )
=> ( ( dvd_dvd @ A @ B2 @ C3 )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 ) ) ) ) ) ).
% divides_mult
thf(fact_5410_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C3 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ C3 @ A3 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_left_left_cancel
thf(fact_5411_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C3: A,A3: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C3 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C3 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_left_right_cancel
thf(fact_5412_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C3 )
=> ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ C3 @ B2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_right_left_cancel
thf(fact_5413_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C3 )
=> ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ B2 @ C3 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_right_right_cancel
thf(fact_5414_gcd__eq__1__imp__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% gcd_eq_1_imp_coprime
thf(fact_5415_coprime__iff__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A4: A,B3: A] :
( ( gcd_gcd @ A @ A4 @ B3 )
= ( one_one @ A ) ) ) ) ) ).
% coprime_iff_gcd_eq_1
thf(fact_5416_coprime__crossproduct,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,D3: A,B2: A,C3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ D3 )
=> ( ( algebr8660921524188924756oprime @ A @ B2 @ C3 )
=> ( ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ C3 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ B2 ) @ ( normal6383669964737779283malize @ A @ D3 ) ) )
= ( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
& ( ( normal6383669964737779283malize @ A @ C3 )
= ( normal6383669964737779283malize @ A @ D3 ) ) ) ) ) ) ) ).
% coprime_crossproduct
thf(fact_5417_invertible__coprime,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C3 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ C3 ) ) ) ).
% invertible_coprime
thf(fact_5418_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,A10: A,B9: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
= ( times_times @ A @ A10 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( ( B2
= ( times_times @ A @ B9 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A10 @ B9 ) ) ) ) ) ).
% gcd_coprime
thf(fact_5419_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ? [A18: A,B10: A] :
( ( A3
= ( times_times @ A @ A18 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( B2
= ( times_times @ A @ B10 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( algebr8660921524188924756oprime @ A @ A18 @ B10 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_5420_coprime__crossproduct_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [B2: A,D3: A,A3: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ B2 )
= ( unit_f5069060285200089521factor @ A @ D3 ) )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( algebr8660921524188924756oprime @ A @ C3 @ D3 )
=> ( ( ( times_times @ A @ A3 @ D3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( ( A3 = C3 )
& ( B2 = D3 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
thf(fact_5421_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( ( semiring_gcd_Lcm_fin @ A @ A5 )
= ( one_one @ A ) )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ A5 )
=> ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) )
& ( finite_finite2 @ A @ A5 ) ) ) ) ).
% Lcm_fin_1_iff
thf(fact_5422_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( ^ [A7: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A7 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A7 ) @ ( zero_zero @ A ) ) ) ) ) ).
% Lcm_fin.eq_fold
thf(fact_5423_gcd__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% gcd_lcm
thf(fact_5424_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_5425_lcm__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_lcm @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% lcm_eq_1_iff
thf(fact_5426_lcm_Otop__left__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ ( one_one @ A ) @ A3 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% lcm.top_left_normalize
thf(fact_5427_lcm_Otop__right__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ A3 @ ( one_one @ A ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% lcm.top_right_normalize
thf(fact_5428_unit__factor__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_lcm
thf(fact_5429_gcd__mult__lcm,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% gcd_mult_lcm
thf(fact_5430_lcm__mult__gcd,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% lcm_mult_gcd
thf(fact_5431_Lcm__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_Lcm @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% Lcm_2
thf(fact_5432_lcm__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ).
% lcm_mult_left
thf(fact_5433_lcm__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C3: A,B2: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_lcm @ A @ B2 @ A3 ) @ C3 ) ) ) ) ).
% lcm_mult_right
thf(fact_5434_lcm__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C3 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( gcd_lcm @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ).
% lcm_mult_distrib'
thf(fact_5435_lcm__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K: A,A3: A,B2: A] :
( ( times_times @ A @ K @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ K @ A3 ) @ ( times_times @ A @ K @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% lcm_mult_distrib
thf(fact_5436_mult__lcm__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ C3 )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) @ ( unit_f5069060285200089521factor @ A @ C3 ) ) ) ) ).
% mult_lcm_right
thf(fact_5437_mult__lcm__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( times_times @ A @ C3 @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C3 ) @ ( gcd_lcm @ A @ ( times_times @ A @ C3 @ A3 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_lcm_left
thf(fact_5438_lcm__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( times_times @ A @ B2 @ A3 ) @ C3 )
= ( gcd_lcm @ A @ B2 @ C3 ) ) ) ) ).
% lcm_mult_unit1
thf(fact_5439_lcm__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( times_times @ A @ C3 @ A3 ) )
= ( gcd_lcm @ A @ B2 @ C3 ) ) ) ) ).
% lcm_mult_unit2
thf(fact_5440_lcm__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C3 )
= ( gcd_lcm @ A @ B2 @ C3 ) ) ) ) ).
% lcm_div_unit1
thf(fact_5441_lcm__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( divide_divide @ A @ C3 @ A3 ) )
= ( gcd_lcm @ A @ B2 @ C3 ) ) ) ) ).
% lcm_div_unit2
thf(fact_5442_lcm__gcd__prod,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% lcm_gcd_prod
thf(fact_5443_lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( gcd_lcm @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% lcm_coprime
thf(fact_5444_lcm__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_lcm @ A )
= ( ^ [A4: A,B3: A] : ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ) ) ).
% lcm_gcd
thf(fact_5445_Lcm__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( member2 @ A @ A3 @ A5 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A5 )
= ( gcd_lcm @ A @ A3 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Lcm_fin.remove
thf(fact_5446_Lcm__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( insert2 @ A @ A3 @ A5 ) )
= ( gcd_lcm @ A @ A3 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Lcm_fin.insert_remove
thf(fact_5447_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_5448_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_5449_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_5450_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_5451_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 ) )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ G ) ) ) ) ).
% acyclic_insert_cyclic
thf(fact_5452_FI__RESULT__def,axiom,
( fI_RESULT
= ( ^ [M7: list @ ( product_prod @ assn @ assn ),UP: assn,UQ: assn,F7: assn] :
( ! [X3: product_prod @ assn @ assn] :
( ( member2 @ ( product_prod @ assn @ assn ) @ X3 @ ( set2 @ ( product_prod @ assn @ assn ) @ M7 ) )
=> ( product_case_prod @ assn @ assn @ $o @ entails @ X3 ) )
=> ( entails @ ( times_times @ assn @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_fst @ assn @ assn ) @ M7 ) @ ( one_one @ assn ) ) @ UP ) @ ( times_times @ assn @ ( times_times @ assn @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_snd @ assn @ assn ) @ M7 ) @ ( one_one @ assn ) ) @ UQ ) @ F7 ) ) ) ) ) ).
% FI_RESULT_def
thf(fact_5453_listsp__mono,axiom,
! [A: $tType,A5: A > $o,B4: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A5 @ B4 )
=> ( ord_less_eq @ ( ( list @ A ) > $o ) @ ( listsp @ A @ A5 ) @ ( listsp @ A @ B4 ) ) ) ).
% listsp_mono
thf(fact_5454_listsp__conj__eq,axiom,
! [A: $tType,A5: A > $o,B4: A > $o] :
( ( listsp @ A
@ ^ [X3: A] :
( ( A5 @ X3 )
& ( B4 @ X3 ) ) )
= ( ^ [X3: list @ A] :
( ( listsp @ A @ A5 @ X3 )
& ( listsp @ A @ B4 @ X3 ) ) ) ) ).
% listsp_conj_eq
thf(fact_5455_listsp__simps_I1_J,axiom,
! [A: $tType,A5: A > $o] : ( listsp @ A @ A5 @ ( nil @ A ) ) ).
% listsp_simps(1)
thf(fact_5456_in__listspI,axiom,
! [A: $tType,Xs: list @ A,A5: A > $o] :
( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( A5 @ X2 ) )
=> ( listsp @ A @ A5 @ Xs ) ) ).
% in_listspI
thf(fact_5457_append__in__listsp__conv,axiom,
! [A: $tType,A5: A > $o,Xs: list @ A,Ys: list @ A] :
( ( listsp @ A @ A5 @ ( append @ A @ Xs @ Ys ) )
= ( ( listsp @ A @ A5 @ Xs )
& ( listsp @ A @ A5 @ Ys ) ) ) ).
% append_in_listsp_conv
thf(fact_5458_acyclic__empty,axiom,
! [A: $tType] : ( transitive_acyclic @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% acyclic_empty
thf(fact_5459_listsp__inf__eq,axiom,
! [A: $tType,A5: A > $o,B4: A > $o] :
( ( listsp @ A @ ( inf_inf @ ( A > $o ) @ A5 @ B4 ) )
= ( inf_inf @ ( ( list @ A ) > $o ) @ ( listsp @ A @ A5 ) @ ( listsp @ A @ B4 ) ) ) ).
% listsp_inf_eq
thf(fact_5460_ent__pure__pre__iff,axiom,
! [P: assn,B2: $o,Q: assn] :
( ( entails @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ Q )
= ( B2
=> ( entails @ P @ Q ) ) ) ).
% ent_pure_pre_iff
thf(fact_5461_ent__pure__pre__iff__sng,axiom,
! [B2: $o,Q: assn] :
( ( entails @ ( pure_assn @ B2 ) @ Q )
= ( B2
=> ( entails @ ( one_one @ assn ) @ Q ) ) ) ).
% ent_pure_pre_iff_sng
thf(fact_5462_cyclic__subset,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ~ ( transitive_acyclic @ A @ R )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ~ ( transitive_acyclic @ A @ S ) ) ) ).
% cyclic_subset
thf(fact_5463_lists__def,axiom,
! [A: $tType] :
( ( lists @ A )
= ( ^ [A7: set @ A] :
( collect @ ( list @ A )
@ ( listsp @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ A7 ) ) ) ) ) ).
% lists_def
thf(fact_5464_listsp__lists__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( listsp @ A
@ ^ [X3: A] : ( member2 @ A @ X3 @ A5 ) )
= ( ^ [X3: list @ A] : ( member2 @ ( list @ A ) @ X3 @ ( lists @ A @ A5 ) ) ) ) ).
% listsp_lists_eq
thf(fact_5465_listsp__infI,axiom,
! [A: $tType,A5: A > $o,L: list @ A,B4: A > $o] :
( ( listsp @ A @ A5 @ L )
=> ( ( listsp @ A @ B4 @ L )
=> ( listsp @ A @ ( inf_inf @ ( A > $o ) @ A5 @ B4 ) @ L ) ) ) ).
% listsp_infI
thf(fact_5466_ent__conjI,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( entails @ A5 @ B4 )
=> ( ( entails @ A5 @ C4 )
=> ( entails @ A5 @ ( inf_inf @ assn @ B4 @ C4 ) ) ) ) ).
% ent_conjI
thf(fact_5467_ent__conjE1,axiom,
! [A5: assn,C4: assn,B4: assn] :
( ( entails @ A5 @ C4 )
=> ( entails @ ( inf_inf @ assn @ A5 @ B4 ) @ C4 ) ) ).
% ent_conjE1
thf(fact_5468_ent__conjE2,axiom,
! [B4: assn,C4: assn,A5: assn] :
( ( entails @ B4 @ C4 )
=> ( entails @ ( inf_inf @ assn @ A5 @ B4 ) @ C4 ) ) ).
% ent_conjE2
thf(fact_5469_in__listsp__conv__set,axiom,
! [A: $tType] :
( ( listsp @ A )
= ( ^ [A7: A > $o,Xs3: list @ A] :
! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs3 ) )
=> ( A7 @ X3 ) ) ) ) ).
% in_listsp_conv_set
thf(fact_5470_in__listspD,axiom,
! [A: $tType,A5: A > $o,Xs: list @ A] :
( ( listsp @ A @ A5 @ Xs )
=> ! [X6: A] :
( ( member2 @ A @ X6 @ ( set2 @ A @ Xs ) )
=> ( A5 @ X6 ) ) ) ).
% in_listspD
thf(fact_5471_listspE,axiom,
! [A: $tType,A5: A > $o,X: A,L: list @ A] :
( ( listsp @ A @ A5 @ ( cons @ A @ X @ L ) )
=> ~ ( ( A5 @ X )
=> ~ ( listsp @ A @ A5 @ L ) ) ) ).
% listspE
thf(fact_5472_listsp_OCons,axiom,
! [A: $tType,A5: A > $o,A3: A,L: list @ A] :
( ( A5 @ A3 )
=> ( ( listsp @ A @ A5 @ L )
=> ( listsp @ A @ A5 @ ( cons @ A @ A3 @ L ) ) ) ) ).
% listsp.Cons
thf(fact_5473_listsp__simps_I2_J,axiom,
! [A: $tType,A5: A > $o,X: A,Xs: list @ A] :
( ( listsp @ A @ A5 @ ( cons @ A @ X @ Xs ) )
= ( ( A5 @ X )
& ( listsp @ A @ A5 @ Xs ) ) ) ).
% listsp_simps(2)
thf(fact_5474_listsp_ONil,axiom,
! [A: $tType,A5: A > $o] : ( listsp @ A @ A5 @ ( nil @ A ) ) ).
% listsp.Nil
thf(fact_5475_ent__trans,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ P @ Q )
=> ( ( entails @ Q @ R )
=> ( entails @ P @ R ) ) ) ).
% ent_trans
thf(fact_5476_ent__refl,axiom,
! [P: assn] : ( entails @ P @ P ) ).
% ent_refl
thf(fact_5477_ent__iffI,axiom,
! [A5: assn,B4: assn] :
( ( entails @ A5 @ B4 )
=> ( ( entails @ B4 @ A5 )
=> ( A5 = B4 ) ) ) ).
% ent_iffI
thf(fact_5478_is__entails,axiom,
! [P: assn,Q: assn] :
( ( entails @ P @ Q )
=> ( entails @ P @ Q ) ) ).
% is_entails
thf(fact_5479_ent__star__mono,axiom,
! [P: assn,P7: assn,Q: assn,Q8: assn] :
( ( entails @ P @ P7 )
=> ( ( entails @ Q @ Q8 )
=> ( entails @ ( times_times @ assn @ P @ Q ) @ ( times_times @ assn @ P7 @ Q8 ) ) ) ) ).
% ent_star_mono
thf(fact_5480_ent__frame__fwd,axiom,
! [P: assn,R: assn,Ps3: assn,F2: assn,Q: assn] :
( ( entails @ P @ R )
=> ( ( entails @ Ps3 @ ( times_times @ assn @ P @ F2 ) )
=> ( ( entails @ ( times_times @ assn @ R @ F2 ) @ Q )
=> ( entails @ Ps3 @ Q ) ) ) ) ).
% ent_frame_fwd
thf(fact_5481_fr__rot__rhs,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( entails @ A5 @ ( times_times @ assn @ B4 @ C4 ) )
=> ( entails @ A5 @ ( times_times @ assn @ C4 @ B4 ) ) ) ).
% fr_rot_rhs
thf(fact_5482_fr__refl,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( entails @ A5 @ B4 )
=> ( entails @ ( times_times @ assn @ A5 @ C4 ) @ ( times_times @ assn @ B4 @ C4 ) ) ) ).
% fr_refl
thf(fact_5483_fr__rot,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( entails @ ( times_times @ assn @ A5 @ B4 ) @ C4 )
=> ( entails @ ( times_times @ assn @ B4 @ A5 ) @ C4 ) ) ).
% fr_rot
thf(fact_5484_ent__true,axiom,
! [P: assn] : ( entails @ P @ ( top_top @ assn ) ) ).
% ent_true
thf(fact_5485_acyclic__union_I1_J,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B4 ) )
=> ( transitive_acyclic @ A @ A5 ) ) ).
% acyclic_union(1)
thf(fact_5486_acyclic__union_I2_J,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B4 ) )
=> ( transitive_acyclic @ A @ B4 ) ) ).
% acyclic_union(2)
thf(fact_5487_ent__disjI2__direct,axiom,
! [B4: assn,A5: assn] : ( entails @ B4 @ ( sup_sup @ assn @ A5 @ B4 ) ) ).
% ent_disjI2_direct
thf(fact_5488_ent__disjI1__direct,axiom,
! [A5: assn,B4: assn] : ( entails @ A5 @ ( sup_sup @ assn @ A5 @ B4 ) ) ).
% ent_disjI1_direct
thf(fact_5489_ent__disjI2_H,axiom,
! [A5: assn,C4: assn,B4: assn] :
( ( entails @ A5 @ C4 )
=> ( entails @ A5 @ ( sup_sup @ assn @ B4 @ C4 ) ) ) ).
% ent_disjI2'
thf(fact_5490_ent__disjI1_H,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( entails @ A5 @ B4 )
=> ( entails @ A5 @ ( sup_sup @ assn @ B4 @ C4 ) ) ) ).
% ent_disjI1'
thf(fact_5491_ent__disjI2,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ ( sup_sup @ assn @ P @ Q ) @ R )
=> ( entails @ Q @ R ) ) ).
% ent_disjI2
thf(fact_5492_ent__disjI1,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ ( sup_sup @ assn @ P @ Q ) @ R )
=> ( entails @ P @ R ) ) ).
% ent_disjI1
thf(fact_5493_ent__disjE,axiom,
! [A5: assn,C4: assn,B4: assn] :
( ( entails @ A5 @ C4 )
=> ( ( entails @ B4 @ C4 )
=> ( entails @ ( sup_sup @ assn @ A5 @ B4 ) @ C4 ) ) ) ).
% ent_disjE
thf(fact_5494_ent__false,axiom,
! [P: assn] : ( entails @ ( bot_bot @ assn ) @ P ) ).
% ent_false
thf(fact_5495_ent__star__mono__true,axiom,
! [A5: assn,A13: assn,B4: assn,B15: assn] :
( ( entails @ A5 @ ( times_times @ assn @ A13 @ ( top_top @ assn ) ) )
=> ( ( entails @ B4 @ ( times_times @ assn @ B15 @ ( top_top @ assn ) ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ A5 @ B4 ) @ ( top_top @ assn ) ) @ ( times_times @ assn @ ( times_times @ assn @ A13 @ B15 ) @ ( top_top @ assn ) ) ) ) ) ).
% ent_star_mono_true
thf(fact_5496_ent__refl__true,axiom,
! [A5: assn] : ( entails @ A5 @ ( times_times @ assn @ A5 @ ( top_top @ assn ) ) ) ).
% ent_refl_true
thf(fact_5497_ent__true__drop_I1_J,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ P @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) )
=> ( entails @ ( times_times @ assn @ P @ R ) @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) ) ) ).
% ent_true_drop(1)
thf(fact_5498_ent__true__drop_I2_J,axiom,
! [P: assn,Q: assn] :
( ( entails @ P @ Q )
=> ( entails @ P @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) ) ) ).
% ent_true_drop(2)
thf(fact_5499_listsp_Ocases,axiom,
! [A: $tType,A5: A > $o,A3: list @ A] :
( ( listsp @ A @ A5 @ A3 )
=> ( ( A3
!= ( nil @ A ) )
=> ~ ! [A6: A,L2: list @ A] :
( ( A3
= ( cons @ A @ A6 @ L2 ) )
=> ( ( A5 @ A6 )
=> ~ ( listsp @ A @ A5 @ L2 ) ) ) ) ) ).
% listsp.cases
thf(fact_5500_listsp_Osimps,axiom,
! [A: $tType] :
( ( listsp @ A )
= ( ^ [A7: A > $o,A4: list @ A] :
( ( A4
= ( nil @ A ) )
| ? [B3: A,L4: list @ A] :
( ( A4
= ( cons @ A @ B3 @ L4 ) )
& ( A7 @ B3 )
& ( listsp @ A @ A7 @ L4 ) ) ) ) ) ).
% listsp.simps
thf(fact_5501_fi__match__entails,axiom,
! [M2: list @ ( product_prod @ assn @ assn )] :
( ! [X2: product_prod @ assn @ assn] :
( ( member2 @ ( product_prod @ assn @ assn ) @ X2 @ ( set2 @ ( product_prod @ assn @ assn ) @ M2 ) )
=> ( product_case_prod @ assn @ assn @ $o @ entails @ X2 ) )
=> ( entails @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_fst @ assn @ assn ) @ M2 ) @ ( one_one @ assn ) ) @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_snd @ assn @ assn ) @ M2 ) @ ( one_one @ assn ) ) ) ) ).
% fi_match_entails
thf(fact_5502_FI__QUERY__def,axiom,
( fI_QUERY
= ( ^ [P3: assn,Q2: assn,F7: assn] : ( entails @ P3 @ ( times_times @ assn @ Q2 @ F7 ) ) ) ) ).
% FI_QUERY_def
thf(fact_5503_frame__inference__init,axiom,
! [P: assn,Q: assn,F2: assn] :
( ( fI_QUERY @ P @ Q @ F2 )
=> ( entails @ P @ ( times_times @ assn @ Q @ F2 ) ) ) ).
% frame_inference_init
thf(fact_5504_cyclicE,axiom,
! [A: $tType,G: set @ ( product_prod @ A @ A )] :
( ~ ( transitive_acyclic @ A @ G )
=> ~ ! [X2: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ G ) ) ) ).
% cyclicE
thf(fact_5505_entails__solve__init_I2_J,axiom,
! [P: assn,Q: assn] :
( ( fI_QUERY @ P @ Q @ ( one_one @ assn ) )
=> ( entails @ P @ Q ) ) ).
% entails_solve_init(2)
thf(fact_5506_wf__set,axiom,
! [A: $tType,Xs: list @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) )
= ( transitive_acyclic @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) ) ).
% wf_set
thf(fact_5507_entails__solve__init_I1_J,axiom,
! [P: assn,Q: assn] :
( ( fI_QUERY @ P @ Q @ ( top_top @ assn ) )
=> ( entails @ P @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) ) ) ).
% entails_solve_init(1)
thf(fact_5508_FI__match,axiom,
! [P4: assn,Q3: assn,M2: list @ ( product_prod @ assn @ assn ),Ps: assn,Up: assn,Qs: assn,Uq: assn,F: assn] :
( ( entails @ P4 @ Q3 )
=> ( ( fi @ ( cons @ ( product_prod @ assn @ assn ) @ ( product_Pair @ assn @ assn @ P4 @ Q3 ) @ M2 ) @ ( times_times @ assn @ Ps @ Up ) @ ( times_times @ assn @ Qs @ Uq ) @ sln @ sln @ F )
=> ( fi @ M2 @ ( times_times @ assn @ Ps @ P4 ) @ ( times_times @ assn @ Qs @ Q3 ) @ Up @ Uq @ F ) ) ) ).
% FI_match
thf(fact_5509_FI__def,axiom,
( fi
= ( ^ [M3: list @ ( product_prod @ assn @ assn ),P6: assn,Q5: assn,Up2: assn,Uq2: assn,F3: assn] :
( ! [X3: product_prod @ assn @ assn] :
( ( member2 @ ( product_prod @ assn @ assn ) @ X3 @ ( set2 @ ( product_prod @ assn @ assn ) @ M3 ) )
=> ( product_case_prod @ assn @ assn @ $o @ entails @ X3 ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_fst @ assn @ assn ) @ M3 ) @ ( one_one @ assn ) ) @ P6 ) @ Up2 ) @ ( times_times @ assn @ ( times_times @ assn @ ( times_times @ assn @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_snd @ assn @ assn ) @ M3 ) @ ( one_one @ assn ) ) @ Q5 ) @ Uq2 ) @ F3 ) ) ) ) ) ).
% FI_def
thf(fact_5510_ent__wand__frameI,axiom,
! [Q: assn,R: assn,F2: assn,S: assn,P: assn,X5: assn] :
( ( entails @ ( times_times @ assn @ ( wand_assn @ Q @ R ) @ F2 ) @ S )
=> ( ( entails @ P @ ( times_times @ assn @ F2 @ X5 ) )
=> ( ( entails @ ( times_times @ assn @ Q @ X5 ) @ R )
=> ( entails @ P @ S ) ) ) ) ).
% ent_wand_frameI
thf(fact_5511_ent__wandI,axiom,
! [Q: assn,P: assn,R: assn] :
( ( entails @ ( times_times @ assn @ Q @ P ) @ R )
=> ( entails @ P @ ( wand_assn @ Q @ R ) ) ) ).
% ent_wandI
thf(fact_5512_ent__mp,axiom,
! [P: assn,Q: assn] : ( entails @ ( times_times @ assn @ P @ ( wand_assn @ P @ Q ) ) @ Q ) ).
% ent_mp
thf(fact_5513_prec__frame,axiom,
! [B: $tType,A: $tType,P: A > B > assn,R12: assn,R22: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X: A,P4: B,F13: assn,Y: A,F24: assn] :
( ( precise @ A @ B @ P )
=> ( ( rep_assn @ ( inf_inf @ assn @ R12 @ R22 ) @ H )
=> ( ( entails @ R12 @ ( times_times @ assn @ ( P @ X @ P4 ) @ F13 ) )
=> ( ( entails @ R22 @ ( times_times @ assn @ ( P @ Y @ P4 ) @ F24 ) )
=> ( X = Y ) ) ) ) ) ).
% prec_frame
thf(fact_5514_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_5515_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_5516_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( ( rep_assn @ X )
= ( rep_assn @ Y ) )
= ( X = Y ) ) ).
% Rep_assn_inject
thf(fact_5517_mod__or__dist,axiom,
! [P: assn,Q: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( sup_sup @ assn @ P @ Q ) @ H )
= ( ( rep_assn @ P @ H )
| ( rep_assn @ Q @ H ) ) ) ).
% mod_or_dist
thf(fact_5518_mod__h__bot__iff_I5_J,axiom,
! [P: assn,Q: assn,H: heap_ext @ product_unit] :
( ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(5)
thf(fact_5519_mod__pure__star__dist,axiom,
! [P: assn,B2: $o,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ H )
= ( ( rep_assn @ P @ H )
& B2 ) ) ).
% mod_pure_star_dist
thf(fact_5520_mod__h__bot__normalize,axiom,
! [A: $tType,H: heap_ext @ product_unit,P: assn] :
( ( syntax7388354845996824322omatch @ A @ ( heap_ext @ product_unit ) @ ( undefined @ A ) @ H )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( 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_5521_mod__h__bot__iff_I7_J,axiom,
! [P: assn,Q: assn,H: heap_ext @ product_unit] :
( ( rep_assn @ ( sup_sup @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
| ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(7)
thf(fact_5522_mod__h__bot__iff_I6_J,axiom,
! [P: assn,Q: assn,H: heap_ext @ product_unit] :
( ( rep_assn @ ( inf_inf @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(6)
thf(fact_5523_mod__pure,axiom,
! [B2: $o,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( pure_assn @ B2 ) @ H )
= ( ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H )
= ( bot_bot @ ( set @ nat ) ) )
& B2 ) ) ).
% mod_pure
thf(fact_5524_mod__h__bot__iff_I1_J,axiom,
! [B2: $o,H: heap_ext @ product_unit] :
( ( rep_assn @ ( pure_assn @ B2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= B2 ) ).
% mod_h_bot_iff(1)
thf(fact_5525_ent__false__iff,axiom,
! [P: assn] :
( ( entails @ P @ ( bot_bot @ assn ) )
= ( ! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H2 ) ) ) ).
% ent_false_iff
thf(fact_5526_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( ( heap @ B )
=> ! [Q3: array @ B,Y: list @ B,H: heap_ext @ product_unit] :
~ ( rep_assn @ ( snga_assn @ B @ Q3 @ Y ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(4)
thf(fact_5527_mod__h__bot__iff_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: ref @ A,X: A,H: heap_ext @ product_unit] :
~ ( rep_assn @ ( sngr_assn @ A @ P4 @ X ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(3)
thf(fact_5528_ent__pure__post__iff,axiom,
! [P: assn,Q: assn,B2: $o] :
( ( entails @ P @ ( times_times @ assn @ Q @ ( pure_assn @ B2 ) ) )
= ( ! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H2 )
=> B2 )
& ( entails @ P @ Q ) ) ) ).
% ent_pure_post_iff
thf(fact_5529_ent__pure__post__iff__sng,axiom,
! [P: assn,B2: $o] :
( ( entails @ P @ ( pure_assn @ B2 ) )
= ( ! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H2 )
=> B2 )
& ( entails @ P @ ( one_one @ assn ) ) ) ) ).
% ent_pure_post_iff_sng
thf(fact_5530_mod__h__bot__indep,axiom,
! [P: assn,H: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_indep
thf(fact_5531_mod__and__dist,axiom,
! [P: assn,Q: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( inf_inf @ assn @ P @ Q ) @ H )
= ( ( rep_assn @ P @ H )
& ( rep_assn @ Q @ H ) ) ) ).
% mod_and_dist
thf(fact_5532_mod__starD,axiom,
! [A5: assn,B4: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A5 @ B4 ) @ H )
=> ? [H1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),H22: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ A5 @ H1 )
& ( rep_assn @ B4 @ H22 ) ) ) ).
% mod_starD
thf(fact_5533_mod__starE,axiom,
! [A3: assn,B2: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A3 @ B2 ) @ H )
=> ~ ( ? [X_1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : ( rep_assn @ A3 @ X_1 )
=> ! [H_2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ B2 @ H_2 ) ) ) ).
% mod_starE
thf(fact_5534_mod__false,axiom,
! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ ( bot_bot @ assn ) @ H ) ).
% mod_false
thf(fact_5535_ent__fwd,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Q: assn] :
( ( rep_assn @ P @ H )
=> ( ( entails @ P @ Q )
=> ( rep_assn @ Q @ H ) ) ) ).
% ent_fwd
thf(fact_5536_entailsD,axiom,
! [P: assn,Q: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( entails @ P @ Q )
=> ( ( rep_assn @ P @ H )
=> ( rep_assn @ Q @ H ) ) ) ).
% entailsD
thf(fact_5537_entailsI,axiom,
! [P: assn,Q: assn] :
( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H4 )
=> ( rep_assn @ Q @ H4 ) )
=> ( entails @ P @ Q ) ) ).
% entailsI
thf(fact_5538_entails__def,axiom,
( entails
= ( ^ [P3: assn,Q2: assn] :
! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P3 @ H2 )
=> ( rep_assn @ Q2 @ H2 ) ) ) ) ).
% entails_def
thf(fact_5539_mod__frame__fwd,axiom,
! [Ps3: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P: assn,R: assn,F2: assn] :
( ( rep_assn @ Ps3 @ H )
=> ( ( entails @ P @ R )
=> ( ( entails @ Ps3 @ ( times_times @ assn @ P @ F2 ) )
=> ( rep_assn @ ( times_times @ assn @ R @ F2 ) @ H ) ) ) ) ).
% mod_frame_fwd
thf(fact_5540_star__assnI,axiom,
! [P: assn,H: heap_ext @ product_unit,As2: set @ nat,Q: assn,As5: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
=> ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As5 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( sup_sup @ ( set @ nat ) @ As2 @ As5 ) ) ) ) ) ) ).
% star_assnI
thf(fact_5541_mod__star__conv,axiom,
! [A5: assn,B4: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A5 @ B4 ) @ H )
= ( ? [Hr: heap_ext @ product_unit,As1: set @ nat,As22: set @ nat] :
( ( H
= ( 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 @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As1 ) )
& ( rep_assn @ B4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As22 ) ) ) ) ) ).
% mod_star_conv
thf(fact_5542_mod__star__trueE_H,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H )
=> ~ ! [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( ( product_fst @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 )
= ( product_fst @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H ) )
=> ( ( ord_less_eq @ ( set @ nat ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H ) )
=> ~ ( rep_assn @ P @ H6 ) ) ) ) ).
% mod_star_trueE'
thf(fact_5543_mod__star__trueI,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H ) ) ).
% mod_star_trueI
thf(fact_5544_mod__star__trueE,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H )
=> ~ ! [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H6 ) ) ).
% mod_star_trueE
thf(fact_5545_mod__h__bot__iff_I2_J,axiom,
! [H: heap_ext @ product_unit] : ( rep_assn @ ( top_top @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_h_bot_iff(2)
thf(fact_5546_mod__emp,axiom,
! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( one_one @ assn ) @ H )
= ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H )
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_emp
thf(fact_5547_mod__emp__simp,axiom,
! [H: heap_ext @ product_unit] : ( rep_assn @ ( one_one @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_emp_simp
thf(fact_5548_preciseD_H,axiom,
! [B: $tType,A: $tType,R: A > B > assn,A3: A,P4: B,F2: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),A10: A,F8: assn] :
( ( precise @ A @ B @ R )
=> ( ( rep_assn @ ( times_times @ assn @ ( R @ A3 @ P4 ) @ F2 ) @ H )
=> ( ( rep_assn @ ( times_times @ assn @ ( R @ A10 @ P4 ) @ F8 ) @ H )
=> ( A3 = A10 ) ) ) ) ).
% preciseD'
thf(fact_5549_prec__frame_H,axiom,
! [A: $tType,P: A > assn,X: A,F13: assn,Y: A,F24: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),R12: assn,R22: assn] :
( ( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( P @ X ) @ F13 ) @ ( times_times @ assn @ ( P @ Y ) @ F24 ) ) @ H )
=> ( X = Y ) )
=> ( ( rep_assn @ ( inf_inf @ assn @ R12 @ R22 ) @ H )
=> ( ( entails @ R12 @ ( times_times @ assn @ ( P @ X ) @ F13 ) )
=> ( ( entails @ R22 @ ( times_times @ assn @ ( P @ Y ) @ F24 ) )
=> ( X = Y ) ) ) ) ) ).
% prec_frame'
thf(fact_5550_prec__frame__expl,axiom,
! [A: $tType,P: A > assn,F13: assn,F24: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),R12: assn,R22: assn,X: A,Y: A] :
( ! [X2: A,Y2: A] :
( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( P @ X2 ) @ F13 ) @ ( times_times @ assn @ ( P @ Y2 ) @ F24 ) ) @ H )
=> ( X2 = Y2 ) )
=> ( ( rep_assn @ ( inf_inf @ assn @ R12 @ R22 ) @ H )
=> ( ( entails @ R12 @ ( times_times @ assn @ ( P @ X ) @ F13 ) )
=> ( ( entails @ R22 @ ( times_times @ assn @ ( P @ Y ) @ F24 ) )
=> ( X = Y ) ) ) ) ) ).
% prec_frame_expl
thf(fact_5551_preciseD,axiom,
! [B: $tType,A: $tType,R: A > B > assn,A3: A,P4: B,F2: assn,A10: A,F8: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( precise @ A @ B @ R )
=> ( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R @ A3 @ P4 ) @ F2 ) @ ( times_times @ assn @ ( R @ A10 @ P4 ) @ F8 ) ) @ H )
=> ( A3 = A10 ) ) ) ).
% preciseD
thf(fact_5552_preciseI,axiom,
! [B: $tType,A: $tType,R: A > B > assn] :
( ! [A6: A,A18: A,H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P8: B,F4: assn,F9: assn] :
( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R @ A6 @ P8 ) @ F4 ) @ ( times_times @ assn @ ( R @ A18 @ P8 ) @ F9 ) ) @ H4 )
=> ( A6 = A18 ) )
=> ( precise @ A @ B @ R ) ) ).
% preciseI
thf(fact_5553_precise__def,axiom,
! [B: $tType,A: $tType] :
( ( precise @ A @ B )
= ( ^ [R6: A > B > assn] :
! [A4: A,A19: A,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P6: B,F7: assn,F10: assn] :
( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R6 @ A4 @ P6 ) @ F7 ) @ ( times_times @ assn @ ( R6 @ A19 @ P6 ) @ F10 ) ) @ H2 )
=> ( A4 = A19 ) ) ) ) ).
% precise_def
thf(fact_5554_comm__monoid__list__set_Oset__conv__list,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,G: B > A,Xs: list @ B] :
( ( groups4802862169904069756st_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( set2 @ B @ Xs ) )
= ( groups_monoid_F @ A @ F @ Z2 @ ( map @ B @ A @ G @ ( remdups @ B @ Xs ) ) ) ) ) ).
% comm_monoid_list_set.set_conv_list
thf(fact_5555_comm__monoid__list__set_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,Xs: list @ B,G: B > A] :
( ( groups4802862169904069756st_set @ A @ F @ Z2 )
=> ( ( distinct @ B @ Xs )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( set2 @ B @ Xs ) )
= ( groups_monoid_F @ A @ F @ Z2 @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% comm_monoid_list_set.distinct_set_conv_list
thf(fact_5556_mod__not__dist,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( uminus_uminus @ assn @ P ) @ H )
= ( ( in_range @ H )
& ~ ( rep_assn @ P @ H ) ) ) ).
% mod_not_dist
thf(fact_5557_in__range__empty,axiom,
! [H: heap_ext @ product_unit] : ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% in_range_empty
thf(fact_5558_mod__true,axiom,
! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( top_top @ assn ) @ H )
= ( in_range @ H ) ) ).
% mod_true
thf(fact_5559_in__range__dist__union,axiom,
! [H: heap_ext @ product_unit,As2: set @ nat,As5: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( sup_sup @ ( set @ nat ) @ As2 @ As5 ) ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) ) ) ) ).
% in_range_dist_union
thf(fact_5560_models__in__range,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> ( in_range @ H ) ) ).
% models_in_range
thf(fact_5561_in__range__subset,axiom,
! [As2: set @ nat,As5: set @ nat,H: heap_ext @ product_unit] :
( ( ord_less_eq @ ( set @ nat ) @ As2 @ As5 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) ) ).
% in_range_subset
thf(fact_5562_one__assn__raw_Ocases,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ).
% one_assn_raw.cases
thf(fact_5563_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 ) ) )] :
~ ! [R7: array @ A,X2: list @ A,H4: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R7 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ) ) ).
% snga_assn_raw.cases
thf(fact_5564_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 ) ) )] :
~ ! [R7: ref @ A,X2: A,H4: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R7 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ) ) ).
% sngr_assn_raw.cases
thf(fact_5565_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 ) ) )] :
~ ! [P2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q9: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H4: heap_ext @ product_unit,As: 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 ) ) ) @ P2 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Q9 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ) ).
% times_assn_raw.cases
thf(fact_5566_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_5567_wand__assnI,axiom,
! [H: heap_ext @ product_unit,As2: set @ nat,Q: assn,R: assn] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ! [H6: heap_ext @ product_unit,As6: set @ nat] :
( ( ( inf_inf @ ( set @ nat ) @ As2 @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( relH @ As2 @ H @ H6 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As2 ) )
=> ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As6 ) )
=> ( rep_assn @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As2 @ As6 ) ) ) ) ) ) )
=> ( rep_assn @ ( wand_assn @ Q @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) ) ).
% wand_assnI
thf(fact_5568_times__assn__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H: heap_ext @ product_unit,As2: set @ nat] :
( ( times_assn_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
= ( ? [As1: set @ nat,As22: set @ nat] :
( ( As2
= ( 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 ) @ H @ As1 ) )
& ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As22 ) ) ) ) ) ).
% times_assn_raw.simps
thf(fact_5569_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( Y
= ( ~ ? [As1: set @ nat,As22: set @ nat] :
( ( As
= ( 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 ) @ H4 @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As22 ) ) ) ) ) ) ) ).
% times_assn_raw.elims(1)
thf(fact_5570_relH__dist__union,axiom,
! [As2: set @ nat,As5: set @ nat,H: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( relH @ ( sup_sup @ ( set @ nat ) @ As2 @ As5 ) @ H @ H5 )
= ( ( relH @ As2 @ H @ H5 )
& ( relH @ As5 @ H @ H5 ) ) ) ).
% relH_dist_union
thf(fact_5571_relH__subset,axiom,
! [Bs: set @ nat,H: heap_ext @ product_unit,H5: heap_ext @ product_unit,As2: set @ nat] :
( ( relH @ Bs @ H @ H5 )
=> ( ( ord_less_eq @ ( set @ nat ) @ As2 @ Bs )
=> ( relH @ As2 @ H @ H5 ) ) ) ).
% relH_subset
thf(fact_5572_relH__trans,axiom,
! [As2: set @ nat,H12: heap_ext @ product_unit,H23: heap_ext @ product_unit,H32: heap_ext @ product_unit] :
( ( relH @ As2 @ H12 @ H23 )
=> ( ( relH @ As2 @ H23 @ H32 )
=> ( relH @ As2 @ H12 @ H32 ) ) ) ).
% relH_trans
thf(fact_5573_relH__sym,axiom,
! [As2: set @ nat,H: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( relH @ As2 @ H @ H5 )
=> ( relH @ As2 @ H5 @ H ) ) ).
% relH_sym
thf(fact_5574_mod__relH,axiom,
! [As2: set @ nat,H: heap_ext @ product_unit,H5: heap_ext @ product_unit,P: assn] :
( ( relH @ As2 @ H @ H5 )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As2 ) ) ) ) ).
% mod_relH
thf(fact_5575_relH__refl,axiom,
! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( relH @ As2 @ H @ H ) ) ).
% relH_refl
thf(fact_5576_relH__in__rangeI_I1_J,axiom,
! [As2: set @ nat,H: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( relH @ As2 @ H @ H5 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) ).
% relH_in_rangeI(1)
thf(fact_5577_relH__in__rangeI_I2_J,axiom,
! [As2: set @ nat,H: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( relH @ As2 @ H @ H5 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As2 ) ) ) ).
% relH_in_rangeI(2)
thf(fact_5578_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ? [As12: set @ nat,As23: set @ nat] :
( ( As
= ( 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 ) @ H4 @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As23 ) ) ) ) ) ).
% times_assn_raw.elims(3)
thf(fact_5579_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ~ ? [As13: set @ nat,As24: set @ nat] :
( ( As
= ( 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 ) @ H4 @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As24 ) ) ) ) ) ).
% times_assn_raw.elims(2)
thf(fact_5580_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( 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 ) @ H4 @ As ) ) ) )
=> ? [As12: set @ nat,As23: set @ nat] :
( ( As
= ( 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 ) @ H4 @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As23 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
thf(fact_5581_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( 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 ) @ H4 @ As ) ) ) )
=> ~ ? [As13: set @ nat,As24: set @ nat] :
( ( As
= ( 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 ) @ H4 @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As24 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
thf(fact_5582_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( Y
= ( ? [As1: set @ nat,As22: set @ nat] :
( ( As
= ( 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 ) @ H4 @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ 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 ) @ H4 @ As ) ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(1)
thf(fact_5583_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
& ! [H6: heap_ext @ product_unit,As6: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H4 @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As6 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As @ As6 ) ) ) ) ) ) ) ).
% wand_raw.elims(3)
thf(fact_5584_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
& ! [H7: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H4 @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As ) )
& ( 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 ) @ As @ As7 ) ) ) ) ) ) ) ).
% wand_raw.elims(2)
thf(fact_5585_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( Y
= ( ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
& ! [H8: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H4 @ H8 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As8 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ ( sup_sup @ ( set @ nat ) @ As @ As8 ) ) ) ) ) ) ) ) ) ).
% wand_raw.elims(1)
thf(fact_5586_wand__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H: heap_ext @ product_unit,As2: set @ nat] :
( ( wand_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
& ! [H8: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As2 @ H @ H8 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As2 ) )
& ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As8 ) ) )
=> ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ ( sup_sup @ ( set @ nat ) @ As2 @ As8 ) ) ) ) ) ) ).
% wand_raw.simps
thf(fact_5587_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( 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 ) @ H4 @ As ) ) ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
& ! [H6: heap_ext @ product_unit,As6: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H4 @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As6 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As @ As6 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
thf(fact_5588_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( 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 ) @ H4 @ As ) ) ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
& ! [H7: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H4 @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As ) )
& ( 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 ) @ As @ As7 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
thf(fact_5589_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( Y
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
& ! [H8: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H4 @ H8 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As8 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ ( sup_sup @ ( set @ nat ) @ As @ As8 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(1)
thf(fact_5590_uminus__assn__def,axiom,
( ( uminus_uminus @ assn )
= ( ^ [P3: assn] :
( abs_assn
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ H2 )
& ~ ( rep_assn @ P3 @ H2 ) ) ) ) ) ).
% uminus_assn_def
thf(fact_5591_comm__monoid__set_OUnion__disjoint,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,C4: set @ ( set @ B ),G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ! [X2: set @ B] :
( ( member2 @ ( set @ B ) @ X2 @ C4 )
=> ( finite_finite2 @ B @ X2 ) )
=> ( ! [X2: set @ B] :
( ( member2 @ ( set @ B ) @ X2 @ C4 )
=> ! [Xa3: set @ B] :
( ( member2 @ ( set @ B ) @ Xa3 @ C4 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X2 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups_comm_monoid_F @ A @ ( set @ B ) @ F @ Z2 ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 ) @ G @ C4 ) ) ) ) ) ).
% comm_monoid_set.Union_disjoint
thf(fact_5592_comm__monoid__set_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > A > A,Z2: A,I5: set @ B,A5: B > ( set @ C ),G: C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ I5 )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ I5 )
=> ( finite_finite2 @ C @ ( A5 @ X2 ) ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ I5 )
=> ! [Xa3: B] :
( ( member2 @ B @ Xa3 @ I5 )
=> ( ( X2 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X2 ) @ ( A5 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z2 @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z2
@ ^ [X3: B] : ( groups_comm_monoid_F @ A @ C @ F @ Z2 @ G @ ( A5 @ X3 ) )
@ I5 ) ) ) ) ) ) ).
% comm_monoid_set.UNION_disjoint
thf(fact_5593_Rep__assn__inverse,axiom,
! [X: assn] :
( ( abs_assn @ ( rep_assn @ X ) )
= X ) ).
% Rep_assn_inverse
thf(fact_5594_comm__monoid__set_Oempty,axiom,
! [B: $tType,A: $tType,F: A > A > A,Z2: A,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ) ).
% comm_monoid_set.empty
thf(fact_5595_pure__assn__def,axiom,
( pure_assn
= ( ^ [B3: $o] : ( abs_assn @ ( pure_assn_raw @ ( heap_ext @ product_unit ) @ nat @ B3 ) ) ) ) ).
% pure_assn_def
thf(fact_5596_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_5597_comm__monoid__set_Oempty_H,axiom,
! [B: $tType,A: $tType,F: A > A > A,Z2: A,P4: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_G @ A @ B @ F @ Z2 @ P4 @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ) ).
% comm_monoid_set.empty'
thf(fact_5598_Abs__assn__eqI_I2_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H4 )
= ( rep_assn @ Pr @ H4 ) )
=> ( Pr
= ( abs_assn @ P ) ) ) ).
% Abs_assn_eqI(2)
thf(fact_5599_Abs__assn__eqI_I1_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H4 )
= ( rep_assn @ Pr @ H4 ) )
=> ( ( abs_assn @ P )
= Pr ) ) ).
% Abs_assn_eqI(1)
thf(fact_5600_bot__assn__def,axiom,
( ( bot_bot @ assn )
= ( abs_assn
@ ^ [Uu3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : $false ) ) ).
% bot_assn_def
thf(fact_5601_comm__monoid__set_Ointer__restrict,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,A5: set @ B,G: B > A,B4: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z2
@ ^ [X3: B] : ( if @ A @ ( member2 @ B @ X3 @ B4 ) @ ( G @ X3 ) @ Z2 )
@ A5 ) ) ) ) ).
% comm_monoid_set.inter_restrict
thf(fact_5602_top__assn__def,axiom,
( ( top_top @ assn )
= ( abs_assn @ in_range ) ) ).
% top_assn_def
thf(fact_5603_sup__assn__def,axiom,
( ( sup_sup @ assn )
= ( ^ [P3: assn,Q2: assn] :
( abs_assn
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P3 @ H2 )
| ( rep_assn @ Q2 @ H2 ) ) ) ) ) ).
% sup_assn_def
thf(fact_5604_inf__assn__def,axiom,
( ( inf_inf @ assn )
= ( ^ [P3: assn,Q2: assn] :
( abs_assn
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P3 @ H2 )
& ( rep_assn @ Q2 @ H2 ) ) ) ) ) ).
% inf_assn_def
thf(fact_5605_comm__monoid__set_Ounion__inter,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,A5: set @ B,B4: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ A5 ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ B4 ) ) ) ) ) ) ).
% comm_monoid_set.union_inter
thf(fact_5606_comm__monoid__set_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,A5: set @ B,B4: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
=> ( ( G @ X2 )
= Z2 ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ A5 ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ B4 ) ) ) ) ) ) ) ).
% comm_monoid_set.union_inter_neutral
thf(fact_5607_comm__monoid__set_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,T: set @ B,S: set @ B,H: B > A,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ T )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T @ S ) )
=> ( ( H @ I3 )
= Z2 ) )
=> ( ! [I3: B] :
( ( member2 @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S @ T ) )
=> ( ( G @ I3 )
= Z2 ) )
=> ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( inf_inf @ ( set @ B ) @ S @ T ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ S )
= ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ H @ T ) ) ) ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_cong
thf(fact_5608_comm__monoid__set_OInt__Diff,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,A5: set @ B,G: B > A,B4: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ A5 )
= ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ).
% comm_monoid_set.Int_Diff
thf(fact_5609_comm__monoid__set_OIf__cases,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,A5: set @ B,P: B > $o,H: B > A,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( H @ X3 ) @ ( G @ X3 ) )
@ A5 )
= ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ H @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% comm_monoid_set.If_cases
thf(fact_5610_comm__monoid__set_Oinsert__remove,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,A5: set @ B,G: B > A,X: B] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( insert2 @ B @ X @ A5 ) )
= ( F @ ( G @ X ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.insert_remove
thf(fact_5611_comm__monoid__set_Oremove,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,A5: set @ B,X: B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( member2 @ B @ X @ A5 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ A5 )
= ( F @ ( G @ X ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.remove
thf(fact_5612_comm__monoid__set_Ounion__disjoint,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,A5: set @ B,B4: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( ( inf_inf @ ( set @ B ) @ A5 @ B4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ A5 ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ B4 ) ) ) ) ) ) ) ).
% comm_monoid_set.union_disjoint
thf(fact_5613_comm__monoid__set_Ounion__diff2,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,A5: set @ B,B4: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( F @ ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ B4 @ A5 ) ) ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ).
% comm_monoid_set.union_diff2
thf(fact_5614_comm__monoid__set_Odelta__remove,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z2: A,S: set @ B,A3: B,B2: B > A,C3: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ S )
=> ( ( ( member2 @ B @ A3 @ S )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B2 @ K2 ) @ ( C3 @ K2 ) )
@ S )
= ( F @ ( B2 @ A3 ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member2 @ B @ A3 @ S )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2
@ ^ [K2: B] : ( if @ A @ ( K2 = A3 ) @ ( B2 @ K2 ) @ ( C3 @ K2 ) )
@ S )
= ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ C3 @ ( minus_minus @ ( set @ B ) @ S @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.delta_remove
thf(fact_5615_times__assn__def,axiom,
( ( times_times @ assn )
= ( ^ [P3: assn,Q2: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P3 ) @ ( rep_assn @ Q2 ) ) ) ) ) ).
% times_assn_def
thf(fact_5616_wand__assn__def,axiom,
( wand_assn
= ( ^ [P3: assn,Q2: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P3 ) @ ( rep_assn @ Q2 ) ) ) ) ) ).
% wand_assn_def
thf(fact_5617_mod__h__bot__iff_I8_J,axiom,
! [C: $tType,R: C > assn,H: heap_ext @ product_unit] :
( ( rep_assn @ ( ex_assn @ C @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ? [X3: C] : ( rep_assn @ ( R @ X3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(8)
thf(fact_5618_sngr__assn__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( sngr_assn @ A )
= ( ^ [R4: ref @ A,X3: A] : ( abs_assn @ ( sngr_assn_raw @ A @ R4 @ X3 ) ) ) ) ) ).
% sngr_assn_def
thf(fact_5619_snga__assn__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( snga_assn @ A )
= ( ^ [R4: array @ A,A4: list @ A] : ( abs_assn @ ( snga_assn_raw @ A @ R4 @ A4 ) ) ) ) ) ).
% snga_assn_def
thf(fact_5620_ex__assn__const,axiom,
! [A: $tType,C3: assn] :
( ( ex_assn @ A
@ ^ [X3: A] : C3 )
= C3 ) ).
% ex_assn_const
thf(fact_5621_norm__assertion__simps_I17_J,axiom,
! [B: $tType,R: assn,Q: B > assn] :
( ( times_times @ assn @ R @ ( ex_assn @ B @ Q ) )
= ( ex_assn @ B
@ ^ [X3: B] : ( times_times @ assn @ R @ ( Q @ X3 ) ) ) ) ).
% norm_assertion_simps(17)
thf(fact_5622_norm__assertion__simps_I16_J,axiom,
! [A: $tType,Q: A > assn,R: assn] :
( ( times_times @ assn @ ( ex_assn @ A @ Q ) @ R )
= ( ex_assn @ A
@ ^ [X3: A] : ( times_times @ assn @ ( Q @ X3 ) @ R ) ) ) ).
% norm_assertion_simps(16)
thf(fact_5623_mod__ex__dist,axiom,
! [A: $tType,P: A > assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H )
= ( ? [X3: A] : ( rep_assn @ ( P @ X3 ) @ H ) ) ) ).
% mod_ex_dist
thf(fact_5624_triv__exI,axiom,
! [A: $tType,Q: A > assn,X: A] : ( entails @ ( Q @ X ) @ ( ex_assn @ A @ Q ) ) ).
% triv_exI
thf(fact_5625_norm__assertion__simps_I21_J,axiom,
! [F6: $tType,Q: assn,P: F6 > assn] :
( ( sup_sup @ assn @ Q @ ( ex_assn @ F6 @ P ) )
= ( ex_assn @ F6
@ ^ [X3: F6] : ( sup_sup @ assn @ Q @ ( P @ X3 ) ) ) ) ).
% norm_assertion_simps(21)
thf(fact_5626_norm__assertion__simps_I20_J,axiom,
! [E4: $tType,Q: E4 > assn,P: assn] :
( ( sup_sup @ assn @ ( ex_assn @ E4 @ Q ) @ P )
= ( ex_assn @ E4
@ ^ [X3: E4] : ( sup_sup @ assn @ ( Q @ X3 ) @ P ) ) ) ).
% norm_assertion_simps(20)
thf(fact_5627_norm__assertion__simps_I18_J,axiom,
! [C: $tType,Q: C > assn,P: assn] :
( ( inf_inf @ assn @ ( ex_assn @ C @ Q ) @ P )
= ( ex_assn @ C
@ ^ [X3: C] : ( inf_inf @ assn @ ( Q @ X3 ) @ P ) ) ) ).
% norm_assertion_simps(18)
thf(fact_5628_norm__assertion__simps_I19_J,axiom,
! [D: $tType,Q: assn,P: D > assn] :
( ( inf_inf @ assn @ Q @ ( ex_assn @ D @ P ) )
= ( ex_assn @ D
@ ^ [X3: D] : ( inf_inf @ assn @ Q @ ( P @ X3 ) ) ) ) ).
% norm_assertion_simps(19)
thf(fact_5629_ex__one__point__gen,axiom,
! [A: $tType,P: A > assn,V: A] :
( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X2: A] :
( ( rep_assn @ ( P @ X2 ) @ H4 )
=> ( X2 = V ) )
=> ( ( ex_assn @ A @ P )
= ( P @ V ) ) ) ).
% ex_one_point_gen
thf(fact_5630_mod__exI,axiom,
! [A: $tType,P: A > assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ? [X6: A] : ( rep_assn @ ( P @ X6 ) @ H )
=> ( rep_assn @ ( ex_assn @ A @ P ) @ H ) ) ).
% mod_exI
thf(fact_5631_mod__exE,axiom,
! [A: $tType,P: A > assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H )
=> ~ ! [X2: A] :
~ ( rep_assn @ ( P @ X2 ) @ H ) ) ).
% mod_exE
thf(fact_5632_ent__ex__postI,axiom,
! [A: $tType,P: assn,Q: A > assn,X: A] :
( ( entails @ P @ ( Q @ X ) )
=> ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ).
% ent_ex_postI
thf(fact_5633_ent__ex__preI,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ! [X2: A] : ( entails @ ( P @ X2 ) @ Q )
=> ( entails @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ent_ex_preI
thf(fact_5634_enorm__exI_H,axiom,
! [A: $tType,Z8: A > $o,P: assn,Q: A > assn] :
( ! [X2: A] :
( ( Z8 @ X2 )
=> ( entails @ P @ ( Q @ X2 ) ) )
=> ( ? [X_12: A] : ( Z8 @ X_12 )
=> ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ) ).
% enorm_exI'
thf(fact_5635_ex__distrib__star,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ( ex_assn @ A
@ ^ [X3: A] : ( times_times @ assn @ ( P @ X3 ) @ Q ) )
= ( times_times @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ex_distrib_star
thf(fact_5636_ex__distrib__and,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ( ex_assn @ A
@ ^ [X3: A] : ( inf_inf @ assn @ ( P @ X3 ) @ Q ) )
= ( inf_inf @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ex_distrib_and
thf(fact_5637_ex__join__or,axiom,
! [A: $tType,P: A > assn,Q: A > assn] :
( ( ex_assn @ A
@ ^ [X3: A] : ( sup_sup @ assn @ ( P @ X3 ) @ ( ex_assn @ A @ Q ) ) )
= ( ex_assn @ A
@ ^ [X3: A] : ( sup_sup @ assn @ ( P @ X3 ) @ ( Q @ X3 ) ) ) ) ).
% ex_join_or
thf(fact_5638_ex__distrib__or,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ( ex_assn @ A
@ ^ [X3: A] : ( sup_sup @ assn @ ( P @ X3 ) @ Q ) )
= ( sup_sup @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ex_distrib_or
thf(fact_5639_ex__assn__def,axiom,
! [A: $tType] :
( ( ex_assn @ A )
= ( ^ [P3: A > assn] :
( abs_assn
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
? [X3: A] : ( rep_assn @ ( P3 @ X3 ) @ H2 ) ) ) ) ).
% ex_assn_def
thf(fact_5640_one__assn__def,axiom,
( ( one_one @ assn )
= ( abs_assn @ one_assn_raw ) ) ).
% one_assn_def
thf(fact_5641_one__assn__raw_Osimps,axiom,
! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( one_assn_raw @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
= ( As2
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% one_assn_raw.simps
thf(fact_5642_one__assn__raw_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( Y
= ( As
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% one_assn_raw.elims(1)
thf(fact_5643_one__assn__raw_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( one_assn_raw @ X )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( As
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(3)
thf(fact_5644_one__assn__raw_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( one_assn_raw @ X )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( As
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(2)
thf(fact_5645_properI,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ! [As: set @ nat,H4: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) )
=> ( ! [As: set @ nat,H4: heap_ext @ product_unit,H6: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( relH @ As @ H4 @ H6 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) ) ) ) )
=> ( proper @ P ) ) ) ).
% properI
thf(fact_5646_properD2,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H: heap_ext @ product_unit,As2: set @ nat,H5: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( ( relH @ As2 @ H @ H5 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As2 ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As2 ) ) ) ) ) ) ).
% properD2
thf(fact_5647_proper__def,axiom,
( proper
= ( ^ [P3: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
! [H2: heap_ext @ product_unit,H8: heap_ext @ product_unit,As3: set @ nat] :
( ( ( P3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As3 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As3 ) ) )
& ( ( ( P3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As3 ) )
& ( relH @ As3 @ H2 @ H8 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As3 ) ) )
=> ( P3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As3 ) ) ) ) ) ) ).
% proper_def
thf(fact_5648_bool__assn__proper_I4_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q )
=> ( proper
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H2 )
& ( Q @ H2 ) ) ) ) ) ).
% bool_assn_proper(4)
thf(fact_5649_bool__assn__proper_I3_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q )
=> ( proper
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H2 )
| ( Q @ H2 ) ) ) ) ) ).
% bool_assn_proper(3)
thf(fact_5650_bool__assn__proper_I2_J,axiom,
( proper
@ ^ [Uu3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : $false ) ).
% bool_assn_proper(2)
thf(fact_5651_bool__assn__proper_I1_J,axiom,
proper @ in_range ).
% bool_assn_proper(1)
thf(fact_5652_ex__assn__proper,axiom,
! [A: $tType,P: A > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ! [X2: A] : ( proper @ ( P @ X2 ) )
=> ( proper
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
? [X3: A] : ( P @ X3 @ H2 ) ) ) ).
% ex_assn_proper
thf(fact_5653_snga__assn__proper,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: array @ A,X: list @ A] : ( proper @ ( snga_assn_raw @ A @ R3 @ X ) ) ) ).
% snga_assn_proper
thf(fact_5654_sngr__assn__proper,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,X: A] : ( proper @ ( sngr_assn_raw @ A @ R3 @ X ) ) ) ).
% sngr_assn_proper
thf(fact_5655_times__assn__proper,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( ( proper @ Q )
=> ( proper @ ( times_assn_raw @ P @ Q ) ) ) ) ).
% times_assn_proper
thf(fact_5656_wand__proper,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] : ( proper @ ( wand_raw @ P @ Q ) ) ).
% wand_proper
thf(fact_5657_one__assn__proper,axiom,
proper @ one_assn_raw ).
% one_assn_proper
thf(fact_5658_bool__assn__proper_I5_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( proper
@ ^ [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ H2 )
& ~ ( P @ H2 ) ) ) ) ).
% bool_assn_proper(5)
thf(fact_5659_pure__assn__proper,axiom,
! [B2: $o] : ( proper @ ( pure_assn_raw @ ( heap_ext @ product_unit ) @ nat @ B2 ) ) ).
% pure_assn_proper
thf(fact_5660_Abs__assn__inject,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Y: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member2 @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ X @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( ( member2 @ ( ( 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_5661_Abs__assn__induct,axiom,
! [P: assn > $o,X: assn] :
( ! [Y2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member2 @ ( ( 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_5662_Abs__assn__cases,axiom,
! [X: assn] :
~ ! [Y2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( X
= ( abs_assn @ Y2 ) )
=> ~ ( member2 @ ( ( 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_5663_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] :
( ( member2 @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( ! [X2: assn] : ( P @ ( rep_assn @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_assn_induct
thf(fact_5664_Rep__assn__cases,axiom,
! [Y: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member2 @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ~ ! [X2: assn] :
( Y
!= ( rep_assn @ X2 ) ) ) ).
% Rep_assn_cases
thf(fact_5665_Rep__assn,axiom,
! [X: assn] : ( member2 @ ( ( 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_5666_properD1,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H: heap_ext @ product_unit,As2: set @ nat] :
( ( proper @ P )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) ) ) ) ).
% properD1
thf(fact_5667_Abs__assn__inverse,axiom,
! [Y: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member2 @ ( ( 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_5668_proper__iff,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,As2: set @ nat,H: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( relH @ As2 @ H @ H5 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As2 ) )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
= ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As2 ) ) ) ) ) ) ).
% proper_iff
thf(fact_5669_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_5670_wo__rel_Osuc__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member2 @ A @ B2 @ B4 )
=> ( ( ( bNF_Wellorder_wo_suc @ A @ R3 @ B4 )
!= B2 )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B4 ) ) @ R3 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_5671_monoid__list_ONil,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( groups_monoid_list @ A @ F @ Z2 )
=> ( ( groups_monoid_F @ A @ F @ Z2 @ ( nil @ A ) )
= Z2 ) ) ).
% monoid_list.Nil
thf(fact_5672_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F: A > A > A,Z2: A,A3: A] :
( ( comm_monoid @ A @ F @ Z2 )
=> ( ( F @ A3 @ Z2 )
= A3 ) ) ).
% comm_monoid.comm_neutral
thf(fact_5673_semilattice__neutr_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( semilattice_neutr @ A @ F @ Z2 )
=> ( comm_monoid @ A @ F @ Z2 ) ) ).
% semilattice_neutr.axioms(2)
thf(fact_5674_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_5675_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_5676_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_5677_inf__top_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( comm_monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.comm_monoid_axioms
thf(fact_5678_wo__rel_Osuc__inField,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B4 ) @ ( field2 @ A @ R3 ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_5679_wo__rel_Osuc__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B4 ) @ ( order_AboveS @ A @ R3 @ B4 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_5680_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( ( order_AboveS @ A @ R3 @ A5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R3 @ A5 ) ) @ R3 )
=> ( ( B2
!= ( bNF_Wellorder_wo_suc @ A @ R3 @ A5 ) )
=> ( member2 @ A @ B2 @ A5 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_5681_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_5682_AboveS__disjoint,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( order_AboveS @ A @ R3 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% AboveS_disjoint
thf(fact_5683_monoid__list_OCons,axiom,
! [A: $tType,F: A > A > A,Z2: A,X: A,Xs: list @ A] :
( ( groups_monoid_list @ A @ F @ Z2 )
=> ( ( groups_monoid_F @ A @ F @ Z2 @ ( cons @ A @ X @ Xs ) )
= ( F @ X @ ( groups_monoid_F @ A @ F @ Z2 @ Xs ) ) ) ) ).
% monoid_list.Cons
thf(fact_5684_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_max @ A )
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% max.semilattice_order_axioms
thf(fact_5685_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_5686_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice_order @ A @ ( sup_sup @ A )
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% sup.semilattice_order_axioms
thf(fact_5687_semilattice__order_Omono,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C3: A,B2: A,D3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ C3 )
=> ( ( Less_eq @ B2 @ D3 )
=> ( Less_eq @ ( F @ A3 @ B2 ) @ ( F @ C3 @ D3 ) ) ) ) ) ).
% semilattice_order.mono
thf(fact_5688_semilattice__order_OorderE,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( A3
= ( F @ A3 @ B2 ) ) ) ) ).
% semilattice_order.orderE
thf(fact_5689_semilattice__order_OorderI,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( A3
= ( F @ A3 @ B2 ) )
=> ( Less_eq @ A3 @ B2 ) ) ) ).
% semilattice_order.orderI
thf(fact_5690_semilattice__order_Oabsorb1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( F @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb1
thf(fact_5691_semilattice__order_Oabsorb2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ A3 )
=> ( ( F @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb2
thf(fact_5692_semilattice__order_Oabsorb3,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( F @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb3
thf(fact_5693_semilattice__order_Oabsorb4,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ B2 @ A3 )
=> ( ( F @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb4
thf(fact_5694_semilattice__order_OboundedE,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ ( F @ B2 @ C3 ) )
=> ~ ( ( Less_eq @ A3 @ B2 )
=> ~ ( Less_eq @ A3 @ C3 ) ) ) ) ).
% semilattice_order.boundedE
thf(fact_5695_semilattice__order_OboundedI,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less_eq @ A3 @ C3 )
=> ( Less_eq @ A3 @ ( F @ B2 @ C3 ) ) ) ) ) ).
% semilattice_order.boundedI
thf(fact_5696_semilattice__order_Oorder__iff,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
= ( A3
= ( F @ A3 @ B2 ) ) ) ) ).
% semilattice_order.order_iff
thf(fact_5697_semilattice__order_Ocobounded1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( Less_eq @ ( F @ A3 @ B2 ) @ A3 ) ) ).
% semilattice_order.cobounded1
thf(fact_5698_semilattice__order_Ocobounded2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( Less_eq @ ( F @ A3 @ B2 ) @ B2 ) ) ).
% semilattice_order.cobounded2
thf(fact_5699_semilattice__order_Oabsorb__iff1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
= ( ( F @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb_iff1
thf(fact_5700_semilattice__order_Oabsorb__iff2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ A3 )
= ( ( F @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb_iff2
thf(fact_5701_semilattice__order_Obounded__iff,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ ( F @ B2 @ C3 ) )
= ( ( Less_eq @ A3 @ B2 )
& ( Less_eq @ A3 @ C3 ) ) ) ) ).
% semilattice_order.bounded_iff
thf(fact_5702_semilattice__order_OcoboundedI1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ C3 )
=> ( Less_eq @ ( F @ A3 @ B2 ) @ C3 ) ) ) ).
% semilattice_order.coboundedI1
thf(fact_5703_semilattice__order_OcoboundedI2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C3: A,A3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ C3 )
=> ( Less_eq @ ( F @ A3 @ B2 ) @ C3 ) ) ) ).
% semilattice_order.coboundedI2
thf(fact_5704_semilattice__order_Ostrict__boundedE,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ A3 @ ( F @ B2 @ C3 ) )
=> ~ ( ( Less @ A3 @ B2 )
=> ~ ( Less @ A3 @ C3 ) ) ) ) ).
% semilattice_order.strict_boundedE
thf(fact_5705_semilattice__order_Ostrict__order__iff,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( A3
= ( F @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% semilattice_order.strict_order_iff
thf(fact_5706_semilattice__order_Ostrict__coboundedI1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ A3 @ C3 )
=> ( Less @ ( F @ A3 @ B2 ) @ C3 ) ) ) ).
% semilattice_order.strict_coboundedI1
thf(fact_5707_semilattice__order_Ostrict__coboundedI2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C3: A,A3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ B2 @ C3 )
=> ( Less @ ( F @ A3 @ B2 ) @ C3 ) ) ) ).
% semilattice_order.strict_coboundedI2
thf(fact_5708_semilattice__neutr__order_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F @ Z2 @ Less_eq @ Less )
=> ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ).
% semilattice_neutr_order.axioms(2)
thf(fact_5709_semilattice__neutr__order__def,axiom,
! [A: $tType] :
( ( semila1105856199041335345_order @ A )
= ( ^ [F3: A > A > A,Z3: A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semilattice_neutr @ A @ F3 @ Z3 )
& ( semilattice_order @ A @ F3 @ Less_eq2 @ Less2 ) ) ) ) ).
% semilattice_neutr_order_def
thf(fact_5710_semilattice__neutr__order_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_neutr @ A @ F @ Z2 )
=> ( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( semila1105856199041335345_order @ A @ F @ Z2 @ Less_eq @ Less ) ) ) ).
% semilattice_neutr_order.intro
thf(fact_5711_inf_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semilattice_order @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% inf.semilattice_order_axioms
thf(fact_5712_min_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% min.semilattice_order_axioms
thf(fact_5713_entt__def__true,axiom,
( entailst
= ( ^ [P3: assn,Q2: assn] : ( entails @ ( times_times @ assn @ P3 @ ( top_top @ assn ) ) @ ( times_times @ assn @ Q2 @ ( top_top @ assn ) ) ) ) ) ).
% entt_def_true
thf(fact_5714_entailst__def,axiom,
( entailst
= ( ^ [A7: assn,B5: assn] : ( entails @ A7 @ ( times_times @ assn @ B5 @ ( top_top @ assn ) ) ) ) ) ).
% entailst_def
thf(fact_5715_enttI__true,axiom,
! [P: assn,Q: assn] :
( ( entails @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ ( times_times @ assn @ Q @ ( top_top @ assn ) ) )
=> ( entailst @ P @ Q ) ) ).
% enttI_true
thf(fact_5716_entt__refl,axiom,
! [A5: assn] : ( entailst @ A5 @ A5 ) ).
% entt_refl
thf(fact_5717_entt__true,axiom,
! [A5: assn] : ( entailst @ A5 @ ( top_top @ assn ) ) ).
% entt_true
thf(fact_5718_entt__emp,axiom,
! [A5: assn] : ( entailst @ A5 @ ( one_one @ assn ) ) ).
% entt_emp
thf(fact_5719_entt__star__true__simp_I2_J,axiom,
! [A5: assn,B4: assn] :
( ( entailst @ ( times_times @ assn @ A5 @ ( top_top @ assn ) ) @ B4 )
= ( entailst @ A5 @ B4 ) ) ).
% entt_star_true_simp(2)
thf(fact_5720_entt__star__true__simp_I1_J,axiom,
! [A5: assn,B4: assn] :
( ( entailst @ A5 @ ( times_times @ assn @ B4 @ ( top_top @ assn ) ) )
= ( entailst @ A5 @ B4 ) ) ).
% entt_star_true_simp(1)
thf(fact_5721_entt__disjI2__direct,axiom,
! [B4: assn,A5: assn] : ( entailst @ B4 @ ( sup_sup @ assn @ A5 @ B4 ) ) ).
% entt_disjI2_direct
thf(fact_5722_entt__disjI1__direct,axiom,
! [A5: assn,B4: assn] : ( entailst @ A5 @ ( sup_sup @ assn @ A5 @ B4 ) ) ).
% entt_disjI1_direct
thf(fact_5723_entt__disjI2_H,axiom,
! [A5: assn,C4: assn,B4: assn] :
( ( entailst @ A5 @ C4 )
=> ( entailst @ A5 @ ( sup_sup @ assn @ B4 @ C4 ) ) ) ).
% entt_disjI2'
thf(fact_5724_entt__disjI1_H,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( entailst @ A5 @ B4 )
=> ( entailst @ A5 @ ( sup_sup @ assn @ B4 @ C4 ) ) ) ).
% entt_disjI1'
thf(fact_5725_entt__disjD2,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( entailst @ ( sup_sup @ assn @ A5 @ B4 ) @ C4 )
=> ( entailst @ B4 @ C4 ) ) ).
% entt_disjD2
thf(fact_5726_entt__disjD1,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( entailst @ ( sup_sup @ assn @ A5 @ B4 ) @ C4 )
=> ( entailst @ A5 @ C4 ) ) ).
% entt_disjD1
thf(fact_5727_entt__disjE,axiom,
! [A5: assn,M: assn,B4: assn] :
( ( entailst @ A5 @ M )
=> ( ( entailst @ B4 @ M )
=> ( entailst @ ( sup_sup @ assn @ A5 @ B4 ) @ M ) ) ) ).
% entt_disjE
thf(fact_5728_entt__frame__fwd,axiom,
! [P: assn,Q: assn,A5: assn,F2: assn,B4: assn] :
( ( entailst @ P @ Q )
=> ( ( entailst @ A5 @ ( times_times @ assn @ P @ F2 ) )
=> ( ( entailst @ ( times_times @ assn @ Q @ F2 ) @ B4 )
=> ( entailst @ A5 @ B4 ) ) ) ) ).
% entt_frame_fwd
thf(fact_5729_entt__star__mono,axiom,
! [A5: assn,B4: assn,C4: assn,D2: assn] :
( ( entailst @ A5 @ B4 )
=> ( ( entailst @ C4 @ D2 )
=> ( entailst @ ( times_times @ assn @ A5 @ C4 ) @ ( times_times @ assn @ B4 @ D2 ) ) ) ) ).
% entt_star_mono
thf(fact_5730_entt__fr__refl,axiom,
! [F2: assn,F8: assn,A5: assn] :
( ( entailst @ F2 @ F8 )
=> ( entailst @ ( times_times @ assn @ F2 @ A5 ) @ ( times_times @ assn @ F8 @ A5 ) ) ) ).
% entt_fr_refl
thf(fact_5731_entt__fr__drop,axiom,
! [F2: assn,F8: assn,A5: assn] :
( ( entailst @ F2 @ F8 )
=> ( entailst @ ( times_times @ assn @ F2 @ A5 ) @ F8 ) ) ).
% entt_fr_drop
thf(fact_5732_entt__trans,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( entailst @ A5 @ B4 )
=> ( ( entailst @ B4 @ C4 )
=> ( entailst @ A5 @ C4 ) ) ) ).
% entt_trans
thf(fact_5733_ent__imp__entt,axiom,
! [P: assn,Q: assn] :
( ( entails @ P @ Q )
=> ( entailst @ P @ Q ) ) ).
% ent_imp_entt
thf(fact_5734_enttD,axiom,
! [A5: assn,B4: assn] :
( ( entailst @ A5 @ B4 )
=> ( entails @ A5 @ ( times_times @ assn @ B4 @ ( top_top @ assn ) ) ) ) ).
% enttD
thf(fact_5735_enttI,axiom,
! [A5: assn,B4: assn] :
( ( entails @ A5 @ ( times_times @ assn @ B4 @ ( top_top @ assn ) ) )
=> ( entailst @ A5 @ B4 ) ) ).
% enttI
thf(fact_5736_comp__fun__commute_Ofoldr__conv__foldl,axiom,
! [B: $tType,A: $tType,F: A > B > B,Xs: list @ A,A3: B] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( ( foldr @ A @ B @ F @ Xs @ A3 )
= ( foldl @ B @ A
@ ^ [A4: B,B3: A] : ( F @ B3 @ A4 )
@ A3
@ Xs ) ) ) ).
% comp_fun_commute.foldr_conv_foldl
thf(fact_5737_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_5738_in__range_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( in_range @ X )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ! [X2: nat] :
( ( member2 @ nat @ X2 @ As )
=> ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H4 ) ) ) ) ) ).
% in_range.elims(3)
thf(fact_5739_comp__fun__commute__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite6289374366891150609ommute @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X3: A] : X3 ) ) ) ).
% comp_fun_commute_insort
thf(fact_5740_comp__fun__commute_Ofoldl__f__commute,axiom,
! [B: $tType,A: $tType,F: A > B > B,A3: A,B2: B,Xs: list @ A] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( ( F @ A3
@ ( foldl @ B @ A
@ ^ [A4: B,B3: A] : ( F @ B3 @ A4 )
@ B2
@ Xs ) )
= ( foldl @ B @ A
@ ^ [A4: B,B3: A] : ( F @ B3 @ A4 )
@ ( F @ A3 @ B2 )
@ Xs ) ) ) ).
% comp_fun_commute.foldl_f_commute
thf(fact_5741_in__range_Osimps,axiom,
! [H: heap_ext @ product_unit,As2: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
= ( ! [X3: nat] :
( ( member2 @ nat @ X3 @ As2 )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H ) ) ) ) ) ).
% in_range.simps
thf(fact_5742_in__range_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( in_range @ X )
= Y )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( Y
= ( ~ ! [X3: nat] :
( ( member2 @ nat @ X3 @ As )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ).
% in_range.elims(1)
thf(fact_5743_in__range_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ X )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ~ ! [X6: nat] :
( ( member2 @ nat @ X6 @ As )
=> ( ord_less @ nat @ X6 @ ( lim @ product_unit @ H4 ) ) ) ) ) ).
% in_range.elims(2)
thf(fact_5744_czeroE,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( field2 @ A @ R3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% czeroE
thf(fact_5745_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_5746_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ).
% sngr_assn_raw.elims(3)
thf(fact_5747_relH__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [As2: set @ nat,H: heap_ext @ product_unit,H5: heap_ext @ product_unit,R3: ref @ A] :
( ( relH @ As2 @ H @ H5 )
=> ( ( member2 @ nat @ ( addr_of_ref @ A @ R3 ) @ As2 )
=> ( ( ref_get @ A @ H @ R3 )
= ( ref_get @ A @ H5 @ R3 ) ) ) ) ) ).
% relH_ref
thf(fact_5748_sngr__assn__raw_Osimps,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,X: A,H: heap_ext @ product_unit,As2: set @ nat] :
( ( sngr_assn_raw @ A @ R3 @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
= ( ( ( ref_get @ A @ H @ R3 )
= X )
& ( As2
= ( insert2 @ nat @ ( addr_of_ref @ A @ R3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ R3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ).
% sngr_assn_raw.simps
thf(fact_5749_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( Y
= ( ~ ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.elims(1)
thf(fact_5750_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ~ ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ).
% sngr_assn_raw.elims(2)
thf(fact_5751_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( Y
= ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) )
=> ~ ( 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 ) @ H4 @ As ) ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(1)
thf(fact_5752_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( 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 ) @ H4 @ As ) ) ) )
=> ~ ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(2)
thf(fact_5753_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( 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 ) @ H4 @ As ) ) ) )
=> ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(3)
thf(fact_5754_relH__set__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,As2: set @ nat,H: heap_ext @ product_unit,X: A] :
( ~ ( member2 @ nat @ ( addr_of_ref @ A @ R3 ) @ As2 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( relH @ As2 @ H @ ( ref_set @ A @ R3 @ X @ H ) ) ) ) ) ).
% relH_set_ref
thf(fact_5755_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ~ ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% Cnotzero_imp_not_empty
thf(fact_5756_czeroI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ A @ R3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% czeroI
thf(fact_5757_Card__order__iff__Restr__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( field2 @ A @ R3 ) )
=> ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ X3 )
@ ^ [Uu3: A] : ( order_underS @ A @ R3 @ X3 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% Card_order_iff_Restr_underS
thf(fact_5758_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member2 @ ( 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 @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( ( ( field2 @ A @ P24 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( field2 @ B @ R2 )
= ( bot_bot @ ( set @ B ) ) ) )
=> ( member2 @ ( 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 @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R2 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2'
thf(fact_5759_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ B @ P4 )
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member2 @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member2 @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R3 )
@ ^ [Uu3: A] : ( field2 @ B @ P4 ) ) )
@ R3 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member2 @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ ( field2 @ B @ P4 )
@ ^ [Uu3: B] : ( field2 @ A @ R3 ) ) )
@ R3 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% Card_order_Times_infinite
thf(fact_5760_card__of__bool,axiom,
! [A: $tType,A1: A,A22: A] :
( ( A1 != A22 )
=> ( member2 @ ( 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_5761_card__of__empty__ordIso,axiom,
! [B: $tType,A: $tType] : ( member2 @ ( 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_5762_card__of__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( member2 @ ( 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty2
thf(fact_5763_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member2 @ ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member2 @ ( 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 @ A5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu3: B] : A5 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(4)
thf(fact_5764_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member2 @ ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member2 @ ( 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 @ B4
@ ^ [Uu3: B] : A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(3)
thf(fact_5765_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member2 @ ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member2 @ ( 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 @ A5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(2)
thf(fact_5766_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member2 @ ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member2 @ ( 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 @ A5
@ ^ [Uu3: A] : B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(1)
thf(fact_5767_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member2 @ ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member2 @ ( 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 @ A5
@ ^ [Uu3: A] : B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member2 @ ( 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 @ B4
@ ^ [Uu3: B] : A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% card_of_Times_infinite
thf(fact_5768_card__of__empty,axiom,
! [B: $tType,A: $tType,A5: set @ B] : ( member2 @ ( 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 @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_empty
thf(fact_5769_card__of__empty3,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( member2 @ ( 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty3
thf(fact_5770_card__of__Times1,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ ( 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 @ B4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu3: B] : A5 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_5771_card__of__Times2,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ ( 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 @ B4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_5772_czero__def,axiom,
! [A: $tType] :
( ( bNF_Cardinal_czero @ A )
= ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% czero_def
thf(fact_5773_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G2: B > A] :
( ( image2 @ B @ A @ G2 @ B4 )
= A5 ) )
= ( member2 @ ( 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_5774_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A5: set @ A,B2: B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_singl_ordLeq
thf(fact_5775_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: set @ B] :
( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F3: B > A] :
( ( image2 @ B @ A @ F3 @ A5 )
= B4 ) )
= ( member2 @ ( 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_5776_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ R3
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R3 )
@ ^ [Uu3: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% Card_order_Times1
thf(fact_5777_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ R3
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu3: B] : ( field2 @ A @ R3 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% Card_order_Times2
thf(fact_5778_Card__order__empty,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member2 @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% Card_order_empty
thf(fact_5779_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( member2 @ ( 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 @ A5 ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_ordIso_czero_iff_empty
thf(fact_5780_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),B2: B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member2 @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% Card_order_singl_ordLeq
thf(fact_5781_card__of__empty1,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
| ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member2 @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_empty1
thf(fact_5782_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R2: set @ ( product_prod @ D @ D )] :
( ( member2 @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member2 @ ( 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 @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( ( field2 @ C @ P24 )
= ( bot_bot @ ( set @ C ) ) )
=> ( ( field2 @ D @ R2 )
= ( bot_bot @ ( set @ D ) ) ) )
=> ( member2 @ ( 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 @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R2 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% cexp_mono'
thf(fact_5783_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A5: set @ A,B4: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) )
=> ( ( member2 @ ( 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member2 @ ( 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 @ A5 @ B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_5784_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A5: set @ A,B1: B,B22: B,B4: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) )
=> ( ( ( B1 != B22 )
& ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ B1 @ ( insert2 @ B @ B22 @ ( bot_bot @ ( set @ B ) ) ) ) @ B4 ) )
=> ( member2 @ ( 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 @ A5 @ B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times
thf(fact_5785_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( ( array_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ).
% snga_assn_raw.elims(3)
thf(fact_5786_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ A] : ( member2 @ ( 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( bot_bot @ ( set @ B ) ) @ A5 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_empty2
thf(fact_5787_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A5: set @ A] : ( member2 @ ( 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ ( bot_bot @ ( set @ B ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus_empty1
thf(fact_5788_relH__array,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [As2: set @ nat,H: heap_ext @ product_unit,H5: heap_ext @ product_unit,R3: array @ A] :
( ( relH @ As2 @ H @ H5 )
=> ( ( member2 @ nat @ ( addr_of_array @ A @ R3 ) @ As2 )
=> ( ( array_get @ A @ H @ R3 )
= ( array_get @ A @ H5 @ R3 ) ) ) ) ) ).
% relH_array
thf(fact_5789_snga__assn__raw_Osimps,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: array @ A,X: list @ A,H: heap_ext @ product_unit,As2: set @ nat] :
( ( snga_assn_raw @ A @ R3 @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
= ( ( ( array_get @ A @ H @ R3 )
= X )
& ( As2
= ( insert2 @ nat @ ( addr_of_array @ A @ R3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ R3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ).
% snga_assn_raw.simps
thf(fact_5790_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( Y
= ( ~ ( ( ( array_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ) ).
% snga_assn_raw.elims(1)
thf(fact_5791_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ~ ( ( ( array_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ).
% snga_assn_raw.elims(2)
thf(fact_5792_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( ( sum_Plus @ A @ B @ A5 @ B4 )
= ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Plus_eq_empty_conv
thf(fact_5793_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( Y
= ( ( ( array_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) )
=> ~ ( 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 ) @ H4 @ As ) ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(1)
thf(fact_5794_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( 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 ) @ H4 @ As ) ) ) )
=> ~ ( ( ( array_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(2)
thf(fact_5795_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 ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( 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 ) @ H4 @ As ) ) ) )
=> ( ( ( array_get @ A @ H4 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(3)
thf(fact_5796_relH__set__array,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: array @ A,As2: set @ nat,H: heap_ext @ product_unit,X: list @ A] :
( ~ ( member2 @ nat @ ( addr_of_array @ A @ R3 ) @ As2 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As2 ) )
=> ( relH @ As2 @ H @ ( array_set @ A @ R3 @ X @ H ) ) ) ) ) ).
% relH_set_array
thf(fact_5797_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 ) )
=> ~ ! [H4: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As ) )
=> ( ( Y
= ( ( As
= ( 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 ) @ H4 @ As ) ) ) ) ) ) ) ).
% pure_assn_raw.pelims(1)
thf(fact_5798_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 ) )
=> ~ ! [H4: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As ) )
=> ( ( 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 ) @ H4 @ As ) ) )
=> ~ ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
thf(fact_5799_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 ) )
=> ~ ! [H4: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As ) )
=> ( ( 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 ) @ H4 @ As ) ) )
=> ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
thf(fact_5800_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_5801_list__all__zip__alt,axiom,
! [B: $tType,A: $tType] :
( ( list_all_zip @ A @ B )
= ( ^ [P3: A > B > $o,As3: list @ A,Bs3: list @ B] :
( ( ( size_size @ ( list @ A ) @ As3 )
= ( size_size @ ( list @ B ) @ Bs3 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ As3 ) )
=> ( P3 @ ( nth @ A @ As3 @ I2 ) @ ( nth @ B @ Bs3 @ I2 ) ) ) ) ) ) ).
% list_all_zip_alt
thf(fact_5802_lazI,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ B2 ) )
=> ( P @ ( nth @ A @ A3 @ I3 ) @ ( nth @ B @ B2 @ I3 ) ) )
=> ( list_all_zip @ A @ B @ P @ A3 @ B2 ) ) ) ).
% lazI
thf(fact_5803_laz__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,A3: list @ A,B2: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
& ( Q @ X3 @ Y3 ) )
@ A3
@ B2 )
= ( ( list_all_zip @ A @ B @ P @ A3 @ B2 )
& ( list_all_zip @ A @ B @ Q @ A3 @ B2 ) ) ) ).
% laz_conj
thf(fact_5804_laz__weak__Pb,axiom,
! [A: $tType,B: $tType,P: B > $o,A5: list @ A,B4: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A4: A] : P
@ A5
@ B4 )
= ( ( ( size_size @ ( list @ A ) @ A5 )
= ( size_size @ ( list @ B ) @ B4 ) )
& ! [X3: B] :
( ( member2 @ B @ X3 @ ( set2 @ B @ B4 ) )
=> ( P @ X3 ) ) ) ) ).
% laz_weak_Pb
thf(fact_5805_laz__weak__Pa,axiom,
! [B: $tType,A: $tType,P: A > $o,A5: list @ A,B4: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A4: A,B3: B] : ( P @ A4 )
@ A5
@ B4 )
= ( ( ( size_size @ ( list @ A ) @ A5 )
= ( size_size @ ( list @ B ) @ B4 ) )
& ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ A5 ) )
=> ( P @ X3 ) ) ) ) ).
% laz_weak_Pa
thf(fact_5806_list__all__zip__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As2: list @ A,F: C > B,Bs: list @ C] :
( ( list_all_zip @ A @ B @ P @ As2 @ ( map @ C @ B @ F @ Bs ) )
= ( list_all_zip @ A @ C
@ ^ [A4: A,B3: C] : ( P @ A4 @ ( F @ B3 ) )
@ As2
@ Bs ) ) ).
% list_all_zip_map2
thf(fact_5807_list__all__zip__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F: C > A,As2: list @ C,Bs: list @ B] :
( ( list_all_zip @ A @ B @ P @ ( map @ C @ A @ F @ As2 ) @ Bs )
= ( list_all_zip @ C @ B
@ ^ [A4: C] : ( P @ ( F @ A4 ) )
@ As2
@ Bs ) ) ).
% list_all_zip_map1
thf(fact_5808_laz__len,axiom,
! [A: $tType,B: $tType,P: A > B > $o,A3: list @ A,B2: list @ B] :
( ( list_all_zip @ A @ B @ P @ A3 @ B2 )
=> ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) ) ) ).
% laz_len
thf(fact_5809_laz__swap__ex,axiom,
! [B: $tType,A: $tType,C: $tType,P: A > B > C > $o,A5: list @ A,B4: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A4: A,B3: B] :
? [X8: C] : ( P @ A4 @ B3 @ X8 )
@ A5
@ B4 )
=> ~ ! [C8: list @ C] :
( ( list_all_zip @ A @ C
@ ^ [A4: A,C6: C] :
? [B3: B] : ( P @ A4 @ B3 @ C6 )
@ A5
@ C8 )
=> ~ ( list_all_zip @ B @ C
@ ^ [B3: B,C6: C] :
? [A4: A] : ( P @ A4 @ B3 @ C6 )
@ B4
@ C8 ) ) ) ).
% laz_swap_ex
thf(fact_5810_laz__eq,axiom,
! [A: $tType,A3: list @ A,B2: list @ A] :
( ( list_all_zip @ A @ A
@ ^ [Y5: A,Z5: A] : Y5 = Z5
@ A3
@ B2 )
= ( A3 = B2 ) ) ).
% laz_eq
thf(fact_5811_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_5812_list__all__zip_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,A3: A,As2: list @ A,B2: B,Bs: list @ B] :
( ( list_all_zip @ A @ B @ P @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs ) )
= ( ( P @ A3 @ B2 )
& ( list_all_zip @ A @ B @ P @ As2 @ Bs ) ) ) ).
% list_all_zip.simps(2)
thf(fact_5813_list__all__zip_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,V: A,Va: list @ A] :
~ ( list_all_zip @ A @ B @ P @ ( cons @ A @ V @ Va ) @ ( nil @ B ) ) ).
% list_all_zip.simps(3)
thf(fact_5814_list__all__zip_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,V: B,Va: list @ B] :
~ ( list_all_zip @ A @ B @ P @ ( nil @ A ) @ ( cons @ B @ V @ Va ) ) ).
% list_all_zip.simps(4)
thf(fact_5815_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,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B6: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B6 @ Bs2 ) )
=> ( Y
= ( ~ ( ( X @ A6 @ B6 )
& ( list_all_zip @ A @ B @ X @ As @ Bs2 ) ) ) ) ) )
=> ( ( ? [V2: A,Va2: list @ A] :
( Xa
= ( cons @ A @ V2 @ Va2 ) )
=> ( ( Xb
= ( nil @ B ) )
=> Y ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V2: B,Va2: list @ B] :
( Xb
= ( cons @ B @ V2 @ Va2 ) )
=> Y ) ) ) ) ) ) ).
% list_all_zip.elims(1)
thf(fact_5816_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,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B6: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B6 @ Bs2 ) )
=> ~ ( ( X @ A6 @ B6 )
& ( list_all_zip @ A @ B @ X @ As @ Bs2 ) ) ) ) ) ) ).
% list_all_zip.elims(2)
thf(fact_5817_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,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B6: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B6 @ Bs2 ) )
=> ( ( X @ A6 @ B6 )
& ( list_all_zip @ A @ B @ X @ As @ Bs2 ) ) ) )
=> ( ( ? [V2: A,Va2: list @ A] :
( Xa
= ( cons @ A @ V2 @ Va2 ) )
=> ( Xb
!= ( nil @ B ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V2: B,Va2: list @ B] :
( Xb
!= ( cons @ B @ V2 @ Va2 ) ) ) ) ) ) ).
% list_all_zip.elims(3)
thf(fact_5818_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,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B6: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B6 @ 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 @ As ) @ ( cons @ B @ B6 @ Bs2 ) ) ) )
=> ( ( X @ A6 @ B6 )
& ( list_all_zip @ A @ B @ X @ As @ Bs2 ) ) ) ) )
=> ( ! [V2: A,Va2: list @ A] :
( ( Xa
= ( cons @ A @ V2 @ Va2 ) )
=> ( ( 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 @ V2 @ Va2 ) @ ( nil @ B ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V2: B,Va2: list @ B] :
( ( Xb
= ( cons @ B @ V2 @ Va2 ) )
=> ~ ( 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 @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(3)
thf(fact_5819_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,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B6: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B6 @ 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 @ As ) @ ( cons @ B @ B6 @ Bs2 ) ) ) )
=> ~ ( ( X @ A6 @ B6 )
& ( list_all_zip @ A @ B @ X @ As @ Bs2 ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(2)
thf(fact_5820_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,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B6: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B6 @ Bs2 ) )
=> ( ( Y
= ( ( X @ A6 @ B6 )
& ( list_all_zip @ A @ B @ X @ As @ 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 @ As ) @ ( cons @ B @ B6 @ Bs2 ) ) ) ) ) ) )
=> ( ! [V2: A,Va2: list @ A] :
( ( Xa
= ( cons @ A @ V2 @ Va2 ) )
=> ( ( 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 @ V2 @ Va2 ) @ ( nil @ B ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V2: B,Va2: list @ B] :
( ( Xb
= ( cons @ B @ V2 @ Va2 ) )
=> ( ~ 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 @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(1)
thf(fact_5821_lexord__asymmetric,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: list @ A,B2: list @ A] :
( ( asym @ A @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A3 @ B2 ) @ ( lexord @ A @ R ) )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ B2 @ A3 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_asymmetric
thf(fact_5822_group_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( group_axioms @ A @ F @ Z2 @ Inverse ) ) ).
% group.axioms(2)
thf(fact_5823_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_5824_asym__lenlex,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ R )
=> ( asym @ ( list @ A ) @ ( lenlex @ A @ R ) ) ) ).
% asym_lenlex
thf(fact_5825_group__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
( ! [A6: A] :
( ( F @ Z2 @ A6 )
= A6 )
=> ( ! [A6: A] :
( ( F @ ( Inverse @ A6 ) @ A6 )
= Z2 )
=> ( group_axioms @ A @ F @ Z2 @ Inverse ) ) ) ).
% group_axioms.intro
thf(fact_5826_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_5827_group__axioms__def,axiom,
! [A: $tType] :
( ( group_axioms @ A )
= ( ^ [F3: A > A > A,Z3: A,Inverse2: A > A] :
( ! [A4: A] :
( ( F3 @ Z3 @ A4 )
= A4 )
& ! [A4: A] :
( ( F3 @ ( Inverse2 @ A4 ) @ A4 )
= Z3 ) ) ) ) ).
% group_axioms_def
thf(fact_5828_ordering__top__axioms__def,axiom,
! [A: $tType] :
( ( ordering_top_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Top2: A] :
! [A4: A] : ( Less_eq2 @ A4 @ Top2 ) ) ) ).
% ordering_top_axioms_def
thf(fact_5829_lexord__asym,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ R )
=> ( asym @ ( list @ A ) @ ( lexord @ A @ R ) ) ) ).
% lexord_asym
thf(fact_5830_asym__lex,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ R )
=> ( asym @ ( list @ A ) @ ( lex @ A @ R ) ) ) ).
% asym_lex
thf(fact_5831_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_5832_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_5833_group__def,axiom,
! [A: $tType] :
( ( group @ A )
= ( ^ [F3: A > A > A,Z3: A,Inverse2: A > A] :
( ( semigroup @ A @ F3 )
& ( group_axioms @ A @ F3 @ Z3 @ Inverse2 ) ) ) ) ).
% group_def
thf(fact_5834_sup_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semigroup @ A @ ( sup_sup @ A ) ) ) ).
% sup.semigroup_axioms
thf(fact_5835_inf_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semigroup @ A @ ( inf_inf @ A ) ) ) ).
% inf.semigroup_axioms
thf(fact_5836_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_5837_group_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( semigroup @ A @ F ) ) ).
% group.axioms(1)
thf(fact_5838_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.semigroup_axioms
thf(fact_5839_min_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_min @ A ) ) ) ).
% min.semigroup_axioms
thf(fact_5840_ordering__dualI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( ordering @ A
@ ^ [A4: A,B3: A] : ( Less_eq @ B3 @ A4 )
@ ^ [A4: A,B3: A] : ( Less @ B3 @ A4 ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ).
% ordering_dualI
thf(fact_5841_ordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A6: A,B6: A] :
( ( Less_eq @ A6 @ B6 )
= ( ( Less @ A6 @ B6 )
| ( A6 = B6 ) ) )
=> ( ! [A6: A,B6: A] :
( ( Less @ A6 @ B6 )
=> ~ ( Less @ B6 @ A6 ) )
=> ( ! [A6: A] :
~ ( Less @ A6 @ A6 )
=> ( ! [A6: A,B6: A,C2: A] :
( ( Less @ A6 @ B6 )
=> ( ( Less @ B6 @ C2 )
=> ( Less @ A6 @ C2 ) ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% ordering_strictI
thf(fact_5842_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F3: A > A > A] :
! [A4: A,B3: A,C6: A] :
( ( F3 @ ( F3 @ A4 @ B3 ) @ C6 )
= ( F3 @ A4 @ ( F3 @ B3 @ C6 ) ) ) ) ) ).
% semigroup_def
thf(fact_5843_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A3 != B2 )
=> ( ( Less_eq @ A3 @ B2 )
=> ( Less @ A3 @ B2 ) ) ) ) ).
% ordering.not_eq_order_implies_strict
thf(fact_5844_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% ordering.strict_implies_not_eq
thf(fact_5845_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( Less_eq @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% ordering.strict_iff_order
thf(fact_5846_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
= ( ( Less @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% ordering.order_iff_strict
thf(fact_5847_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less_eq @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% ordering.antisym
thf(fact_5848_ordering_Oeq__iff,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A3 = B2 )
= ( ( Less_eq @ A3 @ B2 )
& ( Less_eq @ B2 @ A3 ) ) ) ) ).
% ordering.eq_iff
thf(fact_5849_semigroup_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A6: A,B6: A,C2: A] :
( ( F @ ( F @ A6 @ B6 ) @ C2 )
= ( F @ A6 @ ( F @ B6 @ C2 ) ) )
=> ( semigroup @ A @ F ) ) ).
% semigroup.intro
thf(fact_5850_semigroup_Oassoc,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A,C3: A] :
( ( semigroup @ A @ F )
=> ( ( F @ ( F @ A3 @ B2 ) @ C3 )
= ( F @ A3 @ ( F @ B2 @ C3 ) ) ) ) ).
% semigroup.assoc
thf(fact_5851_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ( semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.semigroup_axioms
thf(fact_5852_append_Osemigroup__axioms,axiom,
! [A: $tType] : ( semigroup @ ( list @ A ) @ ( append @ A ) ) ).
% append.semigroup_axioms
thf(fact_5853_max_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_max @ A ) ) ) ).
% max.semigroup_axioms
thf(fact_5854_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( monoid @ A @ F @ Z2 )
=> ( semigroup @ A @ F ) ) ).
% monoid.axioms(1)
thf(fact_5855_order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.ordering_axioms
thf(fact_5856_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% dual_order.ordering_axioms
thf(fact_5857_group_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
( ( semigroup @ A @ F )
=> ( ( group_axioms @ A @ F @ Z2 @ Inverse )
=> ( group @ A @ F @ Z2 @ Inverse ) ) ) ).
% group.intro
thf(fact_5858_monoid__def,axiom,
! [A: $tType] :
( ( monoid @ A )
= ( ^ [F3: A > A > A,Z3: A] :
( ( semigroup @ A @ F3 )
& ( monoid_axioms @ A @ F3 @ Z3 ) ) ) ) ).
% monoid_def
thf(fact_5859_monoid_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( semigroup @ A @ F )
=> ( ( monoid_axioms @ A @ F @ Z2 )
=> ( monoid @ A @ F @ Z2 ) ) ) ).
% monoid.intro
thf(fact_5860_ID_Opred__set,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o ) )
= ( ^ [P3: A > $o,X3: A] :
! [Y3: A] :
( ( member2 @ A @ Y3 @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( P3 @ Y3 ) ) ) ) ).
% ID.pred_set
thf(fact_5861_monoid__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ! [A6: A] :
( ( F @ Z2 @ A6 )
= A6 )
=> ( ! [A6: A] :
( ( F @ A6 @ Z2 )
= A6 )
=> ( monoid_axioms @ A @ F @ Z2 ) ) ) ).
% monoid_axioms.intro
thf(fact_5862_monoid__axioms__def,axiom,
! [A: $tType] :
( ( monoid_axioms @ A )
= ( ^ [F3: A > A > A,Z3: A] :
( ! [A4: A] :
( ( F3 @ Z3 @ A4 )
= A4 )
& ! [A4: A] :
( ( F3 @ A4 @ Z3 )
= A4 ) ) ) ) ).
% monoid_axioms_def
thf(fact_5863_ID_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: A,Pa: A > $o] :
( ( bNF_id_bnf @ ( A > $o ) @ P @ X )
=> ( ! [Z4: A] :
( ( member2 @ 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_5864_ID_Opred__cong,axiom,
! [A: $tType,X: A,Ya: A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z4: A] :
( ( member2 @ A @ Z4 @ ( insert2 @ A @ Ya @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( P @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( bNF_id_bnf @ ( A > $o ) @ P @ X )
= ( bNF_id_bnf @ ( A > $o ) @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_5865_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( monoid @ A @ F @ Z2 )
=> ( monoid_axioms @ A @ F @ Z2 ) ) ).
% monoid.axioms(2)
thf(fact_5866_inj__map__inv__f,axiom,
! [B: $tType,A: $tType,F: A > B,L: list @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( map @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( map @ A @ B @ F @ L ) )
= L ) ) ).
% inj_map_inv_f
thf(fact_5867_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_5868_nth__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,Xs: list @ A,A3: array @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( hoare_hoare_triple @ A @ ( snga_assn @ A @ A3 @ Xs ) @ ( array_nth @ A @ A3 @ I )
@ ^ [R4: A] :
( times_times @ assn @ ( snga_assn @ A @ A3 @ Xs )
@ ( pure_assn
@ ( R4
= ( nth @ A @ Xs @ I ) ) ) ) ) ) ) ).
% nth_rule
thf(fact_5869_false__rule,axiom,
! [A: $tType,C3: heap_Time_Heap @ A,Q: A > assn] : ( hoare_hoare_triple @ A @ ( bot_bot @ assn ) @ C3 @ Q ) ).
% false_rule
thf(fact_5870_norm__pre__pure__iff,axiom,
! [A: $tType,P: assn,B2: $o,F: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ F @ Q )
= ( B2
=> ( hoare_hoare_triple @ A @ P @ F @ Q ) ) ) ).
% norm_pre_pure_iff
thf(fact_5871_norm__pre__pure__iff__sng,axiom,
! [A: $tType,B2: $o,F: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ ( pure_assn @ B2 ) @ F @ Q )
= ( B2
=> ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F @ Q ) ) ) ).
% norm_pre_pure_iff_sng
thf(fact_5872_case__list__rule,axiom,
! [A: $tType,B: $tType,L: list @ A,P: assn,Fn: heap_Time_Heap @ B,Q: B > assn,Fc: A > ( list @ A ) > ( heap_Time_Heap @ B )] :
( ( ( L
= ( nil @ A ) )
=> ( hoare_hoare_triple @ B @ P @ Fn @ Q ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( L
= ( cons @ A @ X2 @ Xs2 ) )
=> ( hoare_hoare_triple @ B @ P @ ( Fc @ X2 @ Xs2 ) @ Q ) )
=> ( hoare_hoare_triple @ B @ P @ ( case_list @ ( heap_Time_Heap @ B ) @ A @ Fn @ Fc @ L ) @ Q ) ) ) ).
% case_list_rule
thf(fact_5873_frame__rule__left,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap @ A,Q: A > assn,R: assn] :
( ( hoare_hoare_triple @ A @ P @ C3 @ Q )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ R @ P ) @ C3
@ ^ [X3: A] : ( times_times @ assn @ R @ ( Q @ X3 ) ) ) ) ).
% frame_rule_left
thf(fact_5874_frame__rule,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap @ A,Q: A > assn,R: assn] :
( ( hoare_hoare_triple @ A @ P @ C3 @ Q )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ R ) @ C3
@ ^ [X3: A] : ( times_times @ assn @ ( Q @ X3 ) @ R ) ) ) ).
% frame_rule
thf(fact_5875_is__hoare__triple,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ P @ C3 @ Q )
=> ( hoare_hoare_triple @ A @ P @ C3 @ Q ) ) ).
% is_hoare_triple
thf(fact_5876_ordering__axioms__def,axiom,
! [A: $tType] :
( ( ordering_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ! [A4: A,B3: A] :
( ( Less2 @ A4 @ B3 )
= ( ( Less_eq2 @ A4 @ B3 )
& ( A4 != B3 ) ) )
& ! [A4: A,B3: A] :
( ( Less_eq2 @ A4 @ B3 )
=> ( ( Less_eq2 @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ) ) ).
% ordering_axioms_def
thf(fact_5877_ordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A6: A,B6: A] :
( ( Less @ A6 @ B6 )
= ( ( Less_eq @ A6 @ B6 )
& ( A6 != B6 ) ) )
=> ( ! [A6: A,B6: A] :
( ( Less_eq @ A6 @ B6 )
=> ( ( Less_eq @ B6 @ A6 )
=> ( A6 = B6 ) ) )
=> ( ordering_axioms @ A @ Less_eq @ Less ) ) ) ).
% ordering_axioms.intro
thf(fact_5878_norm__pre__pure__rule1,axiom,
! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap @ A,Q: A > assn] :
( ( B2
=> ( hoare_hoare_triple @ A @ P @ F @ Q ) )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ F @ Q ) ) ).
% norm_pre_pure_rule1
thf(fact_5879_norm__pre__pure__rule2,axiom,
! [A: $tType,B2: $o,F: heap_Time_Heap @ A,Q: A > assn] :
( ( B2
=> ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F @ Q ) )
=> ( hoare_hoare_triple @ A @ ( pure_assn @ B2 ) @ F @ Q ) ) ).
% norm_pre_pure_rule2
thf(fact_5880_fi__rule,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap @ A,Q: A > assn,Ps3: assn,F2: assn] :
( ( hoare_hoare_triple @ A @ P @ C3 @ Q )
=> ( ( entails @ Ps3 @ ( times_times @ assn @ P @ F2 ) )
=> ( hoare_hoare_triple @ A @ Ps3 @ C3
@ ^ [X3: A] : ( times_times @ assn @ ( Q @ X3 ) @ F2 ) ) ) ) ).
% fi_rule
thf(fact_5881_cons__post__rulet,axiom,
! [A: $tType,P: assn,C3: heap_Time_Heap @ A,Q: A > assn,Q8: A > assn] :
( ( hoare_hoare_triple @ A @ P @ C3
@ ^ [R4: A] : ( times_times @ assn @ ( Q @ R4 ) @ ( top_top @ assn ) ) )
=> ( ! [X2: A] : ( entailst @ ( Q @ X2 ) @ ( Q8 @ X2 ) )
=> ( hoare_hoare_triple @ A @ P @ C3
@ ^ [R4: A] : ( times_times @ assn @ ( Q8 @ R4 ) @ ( top_top @ assn ) ) ) ) ) ).
% cons_post_rulet
thf(fact_5882_cons__pre__rulet,axiom,
! [A: $tType,P: assn,P7: assn,C3: heap_Time_Heap @ A,Q: A > assn] :
( ( entailst @ P @ P7 )
=> ( ( hoare_hoare_triple @ A @ P7 @ C3
@ ^ [R4: A] : ( times_times @ assn @ ( Q @ R4 ) @ ( top_top @ assn ) ) )
=> ( hoare_hoare_triple @ A @ P @ C3
@ ^ [R4: A] : ( times_times @ assn @ ( Q @ R4 ) @ ( top_top @ assn ) ) ) ) ) ).
% cons_pre_rulet
thf(fact_5883_cons__rulet,axiom,
! [A: $tType,P: assn,P7: assn,Q: A > assn,Q8: A > assn,C3: heap_Time_Heap @ A] :
( ( entailst @ P @ P7 )
=> ( ! [X2: A] : ( entailst @ ( Q @ X2 ) @ ( Q8 @ X2 ) )
=> ( ( hoare_hoare_triple @ A @ P7 @ C3
@ ^ [R4: A] : ( times_times @ assn @ ( Q @ R4 ) @ ( top_top @ assn ) ) )
=> ( hoare_hoare_triple @ A @ P @ C3
@ ^ [R4: A] : ( times_times @ assn @ ( Q8 @ R4 ) @ ( top_top @ assn ) ) ) ) ) ) ).
% cons_rulet
thf(fact_5884_if__rule__split,axiom,
! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap @ A,Q1: A > assn,G: heap_Time_Heap @ A,Q22: A > assn,Q: A > assn] :
( ( B2
=> ( hoare_hoare_triple @ A @ P @ F @ Q1 ) )
=> ( ( ~ B2
=> ( hoare_hoare_triple @ A @ P @ G @ Q22 ) )
=> ( ! [X2: A] : ( entails @ ( sup_sup @ assn @ ( times_times @ assn @ ( Q1 @ X2 ) @ ( pure_assn @ B2 ) ) @ ( times_times @ assn @ ( Q22 @ X2 ) @ ( pure_assn @ ~ B2 ) ) ) @ ( Q @ X2 ) )
=> ( hoare_hoare_triple @ A @ P @ ( if @ ( heap_Time_Heap @ A ) @ B2 @ F @ G ) @ Q ) ) ) ) ).
% if_rule_split
thf(fact_5885_upd__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,Xs: list @ A,A3: array @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( hoare_hoare_triple @ ( array @ A ) @ ( snga_assn @ A @ A3 @ Xs ) @ ( array_upd @ A @ I @ X @ A3 )
@ ^ [R4: array @ A] : ( times_times @ assn @ ( snga_assn @ A @ A3 @ ( list_update @ A @ Xs @ I @ X ) ) @ ( pure_assn @ ( R4 = A3 ) ) ) ) ) ) ).
% upd_rule
thf(fact_5886_make__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_make @ A @ N @ F )
@ ^ [R4: array @ A] : ( snga_assn @ A @ R4 @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% make_rule
thf(fact_5887_length__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,Xs: list @ A] :
( hoare_hoare_triple @ nat @ ( snga_assn @ A @ A3 @ Xs ) @ ( array_len @ A @ A3 )
@ ^ [R4: nat] :
( times_times @ assn @ ( snga_assn @ A @ A3 @ Xs )
@ ( pure_assn
@ ( R4
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% length_rule
thf(fact_5888_new__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_new @ A @ N @ X )
@ ^ [R4: array @ A] : ( snga_assn @ A @ R4 @ ( replicate @ A @ N @ X ) ) ) ) ).
% new_rule
thf(fact_5889_lookup__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: ref @ A,X: A] :
( hoare_hoare_triple @ A @ ( sngr_assn @ A @ P4 @ X ) @ ( ref_lookup @ A @ P4 )
@ ^ [R4: A] : ( times_times @ assn @ ( sngr_assn @ A @ P4 @ X ) @ ( pure_assn @ ( R4 = X ) ) ) ) ) ).
% lookup_rule
thf(fact_5890_of__list__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_of_list @ A @ Xs )
@ ^ [R4: array @ A] : ( snga_assn @ A @ R4 @ Xs ) ) ) ).
% of_list_rule
thf(fact_5891_freeze__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,Xs: list @ A] :
( hoare_hoare_triple @ ( list @ A ) @ ( snga_assn @ A @ A3 @ Xs ) @ ( array_freeze @ A @ A3 )
@ ^ [R4: list @ A] : ( times_times @ assn @ ( snga_assn @ A @ A3 @ Xs ) @ ( pure_assn @ ( R4 = Xs ) ) ) ) ) ).
% freeze_rule
thf(fact_5892_ref__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: A] :
( hoare_hoare_triple @ ( ref @ A ) @ ( one_one @ assn ) @ ( ref_ref @ A @ X )
@ ^ [R4: ref @ A] : ( sngr_assn @ A @ R4 @ X ) ) ) ).
% ref_rule
thf(fact_5893_return__sp__rule,axiom,
! [A: $tType,P: assn,X: A] :
( hoare_hoare_triple @ A @ P @ ( heap_Time_return @ A @ X )
@ ^ [R4: A] : ( times_times @ assn @ P @ ( pure_assn @ ( R4 = X ) ) ) ) ).
% return_sp_rule
thf(fact_5894_return__cons__rule,axiom,
! [A: $tType,P: assn,Q: A > assn,X: A] :
( ( entails @ P @ ( Q @ X ) )
=> ( hoare_hoare_triple @ A @ P @ ( heap_Time_return @ A @ X ) @ Q ) ) ).
% return_cons_rule
thf(fact_5895_raise__iff,axiom,
! [A: $tType,P: assn,S4: list @ char,Q: A > assn] :
( ( hoare_hoare_triple @ A @ P @ ( heap_Time_raise @ A @ S4 ) @ Q )
= ( P
= ( bot_bot @ assn ) ) ) ).
% raise_iff
thf(fact_5896_raise__rule,axiom,
! [A: $tType,S4: list @ char,Q: A > assn] : ( hoare_hoare_triple @ A @ ( bot_bot @ assn ) @ ( heap_Time_raise @ A @ S4 ) @ Q ) ).
% raise_rule
thf(fact_5897_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_5898_order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A @ ( ord_less_eq @ A ) ) ) ).
% order.partial_preordering_axioms
thf(fact_5899_partial__preordering__def,axiom,
! [A: $tType] :
( ( partial_preordering @ A )
= ( ^ [Less_eq2: A > A > $o] :
( ! [A4: A] : ( Less_eq2 @ A4 @ A4 )
& ! [A4: A,B3: A,C6: A] :
( ( Less_eq2 @ A4 @ B3 )
=> ( ( Less_eq2 @ B3 @ C6 )
=> ( Less_eq2 @ A4 @ C6 ) ) ) ) ) ) ).
% partial_preordering_def
thf(fact_5900_partial__preordering_Otrans,axiom,
! [A: $tType,Less_eq: A > A > $o,A3: A,B2: A,C3: A] :
( ( partial_preordering @ A @ Less_eq )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less_eq @ B2 @ C3 )
=> ( Less_eq @ A3 @ C3 ) ) ) ) ).
% partial_preordering.trans
thf(fact_5901_partial__preordering_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o] :
( ! [A6: A] : ( Less_eq @ A6 @ A6 )
=> ( ! [A6: A,B6: A,C2: A] :
( ( Less_eq @ A6 @ B6 )
=> ( ( Less_eq @ B6 @ C2 )
=> ( Less_eq @ A6 @ C2 ) ) )
=> ( partial_preordering @ A @ Less_eq ) ) ) ).
% partial_preordering.intro
thf(fact_5902_partial__preordering_Orefl,axiom,
! [A: $tType,Less_eq: A > A > $o,A3: A] :
( ( partial_preordering @ A @ Less_eq )
=> ( Less_eq @ A3 @ A3 ) ) ).
% partial_preordering.refl
thf(fact_5903_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_5904_dual__order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_5905_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_5906_wait__rule,axiom,
! [N: nat] :
( hoare_hoare_triple @ product_unit @ ( one_one @ assn ) @ ( heap_Time_wait @ N )
@ ^ [Uu3: product_unit] : ( one_one @ assn ) ) ).
% wait_rule
thf(fact_5907_update__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: ref @ A,Y: A,X: A] :
( hoare_hoare_triple @ product_unit @ ( sngr_assn @ A @ P4 @ Y ) @ ( ref_update @ A @ P4 @ X )
@ ^ [R4: product_unit] : ( sngr_assn @ A @ P4 @ X ) ) ) ).
% update_rule
thf(fact_5908_semilattice__neutr_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( semilattice @ A @ F )
=> ( ( comm_monoid @ A @ F @ Z2 )
=> ( semilattice_neutr @ A @ F @ Z2 ) ) ) ).
% semilattice_neutr.intro
thf(fact_5909_semilattice__neutr_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( semilattice_neutr @ A @ F @ Z2 )
=> ( semilattice @ A @ F ) ) ).
% semilattice_neutr.axioms(1)
thf(fact_5910_max_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice @ A @ ( ord_max @ A ) ) ) ).
% max.semilattice_axioms
thf(fact_5911_semilattice_Oidem,axiom,
! [A: $tType,F: A > A > A,A3: A] :
( ( semilattice @ A @ F )
=> ( ( F @ A3 @ A3 )
= A3 ) ) ).
% semilattice.idem
thf(fact_5912_semilattice_Oleft__idem,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A] :
( ( semilattice @ A @ F )
=> ( ( F @ A3 @ ( F @ A3 @ B2 ) )
= ( F @ A3 @ B2 ) ) ) ).
% semilattice.left_idem
thf(fact_5913_semilattice_Oright__idem,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A] :
( ( semilattice @ A @ F )
=> ( ( F @ ( F @ A3 @ B2 ) @ B2 )
= ( F @ A3 @ B2 ) ) ) ).
% semilattice.right_idem
thf(fact_5914_min_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice @ A @ ( ord_min @ A ) ) ) ).
% min.semilattice_axioms
thf(fact_5915_inf_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semilattice @ A @ ( inf_inf @ A ) ) ) ).
% inf.semilattice_axioms
thf(fact_5916_sup_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice @ A @ ( sup_sup @ A ) ) ) ).
% sup.semilattice_axioms
thf(fact_5917_semilattice__order_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( semilattice @ A @ F ) ) ).
% semilattice_order.axioms(1)
thf(fact_5918_semilattice__map2,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( semilattice @ ( list @ A )
@ ^ [Xs3: list @ A,Ys3: list @ A] : ( map @ ( product_prod @ A @ A ) @ A @ ( product_case_prod @ A @ A @ A @ F ) @ ( zip @ A @ A @ Xs3 @ Ys3 ) ) ) ) ).
% semilattice_map2
thf(fact_5919_semilattice__neutr__def,axiom,
! [A: $tType] :
( ( semilattice_neutr @ A )
= ( ^ [F3: A > A > A,Z3: A] :
( ( semilattice @ A @ F3 )
& ( comm_monoid @ A @ F3 @ Z3 ) ) ) ) ).
% semilattice_neutr_def
thf(fact_5920_update__wp__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,Y: A,X: A,Q: product_unit > assn] : ( hoare_hoare_triple @ product_unit @ ( times_times @ assn @ ( sngr_assn @ A @ R3 @ Y ) @ ( wand_assn @ ( sngr_assn @ A @ R3 @ X ) @ ( Q @ product_Unity ) ) ) @ ( ref_update @ A @ R3 @ X ) @ Q ) ) ).
% update_wp_rule
thf(fact_5921_semilattice__order__def,axiom,
! [A: $tType] :
( ( semilattice_order @ A )
= ( ^ [F3: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semilattice @ A @ F3 )
& ( semila6385135966242565138axioms @ A @ F3 @ Less_eq2 @ Less2 ) ) ) ) ).
% semilattice_order_def
thf(fact_5922_semilattice__order_Ointro,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice @ A @ F )
=> ( ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less )
=> ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ) ).
% semilattice_order.intro
thf(fact_5923_inf__unit__def,axiom,
( ( inf_inf @ product_unit )
= ( ^ [Uu4: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_5924_semilattice__order__axioms__def,axiom,
! [A: $tType] :
( ( semila6385135966242565138axioms @ A )
= ( ^ [F3: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ! [A4: A,B3: A] :
( ( Less_eq2 @ A4 @ B3 )
= ( A4
= ( F3 @ A4 @ B3 ) ) )
& ! [A4: A,B3: A] :
( ( Less2 @ A4 @ B3 )
= ( ( A4
= ( F3 @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ) ) ).
% semilattice_order_axioms_def
thf(fact_5925_semilattice__order__axioms_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,F: A > A > A,Less: A > A > $o] :
( ! [A6: A,B6: A] :
( ( Less_eq @ A6 @ B6 )
= ( A6
= ( F @ A6 @ B6 ) ) )
=> ( ! [A6: A,B6: A] :
( ( Less @ A6 @ B6 )
= ( ( A6
= ( F @ A6 @ B6 ) )
& ( A6 != B6 ) ) )
=> ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less ) ) ) ).
% semilattice_order_axioms.intro
thf(fact_5926_semilattice__order_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less ) ) ).
% semilattice_order.axioms(2)
thf(fact_5927_CODE__ABORT__def,axiom,
! [A: $tType] :
( ( cODE_ABORT @ A )
= ( ^ [F3: product_unit > A] : ( F3 @ product_Unity ) ) ) ).
% CODE_ABORT_def
thf(fact_5928_comm__monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( comm_monoid @ A @ F @ Z2 )
=> ( comm_monoid_axioms @ A @ F @ Z2 ) ) ).
% comm_monoid.axioms(2)
thf(fact_5929_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ! [A6: A] :
( ( F @ A6 @ Z2 )
= A6 )
=> ( comm_monoid_axioms @ A @ F @ Z2 ) ) ).
% comm_monoid_axioms.intro
thf(fact_5930_comm__monoid__axioms__def,axiom,
! [A: $tType] :
( ( comm_monoid_axioms @ A )
= ( ^ [F3: A > A > A,Z3: A] :
! [A4: A] :
( ( F3 @ A4 @ Z3 )
= A4 ) ) ) ).
% comm_monoid_axioms_def
thf(fact_5931_comm__monoid__def,axiom,
! [A: $tType] :
( ( comm_monoid @ A )
= ( ^ [F3: A > A > A,Z3: A] :
( ( abel_semigroup @ A @ F3 )
& ( comm_monoid_axioms @ A @ F3 @ Z3 ) ) ) ) ).
% comm_monoid_def
thf(fact_5932_comm__monoid_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( abel_semigroup @ A @ F )
=> ( ( comm_monoid_axioms @ A @ F @ Z2 )
=> ( comm_monoid @ A @ F @ Z2 ) ) ) ).
% comm_monoid.intro
thf(fact_5933_semilattice_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( semilattice_axioms @ A @ F ) ) ).
% semilattice.axioms(2)
thf(fact_5934_inf_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( abel_semigroup @ A @ ( inf_inf @ A ) ) ) ).
% inf.abel_semigroup_axioms
thf(fact_5935_abel__semigroup_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( semigroup @ A @ F ) ) ).
% abel_semigroup.axioms(1)
thf(fact_5936_sup_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( abel_semigroup @ A @ ( sup_sup @ A ) ) ) ).
% sup.abel_semigroup_axioms
thf(fact_5937_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( abel_semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.abel_semigroup_axioms
thf(fact_5938_min_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( abel_semigroup @ A @ ( ord_min @ A ) ) ) ).
% min.abel_semigroup_axioms
thf(fact_5939_max_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( abel_semigroup @ A @ ( ord_max @ A ) ) ) ).
% max.abel_semigroup_axioms
thf(fact_5940_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( abel_semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.abel_semigroup_axioms
thf(fact_5941_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F: A > A > A,B2: A,A3: A,C3: A] :
( ( abel_semigroup @ A @ F )
=> ( ( F @ B2 @ ( F @ A3 @ C3 ) )
= ( F @ A3 @ ( F @ B2 @ C3 ) ) ) ) ).
% abel_semigroup.left_commute
thf(fact_5942_abel__semigroup_Ocommute,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A] :
( ( abel_semigroup @ A @ F )
=> ( ( F @ A3 @ B2 )
= ( F @ B2 @ A3 ) ) ) ).
% abel_semigroup.commute
thf(fact_5943_semilattice__axioms__def,axiom,
! [A: $tType] :
( ( semilattice_axioms @ A )
= ( ^ [F3: A > A > A] :
! [A4: A] :
( ( F3 @ A4 @ A4 )
= A4 ) ) ) ).
% semilattice_axioms_def
thf(fact_5944_semilattice__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A6: A] :
( ( F @ A6 @ A6 )
= A6 )
=> ( semilattice_axioms @ A @ F ) ) ).
% semilattice_axioms.intro
thf(fact_5945_abstract__boolean__algebra_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( abel_semigroup @ A @ Disj ) ) ).
% abstract_boolean_algebra.axioms(2)
thf(fact_5946_abstract__boolean__algebra_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( abel_semigroup @ A @ Conj ) ) ).
% abstract_boolean_algebra.axioms(1)
thf(fact_5947_comm__monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( comm_monoid @ A @ F @ Z2 )
=> ( abel_semigroup @ A @ F ) ) ).
% comm_monoid.axioms(1)
thf(fact_5948_semilattice__def,axiom,
! [A: $tType] :
( ( semilattice @ A )
= ( ^ [F3: A > A > A] :
( ( abel_semigroup @ A @ F3 )
& ( semilattice_axioms @ A @ F3 ) ) ) ) ).
% semilattice_def
thf(fact_5949_semilattice_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( ( semilattice_axioms @ A @ F )
=> ( semilattice @ A @ F ) ) ) ).
% semilattice.intro
thf(fact_5950_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( abel_semigroup @ A @ F ) ) ).
% semilattice.axioms(1)
thf(fact_5951_abel__semigroup__def,axiom,
! [A: $tType] :
( ( abel_semigroup @ A )
= ( ^ [F3: A > A > A] :
( ( semigroup @ A @ F3 )
& ( abel_s757365448890700780axioms @ A @ F3 ) ) ) ) ).
% abel_semigroup_def
thf(fact_5952_abel__semigroup_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( semigroup @ A @ F )
=> ( ( abel_s757365448890700780axioms @ A @ F )
=> ( abel_semigroup @ A @ F ) ) ) ).
% abel_semigroup.intro
thf(fact_5953_abstract__boolean__algebra_Ointro,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( abel_semigroup @ A @ Conj )
=> ( ( abel_semigroup @ A @ Disj )
=> ( ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One ) ) ) ) ).
% abstract_boolean_algebra.intro
thf(fact_5954_abstract__boolean__algebra__axioms_Ointro,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,One: A,Zero: A,Compl: A > A] :
( ! [X2: A,Y2: A,Z4: A] :
( ( Conj @ X2 @ ( Disj @ Y2 @ Z4 ) )
= ( Disj @ ( Conj @ X2 @ Y2 ) @ ( Conj @ X2 @ Z4 ) ) )
=> ( ! [X2: A,Y2: A,Z4: A] :
( ( Disj @ X2 @ ( Conj @ Y2 @ Z4 ) )
= ( Conj @ ( Disj @ X2 @ Y2 ) @ ( Disj @ X2 @ Z4 ) ) )
=> ( ! [X2: A] :
( ( Conj @ X2 @ One )
= X2 )
=> ( ! [X2: A] :
( ( Disj @ X2 @ Zero )
= X2 )
=> ( ! [X2: A] :
( ( Conj @ X2 @ ( Compl @ X2 ) )
= Zero )
=> ( ! [X2: A] :
( ( Disj @ X2 @ ( Compl @ X2 ) )
= One )
=> ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One ) ) ) ) ) ) ) ).
% abstract_boolean_algebra_axioms.intro
thf(fact_5955_abstract__boolean__algebra__axioms__def,axiom,
! [A: $tType] :
( ( boolea6902313364301356556axioms @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One2: A] :
( ! [X3: A,Y3: A,Z3: A] :
( ( Conj2 @ X3 @ ( Disj2 @ Y3 @ Z3 ) )
= ( Disj2 @ ( Conj2 @ X3 @ Y3 ) @ ( Conj2 @ X3 @ Z3 ) ) )
& ! [X3: A,Y3: A,Z3: A] :
( ( Disj2 @ X3 @ ( Conj2 @ Y3 @ Z3 ) )
= ( Conj2 @ ( Disj2 @ X3 @ Y3 ) @ ( Disj2 @ X3 @ Z3 ) ) )
& ! [X3: A] :
( ( Conj2 @ X3 @ One2 )
= X3 )
& ! [X3: A] :
( ( Disj2 @ X3 @ Zero2 )
= X3 )
& ! [X3: A] :
( ( Conj2 @ X3 @ ( Compl2 @ X3 ) )
= Zero2 )
& ! [X3: A] :
( ( Disj2 @ X3 @ ( Compl2 @ X3 ) )
= One2 ) ) ) ) ).
% abstract_boolean_algebra_axioms_def
thf(fact_5956_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A6: A,B6: A] :
( ( F @ A6 @ B6 )
= ( F @ B6 @ A6 ) )
=> ( abel_s757365448890700780axioms @ A @ F ) ) ).
% abel_semigroup_axioms.intro
thf(fact_5957_abel__semigroup__axioms__def,axiom,
! [A: $tType] :
( ( abel_s757365448890700780axioms @ A )
= ( ^ [F3: A > A > A] :
! [A4: A,B3: A] :
( ( F3 @ A4 @ B3 )
= ( F3 @ B3 @ A4 ) ) ) ) ).
% abel_semigroup_axioms_def
thf(fact_5958_abel__semigroup_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( abel_s757365448890700780axioms @ A @ F ) ) ).
% abel_semigroup.axioms(2)
thf(fact_5959_abstract__boolean__algebra_Oaxioms_I3_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One ) ) ).
% abstract_boolean_algebra.axioms(3)
thf(fact_5960_abstract__boolean__algebra__def,axiom,
! [A: $tType] :
( ( boolea2506097494486148201lgebra @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One2: A] :
( ( abel_semigroup @ A @ Conj2 )
& ( abel_semigroup @ A @ Disj2 )
& ( boolea6902313364301356556axioms @ A @ Conj2 @ Disj2 @ Compl2 @ Zero2 @ One2 ) ) ) ) ).
% abstract_boolean_algebra_def
thf(fact_5961_dual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ ^ [X3: A,Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% dual_order.preordering_axioms
thf(fact_5962_relH__def,axiom,
( relH
= ( ^ [As3: set @ nat,H2: heap_ext @ product_unit,H8: heap_ext @ product_unit] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As3 ) )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As3 ) )
& ! [T2: typerep,X3: nat] :
( ( member2 @ nat @ X3 @ As3 )
=> ( ( ( refs @ product_unit @ H2 @ T2 @ X3 )
= ( refs @ product_unit @ H8 @ T2 @ X3 ) )
& ( ( arrays @ product_unit @ H2 @ T2 @ X3 )
= ( arrays @ product_unit @ H8 @ T2 @ X3 ) ) ) ) ) ) ) ).
% relH_def
thf(fact_5963_preordering__dualI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( preordering @ A
@ ^ [A4: A,B3: A] : ( Less_eq @ B3 @ A4 )
@ ^ [A4: A,B3: A] : ( Less @ B3 @ A4 ) )
=> ( preordering @ A @ Less_eq @ Less ) ) ).
% preordering_dualI
thf(fact_5964_preordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A6: A,B6: A] :
( ( Less_eq @ A6 @ B6 )
= ( ( Less @ A6 @ B6 )
| ( A6 = B6 ) ) )
=> ( ! [A6: A,B6: A] :
( ( Less @ A6 @ B6 )
=> ~ ( Less @ B6 @ A6 ) )
=> ( ! [A6: A] :
~ ( Less @ A6 @ A6 )
=> ( ! [A6: A,B6: A,C2: A] :
( ( Less @ A6 @ B6 )
=> ( ( Less @ B6 @ C2 )
=> ( Less @ A6 @ C2 ) ) )
=> ( preordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% preordering_strictI
thf(fact_5965_preordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( Less_eq @ A3 @ B2 ) ) ) ).
% preordering.strict_implies_order
thf(fact_5966_preordering_Ostrict__iff__not,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( Less_eq @ A3 @ B2 )
& ~ ( Less_eq @ B2 @ A3 ) ) ) ) ).
% preordering.strict_iff_not
thf(fact_5967_preordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( Less_eq @ B2 @ C3 )
=> ( Less @ A3 @ C3 ) ) ) ) ).
% preordering.strict_trans2
thf(fact_5968_preordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less @ B2 @ C3 )
=> ( Less @ A3 @ C3 ) ) ) ) ).
% preordering.strict_trans1
thf(fact_5969_preordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( Less @ B2 @ C3 )
=> ( Less @ A3 @ C3 ) ) ) ) ).
% preordering.strict_trans
thf(fact_5970_preordering_Oirrefl,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ~ ( Less @ A3 @ A3 ) ) ).
% preordering.irrefl
thf(fact_5971_preordering_Oasym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ~ ( Less @ B2 @ A3 ) ) ) ).
% preordering.asym
thf(fact_5972_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_5973_order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.preordering_axioms
thf(fact_5974_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_5975_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_5976_preordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A6: A,B6: A] :
( ( Less @ A6 @ B6 )
= ( ( Less_eq @ A6 @ B6 )
& ~ ( Less_eq @ B6 @ A6 ) ) )
=> ( preordering_axioms @ A @ Less_eq @ Less ) ) ).
% preordering_axioms.intro
thf(fact_5977_preordering__axioms__def,axiom,
! [A: $tType] :
( ( preordering_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
! [A4: A,B3: A] :
( ( Less2 @ A4 @ B3 )
= ( ( Less_eq2 @ A4 @ B3 )
& ~ ( Less_eq2 @ B3 @ A4 ) ) ) ) ) ).
% preordering_axioms_def
thf(fact_5978_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_5979_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X3: nat] : ( ord_less @ nat @ X3 @ N ) )
@ ^ [Uu3: nat] :
( collect @ nat
@ ^ [X3: nat] : ( ord_less @ nat @ X3 @ N ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less @ nat @ X3 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X3 @ Y3 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_5980_Restr__natLeq2,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
@ ^ [Uu3: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X3: nat,Y3: nat] :
( ( ord_less @ nat @ X3 @ N )
& ( ord_less @ nat @ Y3 @ N )
& ( ord_less_eq @ nat @ X3 @ Y3 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_5981_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( Y
= ( ! [X3: nat] :
( ( member2 @ nat @ X3 @ As )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H4 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) ) ) ) ) ) ).
% in_range.pelims(1)
thf(fact_5982_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ~ ! [X6: nat] :
( ( member2 @ nat @ X6 @ As )
=> ( ord_less @ nat @ X6 @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ).
% in_range.pelims(2)
thf(fact_5983_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ! [X2: nat] :
( ( member2 @ nat @ X2 @ As )
=> ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ).
% in_range.pelims(3)
thf(fact_5984_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( As
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% one_assn_raw.pelims(3)
thf(fact_5985_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( As
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% one_assn_raw.pelims(2)
thf(fact_5986_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 )
=> ~ ! [H4: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As ) )
=> ( ( Y
= ( As
= ( 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 ) @ H4 @ As ) ) ) ) ) ) ).
% one_assn_raw.pelims(1)
thf(fact_5987_lenlex__def,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R4 ) )
@ ^ [Xs3: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs3 ) @ Xs3 ) ) ) ) ).
% lenlex_def
thf(fact_5988_bind__rule_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: assn,F: heap_Time_Heap @ B,R: B > assn,G: B > ( heap_Time_Heap @ C ),Q: C > assn] :
( ! [R7: A] :
( hoare_hoare_triple @ B @ P @ F
@ ^ [S6: B] : ( times_times @ assn @ P @ ( R @ S6 ) ) )
=> ( ! [X2: B] : ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( R @ X2 ) @ P ) @ ( G @ X2 ) @ Q )
=> ( hoare_hoare_triple @ C @ P @ ( heap_Time_bind @ B @ C @ F @ G ) @ Q ) ) ) ).
% bind_rule'
thf(fact_5989_measures__def,axiom,
! [A: $tType] :
( ( measures @ A )
= ( ^ [Fs2: list @ ( A > nat )] :
( inv_image @ ( list @ nat ) @ A @ ( lex @ nat @ less_than )
@ ^ [A4: A] :
( map @ ( A > nat ) @ nat
@ ^ [F3: A > nat] : ( F3 @ A4 )
@ Fs2 ) ) ) ) ).
% measures_def
thf(fact_5990_ccpo_Oadmissible__def,axiom,
! [A: $tType] :
( ( comple1908693960933563346ssible @ A )
= ( ^ [Lub: ( set @ A ) > A,Ord2: A > A > $o,P3: A > $o] :
! [A7: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord2 @ A7 )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member2 @ A @ X3 @ A7 )
=> ( P3 @ X3 ) )
=> ( P3 @ ( Lub @ A7 ) ) ) ) ) ) ) ).
% ccpo.admissible_def
thf(fact_5991_ccpo_OadmissibleD,axiom,
! [A: $tType,Lub2: ( set @ A ) > A,Ord: A > A > $o,P: A > $o,A5: set @ A] :
( ( comple1908693960933563346ssible @ A @ Lub2 @ Ord @ P )
=> ( ( comple1602240252501008431_chain @ A @ Ord @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ A5 )
=> ( P @ X2 ) )
=> ( P @ ( Lub2 @ A5 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_5992_ccpo_OadmissibleI,axiom,
! [A: $tType,Ord: A > A > $o,P: A > $o,Lub2: ( set @ A ) > A] :
( ! [A9: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X6: A] :
( ( member2 @ A @ X6 @ A9 )
=> ( P @ X6 ) )
=> ( P @ ( Lub2 @ A9 ) ) ) ) )
=> ( comple1908693960933563346ssible @ A @ Lub2 @ Ord @ P ) ) ).
% ccpo.admissibleI
thf(fact_5993_fixp__induct,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o,F: A > A] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( ( P @ ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( P @ ( comple115746919287870866o_fixp @ A @ F ) ) ) ) ) ) ) ).
% fixp_induct
thf(fact_5994_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_5995_fixp__mono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A,G: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ G )
=> ( ! [Z9: A] : ( ord_less_eq @ A @ ( F @ Z9 ) @ ( G @ Z9 ) )
=> ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F ) @ ( comple115746919287870866o_fixp @ A @ G ) ) ) ) ) ) ).
% fixp_mono
thf(fact_5996_prod__set__defs_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( basic_fsts @ A @ B )
= ( ^ [P6: product_prod @ A @ B] : ( insert2 @ A @ ( product_fst @ A @ B @ P6 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% prod_set_defs(1)
thf(fact_5997_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_5998_prod__set__defs_I2_J,axiom,
! [D: $tType,C: $tType] :
( ( basic_snds @ C @ D )
= ( ^ [P6: product_prod @ C @ D] : ( insert2 @ D @ ( product_snd @ C @ D @ P6 ) @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% prod_set_defs(2)
thf(fact_5999_enumerate__Suc__eq,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( enumerate @ A @ ( suc @ N ) @ Xs )
= ( map @ ( product_prod @ nat @ A ) @ ( product_prod @ nat @ A ) @ ( product_apfst @ nat @ nat @ A @ suc ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% enumerate_Suc_eq
thf(fact_6000_select,axiom,
! [A: $tType,Xs: list @ A,S4: product_prod @ code_natural @ code_natural] :
( ( Xs
!= ( nil @ A ) )
=> ( member2 @ A @ ( product_fst @ A @ ( product_prod @ code_natural @ code_natural ) @ ( select @ A @ Xs @ S4 ) ) @ ( set2 @ A @ Xs ) ) ) ).
% select
thf(fact_6001_select__weight__select,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( select_weight @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) )
= ( select @ A @ Xs ) ) ) ).
% select_weight_select
thf(fact_6002_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_6003_those_Osimps_I1_J,axiom,
! [A: $tType] :
( ( those @ A @ ( nil @ ( option @ A ) ) )
= ( some @ ( list @ A ) @ ( nil @ A ) ) ) ).
% those.simps(1)
thf(fact_6004_bot__in__iterates,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A] : ( member2 @ A @ ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( comple6359979572994053840erates @ A @ F ) ) ) ).
% bot_in_iterates
thf(fact_6005_those_Osimps_I2_J,axiom,
! [A: $tType,X: option @ A,Xs: list @ ( option @ A )] :
( ( those @ A @ ( cons @ ( option @ A ) @ X @ Xs ) )
= ( case_option @ ( option @ ( list @ A ) ) @ A @ ( none @ ( list @ A ) )
@ ^ [Y3: A] : ( map_option @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( those @ A @ Xs ) )
@ X ) ) ).
% those.simps(2)
thf(fact_6006_sym__Int__converse,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( sym @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( converse @ A @ A @ R3 ) ) ) ).
% sym_Int_converse
thf(fact_6007_listrel__sym,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( sym @ A @ R3 )
=> ( sym @ ( list @ A ) @ ( listrel @ A @ A @ R3 ) ) ) ).
% listrel_sym
thf(fact_6008_sym__Int,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A )] :
( ( sym @ A @ R3 )
=> ( ( sym @ A @ S4 )
=> ( sym @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S4 ) ) ) ) ).
% sym_Int
thf(fact_6009_ac__operator_Osafe__commute,axiom,
! [A: $tType,F: A > A > A,X: A,Y: A,A3: A,B2: A] :
( ( syntax_ac_operator @ A @ F )
=> ( ( syntax7388354845996824322omatch @ A @ A @ ( F @ X @ Y ) @ A3 )
=> ( ( F @ A3 @ B2 )
= ( F @ B2 @ A3 ) ) ) ) ).
% ac_operator.safe_commute
thf(fact_6010_Nitpick_OEx1__unfold,axiom,
! [A: $tType] :
( ( ex1 @ A )
= ( ^ [P3: A > $o] :
? [X3: A] :
( ( collect @ A @ P3 )
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Nitpick.Ex1_unfold
thf(fact_6011_ac__operator_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A6: A,B6: A,C2: A] :
( ( F @ ( F @ A6 @ B6 ) @ C2 )
= ( F @ A6 @ ( F @ B6 @ C2 ) ) )
=> ( ! [A6: A,B6: A] :
( ( F @ A6 @ B6 )
= ( F @ B6 @ A6 ) )
=> ( syntax_ac_operator @ A @ F ) ) ) ).
% ac_operator.intro
thf(fact_6012_ac__operator_Ocommute,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A] :
( ( syntax_ac_operator @ A @ F )
=> ( ( F @ A3 @ B2 )
= ( F @ B2 @ A3 ) ) ) ).
% ac_operator.commute
thf(fact_6013_ac__operator_Oleft__assoc,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A,C3: A] :
( ( syntax_ac_operator @ A @ F )
=> ( ( F @ A3 @ ( F @ B2 @ C3 ) )
= ( F @ ( F @ A3 @ B2 ) @ C3 ) ) ) ).
% ac_operator.left_assoc
thf(fact_6014_ac__operator_Oright__assoc,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A,C3: A] :
( ( syntax_ac_operator @ A @ F )
=> ( ( F @ ( F @ A3 @ B2 ) @ C3 )
= ( F @ A3 @ ( F @ B2 @ C3 ) ) ) ) ).
% ac_operator.right_assoc
thf(fact_6015_ac__operator_Oleft__commute,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A,C3: A] :
( ( syntax_ac_operator @ A @ F )
=> ( ( F @ A3 @ ( F @ B2 @ C3 ) )
= ( F @ B2 @ ( F @ A3 @ C3 ) ) ) ) ).
% ac_operator.left_commute
thf(fact_6016_ac__operator_Oright__commute,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A,C3: A] :
( ( syntax_ac_operator @ A @ F )
=> ( ( F @ ( F @ A3 @ B2 ) @ C3 )
= ( F @ ( F @ A3 @ C3 ) @ B2 ) ) ) ).
% ac_operator.right_commute
thf(fact_6017_ac__operator__def,axiom,
! [A: $tType] :
( ( syntax_ac_operator @ A )
= ( ^ [F3: A > A > A] :
( ! [A4: A,B3: A,C6: A] :
( ( F3 @ ( F3 @ A4 @ B3 ) @ C6 )
= ( F3 @ A4 @ ( F3 @ B3 @ C6 ) ) )
& ! [A4: A,B3: A] :
( ( F3 @ A4 @ B3 )
= ( F3 @ B3 @ A4 ) ) ) ) ) ).
% ac_operator_def
thf(fact_6018_mult_Oac__operator__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( syntax_ac_operator @ A @ ( times_times @ A ) ) ) ).
% mult.ac_operator_axioms
thf(fact_6019_add_Oac__operator__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( syntax_ac_operator @ A @ ( plus_plus @ A ) ) ) ).
% add.ac_operator_axioms
thf(fact_6020_semilattice__neutr__set_Oinsert__remove,axiom,
! [A: $tType,F: A > A > A,Z2: A,A5: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F @ Z2 @ ( insert2 @ A @ X @ A5 ) )
= ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% semilattice_neutr_set.insert_remove
thf(fact_6021_semilattice__neutr__set_Oremove,axiom,
! [A: $tType,F: A > A > A,Z2: A,A5: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( member2 @ A @ X @ A5 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F @ Z2 @ A5 )
= ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% semilattice_neutr_set.remove
thf(fact_6022_semilattice__neutr__set_Oclosed,axiom,
! [A: $tType,F: A > A > A,Z2: A,A5: set @ A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] : ( member2 @ A @ ( F @ X2 @ Y2 ) @ ( insert2 @ A @ X2 @ ( insert2 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member2 @ A @ ( lattic5214292709420241887eutr_F @ A @ F @ Z2 @ A5 ) @ A5 ) ) ) ) ) ).
% semilattice_neutr_set.closed
thf(fact_6023_semilattice__neutr__set_Oempty,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z2 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F @ Z2 @ ( bot_bot @ ( set @ A ) ) )
= Z2 ) ) ).
% semilattice_neutr_set.empty
thf(fact_6024_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_6025_null__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A5 )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5
@ ( null @ A )
@ ( null @ B ) ) ).
% null_transfer
thf(fact_6026_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_6027_of__rat__eq__1__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( field_char_0_of_rat @ A @ A3 )
= ( one_one @ A ) )
= ( A3
= ( one_one @ rat ) ) ) ) ).
% of_rat_eq_1_iff
thf(fact_6028_one__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( one_one @ A )
= ( field_char_0_of_rat @ A @ A3 ) )
= ( ( one_one @ rat )
= A3 ) ) ) ).
% one_eq_of_rat_iff
thf(fact_6029_one__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R3 ) )
= ( ord_less_eq @ rat @ ( one_one @ rat ) @ R3 ) ) ) ).
% one_le_of_rat_iff
thf(fact_6030_of__rat__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ rat @ R3 @ ( one_one @ rat ) ) ) ) ).
% of_rat_le_1_iff
thf(fact_6031_one__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R3 ) )
= ( ord_less @ rat @ ( one_one @ rat ) @ R3 ) ) ) ).
% one_less_of_rat_iff
thf(fact_6032_of__rat__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( one_one @ A ) )
= ( ord_less @ rat @ R3 @ ( one_one @ rat ) ) ) ) ).
% of_rat_less_1_iff
thf(fact_6033_list_Omap__transfer,axiom,
! [A: $tType,B: $tType,F6: $tType,E4: $tType,Rb: A > E4 > $o,Sd: B > F6 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E4 > F6 ) @ ( ( list @ A ) > ( list @ B ) ) @ ( ( list @ E4 ) > ( list @ F6 ) ) @ ( bNF_rel_fun @ A @ E4 @ B @ F6 @ Rb @ Sd ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ E4 ) @ ( list @ B ) @ ( list @ F6 ) @ ( list_all2 @ A @ E4 @ Rb ) @ ( list_all2 @ B @ F6 @ Sd ) ) @ ( map @ A @ B ) @ ( map @ E4 @ F6 ) ) ).
% list.map_transfer
thf(fact_6034_foldr__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A5: A > C > $o,B4: B > D > $o] : ( bNF_rel_fun @ ( A > B > B ) @ ( C > D > D ) @ ( ( list @ A ) > B > B ) @ ( ( list @ C ) > D > D ) @ ( bNF_rel_fun @ A @ C @ ( B > B ) @ ( D > D ) @ A5 @ ( bNF_rel_fun @ B @ D @ B @ D @ B4 @ B4 ) ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ C ) @ ( B > B ) @ ( D > D ) @ ( list_all2 @ A @ C @ A5 ) @ ( bNF_rel_fun @ B @ D @ B @ D @ B4 @ B4 ) ) @ ( foldr @ A @ B ) @ ( foldr @ C @ D ) ) ).
% foldr_transfer
thf(fact_6035_of__rat__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat,B2: rat] :
( ( field_char_0_of_rat @ A @ ( times_times @ rat @ A3 @ B2 ) )
= ( times_times @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% of_rat_mult
thf(fact_6036_foldl__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,B4: A > C > $o,A5: B > D > $o] : ( bNF_rel_fun @ ( A > B > A ) @ ( C > D > C ) @ ( A > ( list @ B ) > A ) @ ( C > ( list @ D ) > C ) @ ( bNF_rel_fun @ A @ C @ ( B > A ) @ ( D > C ) @ B4 @ ( bNF_rel_fun @ B @ D @ A @ C @ A5 @ B4 ) ) @ ( bNF_rel_fun @ A @ C @ ( ( list @ B ) > A ) @ ( ( list @ D ) > C ) @ B4 @ ( bNF_rel_fun @ ( list @ B ) @ ( list @ D ) @ A @ C @ ( list_all2 @ B @ D @ A5 ) @ B4 ) ) @ ( foldl @ A @ B ) @ ( foldl @ C @ D ) ) ).
% foldl_transfer
thf(fact_6037_List_Ofold__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A5: A > C > $o,B4: B > D > $o] : ( bNF_rel_fun @ ( A > B > B ) @ ( C > D > D ) @ ( ( list @ A ) > B > B ) @ ( ( list @ C ) > D > D ) @ ( bNF_rel_fun @ A @ C @ ( B > B ) @ ( D > D ) @ A5 @ ( bNF_rel_fun @ B @ D @ B @ D @ B4 @ B4 ) ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ C ) @ ( B > B ) @ ( D > D ) @ ( list_all2 @ A @ C @ A5 ) @ ( bNF_rel_fun @ B @ D @ B @ D @ B4 @ B4 ) ) @ ( fold @ A @ B ) @ ( fold @ C @ D ) ) ).
% List.fold_transfer
thf(fact_6038_list_Orec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > ( list @ A ) > C > C ) > ( list @ A ) > C ) @ ( ( B > ( list @ B ) > D > D ) > ( list @ B ) > D ) @ S @ ( bNF_rel_fun @ ( A > ( list @ A ) > C > C ) @ ( B > ( list @ B ) > D > D ) @ ( ( list @ A ) > C ) @ ( ( list @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > C > C ) @ ( ( list @ B ) > D > D ) @ R @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( C > C ) @ ( D > D ) @ ( list_all2 @ A @ B @ R ) @ ( bNF_rel_fun @ C @ D @ C @ D @ S @ S ) ) ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ C @ D @ ( list_all2 @ A @ B @ R ) @ S ) ) @ ( rec_list @ C @ A ) @ ( rec_list @ D @ B ) ) ).
% list.rec_transfer
thf(fact_6039_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_mult @ B )
& ( monoid_mult @ A ) )
=> ! [A5: A > B > $o] :
( ( A5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A5 @ ( bNF_rel_fun @ A @ B @ A @ B @ A5 @ A5 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A5 ) @ A5 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% prod_list_transfer
thf(fact_6040_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A5: A > B > $o,B4: C > D > $o] :
( ( A5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A5 @ ( bNF_rel_fun @ A @ B @ A @ B @ A5 @ A5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A5 @ ( bNF_rel_fun @ A @ B @ A @ B @ A5 @ A5 ) @ ( 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 @ B4 @ A5 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A5 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B4 ) @ A5 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_6041_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y5: num,Z5: num] : Y5 = Z5
@ R
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_6042_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R @ R @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y5: int,Z5: int] : Y5 = Z5
@ R
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_6043_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ R )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_6044_list_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5
@ ^ [List2: list @ A] :
( List2
!= ( nil @ A ) )
@ ^ [List2: list @ B] :
( List2
!= ( nil @ B ) ) ) ).
% list.disc_transfer(2)
thf(fact_6045_list_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5
@ ^ [List2: list @ A] :
( List2
= ( nil @ A ) )
@ ^ [List2: list @ B] :
( List2
= ( nil @ B ) ) ) ).
% list.disc_transfer(1)
thf(fact_6046_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ R
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_6047_list__ex__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > $o ) @ ( ( list @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A5 )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( list_ex @ A )
@ ( list_ex @ B ) ) ).
% list_ex_transfer
thf(fact_6048_list_Octr__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ R @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ R ) @ ( list_all2 @ A @ B @ R ) ) @ ( cons @ A ) @ ( cons @ B ) ) ).
% list.ctr_transfer(2)
thf(fact_6049_subseqs__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ ( list @ A ) ) @ ( list @ ( list @ B ) ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( subseqs @ A ) @ ( subseqs @ B ) ) ).
% subseqs_transfer
thf(fact_6050_length__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ nat @ nat @ ( list_all2 @ A @ B @ A5 )
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ( size_size @ ( list @ A ) )
@ ( size_size @ ( list @ B ) ) ) ).
% length_transfer
thf(fact_6051_successively__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( list @ A ) > $o ) @ ( ( list @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A5
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 ) )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A5 )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( successively @ A )
@ ( successively @ B ) ) ).
% successively_transfer
thf(fact_6052_splice__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ ( list_all2 @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( splice @ A ) @ ( splice @ B ) ) ).
% splice_transfer
thf(fact_6053_replicate__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( A > ( list @ A ) ) @ ( B > ( list @ B ) )
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ( bNF_rel_fun @ A @ B @ ( list @ A ) @ ( list @ B ) @ A5 @ ( list_all2 @ A @ B @ A5 ) )
@ ( replicate @ A )
@ ( replicate @ B ) ) ).
% replicate_transfer
thf(fact_6054_rotate__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) )
@ ( rotate @ A )
@ ( rotate @ B ) ) ).
% rotate_transfer
thf(fact_6055_list_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( list @ A ) > ( list @ B ) > $o ) @ ( ( list @ C ) > ( list @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 ) )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ C ) @ ( ( list @ B ) > $o ) @ ( ( list @ D ) > $o ) @ ( list_all2 @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( list @ B ) @ ( list @ D ) @ $o @ $o @ ( list_all2 @ B @ D @ Sc )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 ) )
@ ( list_all2 @ A @ B )
@ ( list_all2 @ C @ D ) ) ).
% list.rel_transfer
thf(fact_6056_dropWhile__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) )
@ ( dropWhile @ A )
@ ( dropWhile @ B ) ) ).
% dropWhile_transfer
thf(fact_6057_list__update__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( nat > A > ( list @ A ) ) @ ( nat > B > ( list @ B ) ) @ ( list_all2 @ A @ B @ A5 )
@ ( bNF_rel_fun @ nat @ nat @ ( A > ( list @ A ) ) @ ( B > ( list @ B ) )
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ( bNF_rel_fun @ A @ B @ ( list @ A ) @ ( list @ B ) @ A5 @ ( list_all2 @ A @ B @ A5 ) ) )
@ ( list_update @ A )
@ ( list_update @ B ) ) ).
% list_update_transfer
thf(fact_6058_take__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) )
@ ( take @ A )
@ ( take @ B ) ) ).
% take_transfer
thf(fact_6059_drop__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) )
@ ( drop @ A )
@ ( drop @ B ) ) ).
% drop_transfer
thf(fact_6060_append__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ ( list_all2 @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( append @ A ) @ ( append @ B ) ) ).
% append_transfer
thf(fact_6061_List_Ofilter__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) )
@ ( filter2 @ A )
@ ( filter2 @ B ) ) ).
% List.filter_transfer
thf(fact_6062_takeWhile__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) )
@ ( takeWhile @ A )
@ ( takeWhile @ B ) ) ).
% takeWhile_transfer
thf(fact_6063_tl__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) @ ( tl @ A ) @ ( tl @ B ) ) ).
% tl_transfer
thf(fact_6064_rev__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) @ ( rev @ A ) @ ( rev @ B ) ) ).
% rev_transfer
thf(fact_6065_concat__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( list @ ( list @ A ) ) @ ( list @ ( list @ B ) ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( list_all2 @ A @ B @ A5 ) @ ( concat @ A ) @ ( concat @ B ) ) ).
% concat_transfer
thf(fact_6066_butlast__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) @ ( butlast @ A ) @ ( butlast @ B ) ) ).
% butlast_transfer
thf(fact_6067_rotate1__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) @ ( rotate1 @ A ) @ ( rotate1 @ B ) ) ).
% rotate1_transfer
thf(fact_6068_list_Ocase__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S: C > D > $o,R: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > ( list @ A ) > C ) > ( list @ A ) > C ) @ ( ( B > ( list @ B ) > D ) > ( list @ B ) > D ) @ S @ ( bNF_rel_fun @ ( A > ( list @ A ) > C ) @ ( B > ( list @ B ) > D ) @ ( ( list @ A ) > C ) @ ( ( list @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > C ) @ ( ( list @ B ) > D ) @ R @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ C @ D @ ( list_all2 @ A @ B @ R ) @ S ) ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ C @ D @ ( list_all2 @ A @ B @ R ) @ S ) ) @ ( case_list @ C @ A ) @ ( case_list @ D @ B ) ) ).
% list.case_transfer
thf(fact_6069_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5
@ R
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_6070_product__lists__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( list @ ( list @ A ) ) @ ( list @ ( list @ B ) ) @ ( list @ ( list @ A ) ) @ ( list @ ( list @ B ) ) @ ( list_all2 @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( list_all2 @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( product_lists @ A ) @ ( product_lists @ B ) ) ).
% product_lists_transfer
thf(fact_6071_list_Opred__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > $o ) @ ( ( list @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( list_all @ A )
@ ( list_all @ B ) ) ).
% list.pred_transfer
thf(fact_6072_list__all__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > $o ) @ ( ( list @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A5 )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( list_all @ A )
@ ( list_all @ B ) ) ).
% list_all_transfer
thf(fact_6073_List_Oinsert__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ A5 @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( insert @ A ) @ ( insert @ B ) ) ) ).
% List.insert_transfer
thf(fact_6074_removeAll__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ A5 @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( removeAll @ A ) @ ( removeAll @ B ) ) ) ).
% removeAll_transfer
thf(fact_6075_list_Obi__unique__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_unique @ A @ B @ R )
=> ( bi_unique @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ R ) ) ) ).
% list.bi_unique_rel
thf(fact_6076_remdups__adj__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) @ ( remdups_adj @ A ) @ ( remdups_adj @ B ) ) ) ).
% remdups_adj_transfer
thf(fact_6077_remdups__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) @ ( remdups @ A ) @ ( remdups @ B ) ) ) ).
% remdups_transfer
thf(fact_6078_distinct__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A5 )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5
@ ( distinct @ A )
@ ( distinct @ B ) ) ) ).
% distinct_transfer
thf(fact_6079_distinct__adj__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A5 )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5
@ ( distinct_adj @ A )
@ ( distinct_adj @ B ) ) ) ).
% distinct_adj_transfer
thf(fact_6080_remove1__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ A5 @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( remove1 @ A ) @ ( remove1 @ B ) ) ) ).
% remove1_transfer
thf(fact_6081_monotone__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B4: C > D > $o] :
( ( bi_total @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( A > A > $o ) @ ( B > B > $o ) @ ( ( C > C > $o ) > ( A > C ) > $o ) @ ( ( D > D > $o ) > ( B > D ) > $o )
@ ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A5
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 ) )
@ ( bNF_rel_fun @ ( C > C > $o ) @ ( D > D > $o ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o )
@ ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B4
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B4
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B4 )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 ) )
@ ( comple7038119648293358887notone @ A @ C )
@ ( comple7038119648293358887notone @ B @ D ) ) ) ).
% monotone_parametric
thf(fact_6082_partition__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( ( list @ B ) > ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( list_all2 @ A @ B @ A5 ) @ ( basic_rel_prod @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( list_all2 @ A @ B @ A5 ) ) )
@ ( partition @ A )
@ ( partition @ B ) ) ).
% partition_transfer
thf(fact_6083_zip__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B4: C > D > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( list @ C ) > ( list @ ( product_prod @ A @ C ) ) ) @ ( ( list @ D ) > ( list @ ( product_prod @ B @ D ) ) ) @ ( list_all2 @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ ( list @ ( product_prod @ A @ C ) ) @ ( list @ ( product_prod @ B @ D ) ) @ ( list_all2 @ C @ D @ B4 ) @ ( list_all2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( basic_rel_prod @ A @ B @ C @ D @ A5 @ B4 ) ) ) @ ( zip @ A @ C ) @ ( zip @ B @ D ) ) ).
% zip_transfer
thf(fact_6084_list_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( bi_total @ A @ B @ R )
=> ( bi_total @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ R ) ) ) ).
% list.bi_total_rel
thf(fact_6085_product__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B4: C > D > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( list @ C ) > ( list @ ( product_prod @ A @ C ) ) ) @ ( ( list @ D ) > ( list @ ( product_prod @ B @ D ) ) ) @ ( list_all2 @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ ( list @ ( product_prod @ A @ C ) ) @ ( list @ ( product_prod @ B @ D ) ) @ ( list_all2 @ C @ D @ B4 ) @ ( list_all2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( basic_rel_prod @ A @ B @ C @ D @ A5 @ B4 ) ) ) @ ( product @ A @ C ) @ ( product @ B @ D ) ) ).
% product_transfer
thf(fact_6086_fun__ord__parametric,axiom,
! [C: $tType,D: $tType,A: $tType,B: $tType,F6: $tType,E4: $tType,C4: A > B > $o,A5: C > E4 > $o,B4: D > F6 > $o] :
( ( bi_total @ A @ B @ C4 )
=> ( bNF_rel_fun @ ( C > D > $o ) @ ( E4 > F6 > $o ) @ ( ( A > C ) > ( A > D ) > $o ) @ ( ( B > E4 ) > ( B > F6 ) > $o )
@ ( bNF_rel_fun @ C @ E4 @ ( D > $o ) @ ( F6 > $o ) @ A5
@ ( bNF_rel_fun @ D @ F6 @ $o @ $o @ B4
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > E4 ) @ ( ( A > D ) > $o ) @ ( ( B > F6 ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ E4 @ C4 @ A5 )
@ ( bNF_rel_fun @ ( A > D ) @ ( B > F6 ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ D @ F6 @ C4 @ B4 )
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 ) )
@ ( partial_fun_ord @ C @ D @ A )
@ ( partial_fun_ord @ E4 @ F6 @ B ) ) ) ).
% fun_ord_parametric
thf(fact_6087_rtrancl__parametric,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( ( bi_total @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_rel_set @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( basic_rel_prod @ A @ B @ A @ B @ A5 @ A5 ) ) @ ( bNF_rel_set @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( basic_rel_prod @ A @ B @ A @ B @ A5 @ A5 ) ) @ ( transitive_rtrancl @ A ) @ ( transitive_rtrancl @ B ) ) ) ) ).
% rtrancl_parametric
thf(fact_6088_list_Oset__transfer,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( set @ A ) @ ( set @ B ) @ ( list_all2 @ A @ B @ R ) @ ( bNF_rel_set @ A @ B @ R ) @ ( set2 @ A ) @ ( set2 @ B ) ) ).
% list.set_transfer
thf(fact_6089_listset__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( list @ ( set @ A ) ) @ ( list @ ( set @ B ) ) @ ( set @ ( list @ A ) ) @ ( set @ ( list @ B ) ) @ ( list_all2 @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A5 ) ) @ ( bNF_rel_set @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( listset @ A ) @ ( listset @ B ) ) ).
% listset_transfer
thf(fact_6090_set__Cons__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) @ ( ( set @ ( list @ B ) ) > ( set @ ( list @ B ) ) ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( set @ ( list @ A ) ) @ ( set @ ( list @ B ) ) @ ( set @ ( list @ A ) ) @ ( set @ ( list @ B ) ) @ ( bNF_rel_set @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( bNF_rel_set @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) ) ) @ ( set_Cons @ A ) @ ( set_Cons @ B ) ) ).
% set_Cons_transfer
thf(fact_6091_lists__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ ( list @ A ) ) @ ( set @ ( list @ B ) ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_set @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) ) @ ( lists @ A ) @ ( lists @ B ) ) ).
% lists_transfer
thf(fact_6092_inter__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( set @ A ) > ( set @ A ) ) @ ( ( set @ B ) > ( set @ B ) ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_set @ A @ B @ A5 ) ) @ ( inf_inf @ ( set @ A ) ) @ ( inf_inf @ ( set @ B ) ) ) ) ).
% inter_transfer
thf(fact_6093_fun__lub__parametric,axiom,
! [A: $tType,C: $tType,E4: $tType,F6: $tType,D: $tType,B: $tType,A5: A > B > $o,B4: C > D > $o,C4: E4 > F6 > $o] :
( ( bi_total @ A @ B @ A5 )
=> ( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( ( set @ A ) > C ) @ ( ( set @ B ) > D ) @ ( ( set @ ( E4 > A ) ) > E4 > C ) @ ( ( set @ ( F6 > B ) ) > F6 > D ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ C @ D @ ( bNF_rel_set @ A @ B @ A5 ) @ B4 ) @ ( bNF_rel_fun @ ( set @ ( E4 > A ) ) @ ( set @ ( F6 > B ) ) @ ( E4 > C ) @ ( F6 > D ) @ ( bNF_rel_set @ ( E4 > A ) @ ( F6 > B ) @ ( bNF_rel_fun @ E4 @ F6 @ A @ B @ C4 @ A5 ) ) @ ( bNF_rel_fun @ E4 @ F6 @ C @ D @ C4 @ B4 ) ) @ ( partial_fun_lub @ A @ C @ E4 ) @ ( partial_fun_lub @ B @ D @ F6 ) ) ) ) ).
% fun_lub_parametric
thf(fact_6094_nths__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( set @ nat ) > ( list @ A ) ) @ ( ( set @ nat ) > ( list @ B ) ) @ ( list_all2 @ A @ B @ A5 )
@ ( bNF_rel_fun @ ( set @ nat ) @ ( set @ nat ) @ ( list @ A ) @ ( list @ B )
@ ( bNF_rel_set @ nat @ nat
@ ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
@ ( list_all2 @ A @ B @ A5 ) )
@ ( nths @ A )
@ ( nths @ B ) ) ).
% nths_transfer
thf(fact_6095_shuffles__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( list @ A ) > ( set @ ( list @ A ) ) ) @ ( ( list @ B ) > ( set @ ( list @ B ) ) ) @ ( list_all2 @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( set @ ( list @ A ) ) @ ( set @ ( list @ B ) ) @ ( list_all2 @ A @ B @ A5 ) @ ( bNF_rel_set @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A5 ) ) ) @ ( shuffles @ A ) @ ( shuffles @ B ) ) ).
% shuffles_transfer
thf(fact_6096_empty__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_set @ A @ B @ A5 @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ B ) ) ) ).
% empty_transfer
thf(fact_6097_vimage__right__total__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B4: A > B > $o,A5: C > D > $o] :
( ( bi_unique @ A @ B @ B4 )
=> ( ( right_total @ C @ D @ A5 )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( set @ A ) > ( set @ C ) ) @ ( ( set @ B ) > ( set @ D ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ A5 @ B4 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ C ) @ ( set @ D ) @ ( bNF_rel_set @ A @ B @ B4 ) @ ( bNF_rel_set @ C @ D @ A5 ) )
@ ^ [F3: C > A,X8: set @ A] : ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F3 @ X8 ) @ ( collect @ C @ ( domainp @ C @ D @ A5 ) ) )
@ ( vimage @ D @ B ) ) ) ) ).
% vimage_right_total_transfer
thf(fact_6098_find__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > ( option @ A ) ) @ ( ( list @ B ) > ( option @ B ) )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y5: $o,Z5: $o] : Y5 = Z5 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( option @ A ) @ ( option @ B ) @ ( list_all2 @ A @ B @ A5 ) @ ( rel_option @ A @ B @ A5 ) )
@ ( find @ A )
@ ( find @ B ) ) ).
% find_transfer
thf(fact_6099_right__total__Inter__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( ( right_total @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( set @ ( set @ A ) ) @ ( set @ ( set @ B ) ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A5 ) ) @ ( bNF_rel_set @ A @ B @ A5 )
@ ^ [S7: set @ ( set @ A )] : ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ S7 ) @ ( collect @ A @ ( domainp @ A @ B @ A5 ) ) )
@ ( complete_Inf_Inf @ ( set @ B ) ) ) ) ) ).
% right_total_Inter_transfer
thf(fact_6100_those__transfer,axiom,
! [A: $tType,B: $tType,P: A > B > $o] : ( bNF_rel_fun @ ( list @ ( option @ A ) ) @ ( list @ ( option @ B ) ) @ ( option @ ( list @ A ) ) @ ( option @ ( list @ B ) ) @ ( list_all2 @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ P ) ) @ ( rel_option @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ P ) ) @ ( those @ A ) @ ( those @ B ) ) ).
% those_transfer
thf(fact_6101_list_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( domainp @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ R ) )
= ( list_all @ A @ ( domainp @ A @ B @ R ) ) ) ).
% list.Domainp_rel
thf(fact_6102_list_Oright__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( right_total @ A @ B @ R )
=> ( right_total @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ R ) ) ) ).
% list.right_total_rel
thf(fact_6103_right__total__Compl__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( ( right_total @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_set @ A @ B @ A5 )
@ ^ [S7: set @ A] : ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ S7 ) @ ( collect @ A @ ( domainp @ A @ B @ A5 ) ) )
@ ( uminus_uminus @ ( set @ B ) ) ) ) ) ).
% right_total_Compl_transfer
thf(fact_6104_rel__option__inf,axiom,
! [B: $tType,A: $tType,A5: A > B > $o,B4: A > B > $o] :
( ( inf_inf @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ A5 ) @ ( rel_option @ A @ B @ B4 ) )
= ( rel_option @ A @ B @ ( inf_inf @ ( A > B > $o ) @ A5 @ B4 ) ) ) ).
% rel_option_inf
thf(fact_6105_wfP__SUP,axiom,
! [B: $tType,A: $tType,R3: A > B > B > $o] :
( ! [I3: A] : ( wfP @ B @ ( R3 @ I3 ) )
=> ( ! [I3: A,J3: A] :
( ( ( R3 @ I3 )
!= ( R3 @ J3 ) )
=> ( ( inf_inf @ ( B > $o ) @ ( domainp @ B @ B @ ( R3 @ I3 ) ) @ ( rangep @ B @ B @ ( R3 @ J3 ) ) )
= ( bot_bot @ ( B > $o ) ) ) )
=> ( wfP @ B @ ( complete_Sup_Sup @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% wfP_SUP
thf(fact_6106_option_Osimps_I15_J,axiom,
! [A: $tType,X24: A] :
( ( set_option @ A @ ( some @ A @ X24 ) )
= ( insert2 @ A @ X24 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% option.simps(15)
thf(fact_6107_these__not__empty__eq,axiom,
! [A: $tType,B4: set @ ( option @ A )] :
( ( ( these @ A @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B4
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B4
!= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_6108_set__empty__eq,axiom,
! [A: $tType,Xo2: option @ A] :
( ( ( set_option @ A @ Xo2 )
= ( bot_bot @ ( set @ A ) ) )
= ( Xo2
= ( none @ A ) ) ) ).
% set_empty_eq
thf(fact_6109_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_6110_option_Osimps_I14_J,axiom,
! [A: $tType] :
( ( set_option @ A @ ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% option.simps(14)
thf(fact_6111_these__empty__eq,axiom,
! [A: $tType,B4: set @ ( option @ A )] :
( ( ( these @ A @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B4
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B4
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_6112_partial__function__definitions_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,Leq: A > A > $o,Lub2: ( set @ A ) > A,U2: C > B > A,F2: C > C,C4: ( B > A ) > C,F: C,P: ( B > A ) > $o] :
( ( partia7178651479351089652itions @ A @ Leq @ Lub2 )
=> ( ! [X2: B] :
( comple7038119648293358887notone @ ( B > A ) @ A @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ Leq
@ ^ [F3: B > A] : ( U2 @ ( F2 @ ( C4 @ F3 ) ) @ X2 ) )
=> ( ( F
= ( C4
@ ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub2 ) @ ( partial_fun_ord @ A @ A @ B @ Leq )
@ ^ [F3: B > A] : ( U2 @ ( F2 @ ( C4 @ F3 ) ) ) ) ) )
=> ( ! [F5: B > A] :
( ( U2 @ ( C4 @ F5 ) )
= F5 )
=> ( ( comple1908693960933563346ssible @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub2 ) @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ P )
=> ( ( P
@ ^ [Uu3: B] : ( Lub2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [F5: C] :
( ( P @ ( U2 @ F5 ) )
=> ( P @ ( U2 @ ( F2 @ F5 ) ) ) )
=> ( P @ ( U2 @ F ) ) ) ) ) ) ) ) ) ).
% partial_function_definitions.fixp_induct_uc
thf(fact_6113_inf__filter__parametric,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( ( bi_total @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > ( filter @ A ) ) @ ( ( filter @ B ) > ( filter @ B ) ) @ ( rel_filter @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A5 ) @ ( rel_filter @ A @ B @ A5 ) ) @ ( inf_inf @ ( filter @ A ) ) @ ( inf_inf @ ( filter @ B ) ) ) ) ) ).
% inf_filter_parametric
thf(fact_6114_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I5: set @ A,F2: A > ( filter @ B )] :
( ! [I3: A] :
( ( member2 @ A @ I3 @ I5 )
=> ! [J3: A] :
( ( member2 @ A @ J3 @ I5 )
=> ? [X6: A] :
( ( member2 @ A @ X6 @ I5 )
& ( ord_less_eq @ ( filter @ B ) @ ( F2 @ X6 ) @ ( inf_inf @ ( filter @ B ) @ ( F2 @ I3 ) @ ( F2 @ J3 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F2 @ I5 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ I5 )
& ( ( F2 @ X3 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_6115_inf__principal,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( filter @ A ) @ ( principal @ A @ A5 ) @ ( principal @ A @ B4 ) )
= ( principal @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% inf_principal
thf(fact_6116_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B4: set @ A,F2: A > ( filter @ B ),P: B > $o] :
( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member2 @ A @ A6 @ B4 )
=> ! [B6: A] :
( ( member2 @ A @ B6 @ B4 )
=> ? [X6: A] :
( ( member2 @ A @ X6 @ B4 )
& ( ord_less_eq @ ( filter @ B ) @ ( F2 @ X6 ) @ ( inf_inf @ ( filter @ B ) @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F2 @ B4 ) ) )
= ( ? [X3: A] :
( ( member2 @ A @ X3 @ B4 )
& ( eventually @ B @ P @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_6117_eventually__Inf__base,axiom,
! [A: $tType,B4: set @ ( filter @ A ),P: A > $o] :
( ( B4
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F4: filter @ A] :
( ( member2 @ ( filter @ A ) @ F4 @ B4 )
=> ! [G5: filter @ A] :
( ( member2 @ ( filter @ A ) @ G5 @ B4 )
=> ? [X6: filter @ A] :
( ( member2 @ ( filter @ A ) @ X6 @ B4 )
& ( ord_less_eq @ ( filter @ A ) @ X6 @ ( inf_inf @ ( filter @ A ) @ F4 @ G5 ) ) ) ) )
=> ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B4 ) )
= ( ? [X3: filter @ A] :
( ( member2 @ ( filter @ A ) @ X3 @ B4 )
& ( eventually @ A @ P @ X3 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_6118_eventually__inf,axiom,
! [A: $tType,P: A > $o,F2: filter @ A,F8: filter @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F2 @ F8 ) )
= ( ? [Q2: A > $o,R6: A > $o] :
( ( eventually @ A @ Q2 @ F2 )
& ( eventually @ A @ R6 @ F8 )
& ! [X3: A] :
( ( ( Q2 @ X3 )
& ( R6 @ X3 ) )
=> ( P @ X3 ) ) ) ) ) ).
% eventually_inf
thf(fact_6119_eventually__inf__principal,axiom,
! [A: $tType,P: A > $o,F2: filter @ A,S4: set @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F2 @ ( principal @ A @ S4 ) ) )
= ( eventually @ A
@ ^ [X3: A] :
( ( member2 @ A @ X3 @ S4 )
=> ( P @ X3 ) )
@ F2 ) ) ).
% eventually_inf_principal
thf(fact_6120_principal__eq__bot__iff,axiom,
! [A: $tType,X5: set @ A] :
( ( ( principal @ A @ X5 )
= ( bot_bot @ ( filter @ A ) ) )
= ( X5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% principal_eq_bot_iff
thf(fact_6121_bot__eq__principal__empty,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_eq_principal_empty
thf(fact_6122_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I5: set @ A,F2: A > ( set @ B ),F: B > C,G3: D > ( set @ C ),J4: set @ D] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member2 @ A @ I3 @ I5 )
=> ! [J3: A] :
( ( member2 @ A @ J3 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F2 @ I3 ) @ ( F2 @ J3 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F2 @ J3 ) @ ( F2 @ I3 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J2: D] : ( principal @ C @ ( G3 @ J2 ) )
@ J4 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I2: A] : ( principal @ B @ ( F2 @ I2 ) )
@ I5 ) ) )
= ( ! [X3: D] :
( ( member2 @ D @ X3 @ J4 )
=> ? [Y3: A] :
( ( member2 @ A @ Y3 @ I5 )
& ! [Z3: B] :
( ( member2 @ B @ Z3 @ ( F2 @ Y3 ) )
=> ( member2 @ C @ ( F @ Z3 ) @ ( G3 @ X3 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_6123_inf__filter__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F10: filter @ A] :
( abs_filter @ A
@ ^ [P3: A > $o] :
? [Q2: A > $o,R6: A > $o] :
( ( eventually @ A @ Q2 @ F7 )
& ( eventually @ A @ R6 @ F10 )
& ! [X3: A] :
( ( ( Q2 @ X3 )
& ( R6 @ X3 ) )
=> ( P3 @ X3 ) ) ) ) ) ) ).
% inf_filter_def
thf(fact_6124_filterlim__inf,axiom,
! [B: $tType,A: $tType,F: A > B,F24: filter @ B,F32: filter @ B,F13: filter @ A] :
( ( filterlim @ A @ B @ F @ ( inf_inf @ ( filter @ B ) @ F24 @ F32 ) @ F13 )
= ( ( filterlim @ A @ B @ F @ F24 @ F13 )
& ( filterlim @ A @ B @ F @ F32 @ F13 ) ) ) ).
% filterlim_inf
thf(fact_6125_filterlim__If,axiom,
! [B: $tType,A: $tType,F: A > B,G3: filter @ B,F2: filter @ A,P: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F @ G3 @ ( inf_inf @ ( filter @ A ) @ F2 @ ( principal @ A @ ( collect @ A @ P ) ) ) )
=> ( ( filterlim @ A @ B @ G @ G3
@ ( inf_inf @ ( filter @ A ) @ F2
@ ( principal @ A
@ ( collect @ A
@ ^ [X3: A] :
~ ( P @ X3 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X3: A] : ( if @ B @ ( P @ X3 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ G3
@ F2 ) ) ) ).
% filterlim_If
thf(fact_6126_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I5: set @ A,J4: set @ B,A5: A > ( filter @ C ),B4: B > ( filter @ D )] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ A5 @ I5 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B4 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I2: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J2: B] : ( prod_filter @ C @ D @ ( A5 @ I2 ) @ ( B4 @ J2 ) )
@ J4 ) )
@ I5 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_6127_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I5: set @ A,A5: A > ( filter @ B ),B4: filter @ C] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A5 @ I5 ) ) @ B4 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I2: A] : ( prod_filter @ B @ C @ ( A5 @ I2 ) @ B4 )
@ I5 ) ) ) ) ).
% prod_filter_INF1
thf(fact_6128_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J4: set @ A,A5: filter @ B,B4: A > ( filter @ C )] :
( ( J4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A5 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B4 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I2: A] : ( prod_filter @ B @ C @ A5 @ ( B4 @ I2 ) )
@ J4 ) ) ) ) ).
% prod_filter_INF2
thf(fact_6129_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F2: filter @ A,X: B] :
( ( prod_filter @ A @ B @ F2 @ ( principal @ B @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A4: A] : ( product_Pair @ A @ B @ A4 @ X )
@ F2 ) ) ).
% prod_filter_principal_singleton2
thf(fact_6130_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X: A,F2: filter @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ F2 )
= ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ F2 ) ) ).
% prod_filter_principal_singleton
thf(fact_6131_filtermap__inf,axiom,
! [A: $tType,B: $tType,F: B > A,F13: filter @ B,F24: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F @ ( inf_inf @ ( filter @ B ) @ F13 @ F24 ) ) @ ( inf_inf @ ( filter @ A ) @ ( filtermap @ B @ A @ F @ F13 ) @ ( filtermap @ B @ A @ F @ F24 ) ) ) ).
% filtermap_inf
thf(fact_6132_ID_Orel__cong,axiom,
! [A: $tType,B: $tType,X: A,Ya: A,Y: B,Xa: B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z4: A,Yb: B] :
( ( member2 @ A @ Z4 @ ( insert2 @ A @ Ya @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( member2 @ B @ Yb @ ( insert2 @ B @ Xa @ ( bot_bot @ ( set @ B ) ) ) )
=> ( ( R @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_id_bnf @ ( A > B > $o ) @ R @ X @ Y )
= ( bNF_id_bnf @ ( A > B > $o ) @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_6133_ID_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: A,Y: B,Ra: A > B > $o] :
( ( bNF_id_bnf @ ( A > B > $o ) @ R @ X @ Y )
=> ( ! [Z4: A,Yb: B] :
( ( member2 @ A @ Z4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( member2 @ B @ Yb @ ( insert2 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) )
=> ( ( R @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_id_bnf @ ( A > B > $o ) @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_6134_ID_Orel__refl__strong,axiom,
! [A: $tType,X: A,Ra: A > A > $o] :
( ! [Z4: A] :
( ( member2 @ A @ Z4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( bNF_id_bnf @ ( A > A > $o ) @ Ra @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_6135_pred__on_Ochain__extend,axiom,
! [A: $tType,A5: set @ A,P: A > A > $o,C4: set @ A,Z2: A] :
( ( pred_chain @ A @ A5 @ P @ C4 )
=> ( ( member2 @ A @ Z2 @ A5 )
=> ( ! [X2: A] :
( ( member2 @ A @ X2 @ C4 )
=> ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z5: A] : Y5 = Z5
@ X2
@ Z2 ) )
=> ( pred_chain @ A @ A5 @ P @ ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) @ C4 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_6136_sup2CI,axiom,
! [A: $tType,B: $tType,B4: A > B > $o,X: A,Y: B,A5: A > B > $o] :
( ( ~ ( B4 @ X @ Y )
=> ( A5 @ X @ Y ) )
=> ( sup_sup @ ( A > B > $o ) @ A5 @ B4 @ X @ Y ) ) ).
% sup2CI
thf(fact_6137_sup2I2,axiom,
! [A: $tType,B: $tType,B4: A > B > $o,X: A,Y: B,A5: A > B > $o] :
( ( B4 @ X @ Y )
=> ( sup_sup @ ( A > B > $o ) @ A5 @ B4 @ X @ Y ) ) ).
% sup2I2
thf(fact_6138_sup2I1,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,X: A,Y: B,B4: A > B > $o] :
( ( A5 @ X @ Y )
=> ( sup_sup @ ( A > B > $o ) @ A5 @ B4 @ X @ Y ) ) ).
% sup2I1
thf(fact_6139_sup2E,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,B4: A > B > $o,X: A,Y: B] :
( ( sup_sup @ ( A > B > $o ) @ A5 @ B4 @ X @ Y )
=> ( ~ ( A5 @ X @ Y )
=> ( B4 @ X @ Y ) ) ) ).
% sup2E
thf(fact_6140_pred__on_Ochain__empty,axiom,
! [A: $tType,A5: set @ A,P: A > A > $o] : ( pred_chain @ A @ A5 @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pred_on.chain_empty
% Type constructors (565)
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A20: $tType,A26: $tType] :
( ( comple6319245703460814977attice @ A26 )
=> ( condit1219197933456340205attice @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A20: $tType,A26: $tType] :
( ( comple592849572758109894attice @ A26 )
=> ( comple592849572758109894attice @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A20: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounde4967611905675639751up_bot @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A20: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounde4346867609351753570nf_top @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A20: $tType,A26: $tType] :
( ( comple6319245703460814977attice @ A26 )
=> ( comple6319245703460814977attice @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A20: $tType,A26: $tType] :
( ( boolea8198339166811842893lgebra @ A26 )
=> ( boolea8198339166811842893lgebra @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A20: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounded_lattice_top @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A20: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounded_lattice_bot @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A20: $tType,A26: $tType] :
( ( comple6319245703460814977attice @ A26 )
=> ( comple9053668089753744459l_ccpo @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A20: $tType,A26: $tType] :
( ( semilattice_sup @ A26 )
=> ( semilattice_sup @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A20: $tType,A26: $tType] :
( ( semilattice_inf @ A26 )
=> ( semilattice_inf @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A20: $tType,A26: $tType] :
( ( distrib_lattice @ A26 )
=> ( distrib_lattice @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A20: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounded_lattice @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A20: $tType,A26: $tType] :
( ( order_top @ A26 )
=> ( order_top @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A20: $tType,A26: $tType] :
( ( order_bot @ A26 )
=> ( order_bot @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A20: $tType,A26: $tType] :
( ( preorder @ A26 )
=> ( preorder @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A20: $tType,A26: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A26 ) )
=> ( finite_finite @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A20: $tType,A26: $tType] :
( ( lattice @ A26 )
=> ( lattice @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A20: $tType,A26: $tType] :
( ( order @ A26 )
=> ( order @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A20: $tType,A26: $tType] :
( ( top @ A26 )
=> ( top @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A20: $tType,A26: $tType] :
( ( ord @ A26 )
=> ( ord @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A20: $tType,A26: $tType] :
( ( bot @ A26 )
=> ( bot @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A20: $tType,A26: $tType] :
( ( uminus @ A26 )
=> ( uminus @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Oinf,axiom,
! [A20: $tType,A26: $tType] :
( ( semilattice_inf @ A26 )
=> ( inf @ ( A20 > A26 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A20: $tType,A26: $tType] :
( ( minus @ A26 )
=> ( minus @ ( A20 > A26 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,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_2,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_3,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_4,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_5,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_6,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_7,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_8,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_9,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Lattices_Oinf_10,axiom,
inf @ int ).
thf(tcon_Int_Oint___Groups_Otimes,axiom,
times @ int ).
thf(tcon_Int_Oint___Groups_Ominus_11,axiom,
minus @ int ).
thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
ring_gcd @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Int_Oint___Heap_Oheap,axiom,
heap @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_12,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_13,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_14,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_15,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_16,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_17,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_18,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_19,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_20,axiom,
normal6328177297339901930cative @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_21,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_22,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_23,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_24,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_25,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_26,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_27,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_28,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_29,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_30,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_31,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_32,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_33,axiom,
semido2269285787275462019factor @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_34,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_35,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_36,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_37,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_38,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_39,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_40,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_41,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_42,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_43,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom_44,axiom,
normal8620421768224518004emidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_45,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_46,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_47,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_48,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_49,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_50,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_51,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_52,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_53,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_54,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_55,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_56,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_57,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_58,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_59,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_60,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_61,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_62,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_63,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_64,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_65,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_66,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_67,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_68,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_69,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_70,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_71,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_72,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_73,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_74,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_75,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_76,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_77,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_78,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_79,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_80,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_81,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_82,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_83,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_84,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_85,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_86,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Lattices_Oinf_87,axiom,
inf @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes_88,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_89,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_90,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_91,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_92,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_93,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_94,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_95,axiom,
heap @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_96,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_97,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_98,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_99,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Otimes_100,axiom,
times @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_101,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_102,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_103,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_104,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_105,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_106,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_107,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_108,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_109,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_110,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_111,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_112,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_113,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_114,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_115,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_116,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_117,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_118,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_119,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_120,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_121,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_122,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_123,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_124,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_125,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_126,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_127,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_128,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_129,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_130,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_131,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_132,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_133,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_134,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_135,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_136,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_137,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_138,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_139,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_140,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_141,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_142,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_143,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_144,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_145,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_146,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_147,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_148,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_149,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_150,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_151,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_152,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_153,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_154,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_155,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_156,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_157,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_158,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_159,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_160,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_161,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_162,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_163,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_164,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_165,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_166,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_167,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_168,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_169,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_170,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_171,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_172,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_173,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_174,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Lattices_Oinf_175,axiom,
inf @ rat ).
thf(tcon_Rat_Orat___Groups_Otimes_176,axiom,
times @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_177,axiom,
minus @ 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___Lattices_Obounded__semilattice__inf__top_188,axiom,
! [A20: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_189,axiom,
! [A20: $tType] : ( comple6319245703460814977attice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_190,axiom,
! [A20: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_191,axiom,
! [A20: $tType] : ( bounded_lattice_top @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_192,axiom,
! [A20: $tType] : ( bounded_lattice_bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_193,axiom,
! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_194,axiom,
! [A20: $tType] : ( semilattice_sup @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_195,axiom,
! [A20: $tType] : ( semilattice_inf @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_196,axiom,
! [A20: $tType] : ( distrib_lattice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_197,axiom,
! [A20: $tType] : ( bounded_lattice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_198,axiom,
! [A20: $tType] : ( order_top @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_199,axiom,
! [A20: $tType] : ( order_bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_200,axiom,
! [A20: $tType] : ( preorder @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_201,axiom,
! [A20: $tType] :
( ( finite_finite @ A20 )
=> ( finite_finite @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_202,axiom,
! [A20: $tType] : ( lattice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_203,axiom,
! [A20: $tType] : ( order @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_204,axiom,
! [A20: $tType] : ( top @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_205,axiom,
! [A20: $tType] : ( ord @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_206,axiom,
! [A20: $tType] : ( bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_207,axiom,
! [A20: $tType] : ( uminus @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Oinf_208,axiom,
! [A20: $tType] : ( inf @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_209,axiom,
! [A20: $tType] : ( minus @ ( set @ A20 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_210,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_211,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_212,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_213,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_214,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_215,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_216,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_217,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_218,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_219,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_220,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_221,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_222,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_223,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_224,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_225,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_226,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_227,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_228,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_229,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_230,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_231,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_232,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_233,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Lattices_Oinf_234,axiom,
inf @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_235,axiom,
minus @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_236,axiom,
heap @ $o ).
thf(tcon_Heap_Oref___Heap_Oheap_237,axiom,
! [A20: $tType] : ( heap @ ( ref @ A20 ) ) ).
thf(tcon_List_Olist___Heap_Oheap_238,axiom,
! [A20: $tType] :
( ( heap @ A20 )
=> ( heap @ ( list @ A20 ) ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_239,axiom,
! [A20: $tType] : ( heap @ ( array @ A20 ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_240,axiom,
! [A20: $tType,A26: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A26 ) )
=> ( finite_finite @ ( sum_sum @ A20 @ A26 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_241,axiom,
! [A20: $tType,A26: $tType] :
( ( ( heap @ A20 )
& ( heap @ A26 ) )
=> ( heap @ ( sum_sum @ A20 @ A26 ) ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_242,axiom,
! [A20: $tType] : ( condit1219197933456340205attice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_243,axiom,
! [A20: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_244,axiom,
! [A20: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_245,axiom,
! [A20: $tType] : ( comple6319245703460814977attice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_246,axiom,
! [A20: $tType] : ( bounded_lattice_top @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_247,axiom,
! [A20: $tType] : ( bounded_lattice_bot @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_248,axiom,
! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_249,axiom,
! [A20: $tType] : ( semilattice_sup @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_250,axiom,
! [A20: $tType] : ( semilattice_inf @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_251,axiom,
! [A20: $tType] : ( distrib_lattice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_252,axiom,
! [A20: $tType] : ( bounded_lattice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_253,axiom,
! [A20: $tType] : ( order_top @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_254,axiom,
! [A20: $tType] : ( order_bot @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_255,axiom,
! [A20: $tType] : ( preorder @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_256,axiom,
! [A20: $tType] : ( lattice @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_257,axiom,
! [A20: $tType] : ( order @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_258,axiom,
! [A20: $tType] : ( top @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_259,axiom,
! [A20: $tType] : ( ord @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_260,axiom,
! [A20: $tType] : ( bot @ ( filter @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Oinf_261,axiom,
! [A20: $tType] : ( inf @ ( filter @ A20 ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_262,axiom,
! [A20: $tType] :
( ( comple5582772986160207858norder @ A20 )
=> ( condit6923001295902523014norder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_263,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( condit1219197933456340205attice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_264,axiom,
! [A20: $tType] :
( ( comple592849572758109894attice @ A20 )
=> ( comple592849572758109894attice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_265,axiom,
! [A20: $tType] :
( ( lattice @ A20 )
=> ( bounde4967611905675639751up_bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_266,axiom,
! [A20: $tType] :
( ( bounded_lattice_top @ A20 )
=> ( bounde4346867609351753570nf_top @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
! [A20: $tType] :
( ( comple5582772986160207858norder @ A20 )
=> ( comple5582772986160207858norder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_267,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple6319245703460814977attice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_268,axiom,
! [A20: $tType] :
( ( bounded_lattice_top @ A20 )
=> ( bounded_lattice_top @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_269,axiom,
! [A20: $tType] :
( ( lattice @ A20 )
=> ( bounded_lattice_bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_270,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple9053668089753744459l_ccpo @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_271,axiom,
! [A20: $tType] :
( ( semilattice_sup @ A20 )
=> ( semilattice_sup @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_272,axiom,
! [A20: $tType] :
( ( semilattice_inf @ A20 )
=> ( semilattice_inf @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Odistrib__lattice_273,axiom,
! [A20: $tType] :
( ( distrib_lattice @ A20 )
=> ( distrib_lattice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice_274,axiom,
! [A20: $tType] :
( ( bounded_lattice_top @ A20 )
=> ( bounded_lattice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_275,axiom,
! [A20: $tType] :
( ( wellorder @ A20 )
=> ( wellorder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_276,axiom,
! [A20: $tType] :
( ( order_top @ A20 )
=> ( order_top @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_277,axiom,
! [A20: $tType] :
( ( order @ A20 )
=> ( order_bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_278,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_279,axiom,
! [A20: $tType] :
( ( linorder @ A20 )
=> ( linorder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_280,axiom,
! [A20: $tType] :
( ( finite_finite @ A20 )
=> ( finite_finite @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Olattice_281,axiom,
! [A20: $tType] :
( ( lattice @ A20 )
=> ( lattice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_282,axiom,
! [A20: $tType] :
( ( order @ A20 )
=> ( order @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Otop_283,axiom,
! [A20: $tType] :
( ( order_top @ A20 )
=> ( top @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_284,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ord @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Obot_285,axiom,
! [A20: $tType] :
( ( order @ A20 )
=> ( bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Oinf_286,axiom,
! [A20: $tType] :
( ( inf @ A20 )
=> ( inf @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_287,axiom,
! [A20: $tType] :
( ( heap @ A20 )
=> ( heap @ ( option @ A20 ) ) ) ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_288,axiom,
bounde4967611905675639751up_bot @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_289,axiom,
bounde4346867609351753570nf_top @ assn ).
thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_290,axiom,
boolea8198339166811842893lgebra @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_291,axiom,
bounded_lattice_top @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_292,axiom,
bounded_lattice_bot @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_293,axiom,
semilattice_sup @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_294,axiom,
semilattice_inf @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_295,axiom,
distrib_lattice @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice_296,axiom,
bounded_lattice @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_297,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_298,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_299,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__top_300,axiom,
order_top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_301,axiom,
order_bot @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Opreorder_302,axiom,
preorder @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_303,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Olattice_304,axiom,
lattice @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder_305,axiom,
order @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Otop_306,axiom,
top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oord_307,axiom,
ord @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Obot_308,axiom,
bot @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ouminus_309,axiom,
uminus @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Oinf_310,axiom,
inf @ assn ).
thf(tcon_Assertions_Oassn___Groups_Otimes_311,axiom,
times @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ominus_312,axiom,
minus @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_313,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_314,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_315,axiom,
dvd @ assn ).
thf(tcon_Typerep_Otyperep___Heap_Oheap_316,axiom,
heap @ typerep ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_317,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_318,axiom,
! [A20: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_319,axiom,
! [A20: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_320,axiom,
! [A20: $tType] : ( cancel_semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_321,axiom,
! [A20: $tType] : ( comm_monoid_diff @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_322,axiom,
! [A20: $tType] : ( ab_semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_323,axiom,
! [A20: $tType] : ( comm_monoid_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_324,axiom,
! [A20: $tType] : ( semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_325,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_326,axiom,
! [A20: $tType] : ( monoid_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_327,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( order @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_328,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ord @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ominus_329,axiom,
! [A20: $tType] : ( minus @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_330,axiom,
! [A20: $tType] : ( zero @ ( multiset @ A20 ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_331,axiom,
! [A20: $tType,A26: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A26 ) )
=> ( finite_finite @ ( product_prod @ A20 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_332,axiom,
! [A20: $tType,A26: $tType] :
( ( ( heap @ A20 )
& ( heap @ A26 ) )
=> ( heap @ ( product_prod @ A20 @ A26 ) ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_333,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_334,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_335,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_336,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_337,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_338,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_339,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_340,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_341,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_342,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_343,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_344,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_345,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_346,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_347,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_348,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_349,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_350,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_351,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_352,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_353,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_354,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_355,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_356,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_357,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_358,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_359,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Oinf_360,axiom,
inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_361,axiom,
minus @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_362,axiom,
heap @ product_unit ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_363,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_364,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_365,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_366,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_367,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_368,axiom,
semiri6575147826004484403cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_369,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_370,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_371,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_372,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_373,axiom,
linord2810124833399127020strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_374,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_375,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_376,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_377,axiom,
linord4140545234300271783up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_378,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_379,axiom,
linord8928482502909563296strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_380,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_381,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_382,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_383,axiom,
cancel2418104881723323429up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_384,axiom,
cancel1802427076303600483id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_385,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_386,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_387,axiom,
ordere2520102378445227354miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_388,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_389,axiom,
linordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_390,axiom,
ordered_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_391,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_392,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_393,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_394,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_395,axiom,
comm_monoid_diff @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_396,axiom,
ab_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_397,axiom,
ordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_398,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_399,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_400,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_401,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_402,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_403,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_404,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_405,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_406,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_407,axiom,
semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_408,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_409,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_410,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_411,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_412,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_413,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_414,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_415,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_416,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_417,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_418,axiom,
mult_zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_419,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_420,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_421,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Otimes_422,axiom,
times @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ominus_423,axiom,
minus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_424,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_425,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_426,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_427,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_428,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 ) )
= ( ? [X8: A] : ( P @ X8 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( vEBT_List_list_assn @ a @ b @ p @ ( nil @ a ) @ l )
= ( pure_assn
@ ( l
= ( nil @ b ) ) ) ) ).
%------------------------------------------------------------------------------