TPTP Problem File: ITP219^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP219^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_List_Assn 00252_007651
% 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_00252_007651 [Des22]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.33 v8.1.0
% Syntax : Number of formulae : 9745 (3208 unt; 904 typ; 0 def)
% Number of atoms : 27143 (9591 equ; 9 cnn)
% Maximal formula atoms : 28 ( 3 avg)
% Number of connectives : 192390 (2673 ~; 309 |;1916 &;175156 @)
% ( 0 <=>;12336 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 8 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 7767 (7767 >; 0 *; 0 +; 0 <<)
% Number of symbols : 896 ( 892 usr; 29 con; 0-9 aty)
% Number of variables : 34710 (3248 ^;29633 !; 836 ?;34710 :)
% ( 993 !>; 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 17:06:47.478
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
thf(ty_t_Heap__Time__Monad_OHeap,type,
heap_Time_Heap: $tType > $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
thf(ty_t_Code__Evaluation_Oterm,type,
code_term: $tType ).
thf(ty_t_Heap_Oheap_Oheap__ext,type,
heap_ext: $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Multiset_Omultiset,type,
multiset: $tType > $tType ).
thf(ty_t_Assertions_Oassn,type,
assn: $tType ).
thf(ty_t_Predicate_Opred,type,
pred: $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Predicate_Oseq,type,
seq: $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_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_d,type,
d: $tType ).
thf(ty_tf_c,type,
c: $tType ).
% Explicit typings (878)
thf(sy_cl_Typerep_Otyperep,type,
typerep:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Oenum,type,
enum:
!>[A: $tType] : $o ).
thf(sy_cl_Code__Evaluation_Oterm__of,type,
code_term_of:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Oequal,type,
cl_HOL_Oequal:
!>[A: $tType] : $o ).
thf(sy_cl_Heap_Oheap,type,
heap:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[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_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__modulo,type,
idom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_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_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Quickcheck__Random_Orandom,type,
quickcheck_random:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
semiri6843258321239162965malize:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Quickcheck__Exhaustive_Oexhaustive,type,
quickc658316121487927005ustive:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring,type,
euclid5891614535332579305n_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri2026040879449505780visors:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ounbounded__dense__linorder,type,
unboun7993243217541854897norder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde4346867609351753570nf_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde4967611905675639751up_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Quickcheck__Exhaustive_Ofull__exhaustive,type,
quickc3360725361186068524ustive:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__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_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit5016429287641298734tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Array__Time_Oalloc,type,
array_alloc:
!>[A: $tType] : ( ( list @ A ) > ( heap_ext @ product_unit ) > ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ).
thf(sy_c_Array__Time_Ofreeze,type,
array_freeze:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
thf(sy_c_Array__Time_Oget,type,
array_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > ( list @ A ) ) ).
thf(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
thf(sy_c_Array__Time_Olength,type,
array_length:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > nat ) ).
thf(sy_c_Array__Time_Omake,type,
array_make:
!>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Omap__entry,type,
array_map_entry:
!>[A: $tType] : ( nat > ( A > A ) > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onew,type,
array_new:
!>[A: $tType] : ( nat > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onoteq,type,
array_noteq:
!>[A: $tType,B: $tType] : ( ( array @ A ) > ( array @ B ) > $o ) ).
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_Opresent,type,
array_present:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > $o ) ).
thf(sy_c_Array__Time_Oset,type,
array_set:
!>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Array__Time_Oswap,type,
array_swap:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oupd,type,
array_upd:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Oupdate,type,
array_update:
!>[A: $tType] : ( ( array @ A ) > nat > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Assertions_Oassn_OAbs__assn,type,
abs_assn: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > assn ).
thf(sy_c_Assertions_Oassn_ORep__assn,type,
rep_assn: assn > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oentails,type,
entails: assn > assn > $o ).
thf(sy_c_Assertions_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_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_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_OCsum,type,
bNF_Cardinal_Csum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
bNF_Cardinal_cexp:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocfinite,type,
bNF_Cardinal_cfinite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
bNF_Cardinal_cone: set @ ( product_prod @ product_unit @ product_unit ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
bNF_Cardinal_cprod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
bNF_Cardinal_csum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
bNF_Cardinal_ctwo: set @ ( product_prod @ $o @ $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Oczero,type,
bNF_Cardinal_czero:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
bNF_Ca8387033319878233205ardSuc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
bNF_Ca6860139660246222851ard_of:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
bNF_Ca8970107618336181345der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
bNF_Ca7293521722713021262ofinal:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OisCardSuc,type,
bNF_Ca6246979054910435723ardSuc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
bNF_Ca7133664381575040944arCard:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
thf(sy_c_BNF__Def_Ocollect,type,
bNF_collect:
!>[B: $tType,A: $tType] : ( ( set @ ( B > ( set @ A ) ) ) > B > ( set @ A ) ) ).
thf(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_BNF__Def_Orel__set,type,
bNF_rel_set:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
bNF_Gr5436034075474128252omCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > B > A ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr4221423524335903396lImage:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
bNF_Gr7122648621184425601vImage:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
bNF_Greatest_toCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > A > B ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
bNF_Gr1419584066657907630d_pred:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
bNF_We4925052301507509544nc_map:
!>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set @ B ) > ( C > A ) > ( B > D ) > ( D > C ) > B > A ) ).
thf(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
bNF_Wellorder_bsqr:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
bNF_We2720479622203943262_image:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > A2 ) > ( set @ ( product_prod @ A2 @ A2 ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
bNF_We413866401316099525erIncl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordIso,type,
bNF_Wellorder_ordIso:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
bNF_Wellorder_ordLeq:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordLess,type,
bNF_We4044943003108391690rdLess:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__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_Opred__fun,type,
basic_pred_fun:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > B > $o ) > ( C > D > $o ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) > $o ) ).
thf(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: code_natural > nat ).
thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: nat > code_natural ).
thf(sy_c_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_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_Conditionally__Complete__Lattices_Opreordering__bdd,type,
condit622319405099724424ng_bdd:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd,type,
condit16957441358409770ng_bdd:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequivp,type,
equiv_equivp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
thf(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oabstract__filter,type,
abstract_filter:
!>[A: $tType] : ( ( product_unit > ( filter @ A ) ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Ocofinite,type,
cofinite:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofilter_OAbs__filter,type,
abs_filter:
!>[A: $tType] : ( ( ( A > $o ) > $o ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofilter_ORep__filter,type,
rep_filter:
!>[A: $tType] : ( ( filter @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Filter_Ofiltercomap,type,
filtercomap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Ofinite__subsets__at__top,type,
finite5375528669736107172at_top:
!>[A: $tType] : ( ( set @ A ) > ( filter @ ( set @ A ) ) ) ).
thf(sy_c_Filter_Ofrequently,type,
frequently:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ois__filter,type,
is_filter:
!>[A: $tType] : ( ( ( A > $o ) > $o ) > $o ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Filter_Orel__filter,type,
rel_filter:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( filter @ A ) > ( filter @ B ) > $o ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on,type,
finite1890593828518410140dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
thf(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Fun__Def_Omax__strict,type,
fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omax__weak,type,
fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__strict,type,
fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__weak,type,
fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_OGcd__class_OLcm,type,
gcd_Lcm:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__class_Olcm,type,
gcd_lcm:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
semiri4206861660011772517g_char:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
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_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum,type,
groups3894954378712506084id_sum:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set,type,
groups778175481326437816id_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
groups_comm_monoid_F:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
groups_comm_monoid_G:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list,type,
groups1828464146339083142d_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set,type,
groups4802862169904069756st_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add_Osum__list,type,
groups4543113879258116180m_list:
!>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > 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_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_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Heap_Oaddr__of__array,type,
addr_of_array:
!>[A: $tType] : ( ( array @ A ) > nat ) ).
thf(sy_c_Heap_Oaddr__of__ref,type,
addr_of_ref:
!>[A: $tType] : ( ( ref @ A ) > nat ) ).
thf(sy_c_Heap_Oarray_OArray,type,
array2:
!>[A: $tType] : ( nat > ( array @ A ) ) ).
thf(sy_c_Heap_Oheap_Olim,type,
lim:
!>[Z: $tType] : ( ( heap_ext @ Z ) > nat ) ).
thf(sy_c_Heap_Oheap_Olim__update,type,
lim_update:
!>[Z: $tType] : ( ( nat > nat ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
thf(sy_c_Heap_Oref_ORef,type,
ref2:
!>[A: $tType] : ( nat > ( ref @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
heap_Time_Heap2:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oassert,type,
heap_Time_assert:
!>[A: $tType] : ( ( A > $o ) > A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Obind,type,
heap_Time_bind:
!>[A: $tType,B: $tType] : ( ( heap_Time_Heap @ A ) > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ B ) ) ).
thf(sy_c_Heap__Time__Monad_Oeffect,type,
heap_Time_effect:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o ) ).
thf(sy_c_Heap__Time__Monad_Oexecute,type,
heap_Time_execute:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
thf(sy_c_Heap__Time__Monad_Oguard,type,
heap_Time_guard:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > $o ) > ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oheap,type,
heap_Time_heap:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_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_Osuccess,type,
heap_Time_success:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
thf(sy_c_Heap__Time__Monad_Otap,type,
heap_Time_tap:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OtimeFrame,type,
heap_Time_timeFrame:
!>[A: $tType] : ( nat > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
thf(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
heap_T5500966940807335491me_rel:
!>[A: $tType] : ( ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > $o ) ).
thf(sy_c_Heap__Time__Monad_Oureturn,type,
heap_Time_ureturn:
!>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Owait,type,
heap_Time_wait: nat > ( heap_Time_Heap @ product_unit ) ).
thf(sy_c_Hoare__Triple_Ohoare__triple,type,
hoare_hoare_triple:
!>[A: $tType] : ( assn > ( heap_Time_Heap @ A ) > ( A > assn ) > $o ) ).
thf(sy_c_Hoare__Triple_Onew__addrs,type,
hoare_new_addrs: ( heap_ext @ product_unit ) > ( set @ nat ) > ( heap_ext @ product_unit ) > ( set @ nat ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
thf(sy_c_Int_Ointrel,type,
intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opcr__int,type,
pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( A > int > A ) ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__order,type,
semilattice_order:
!>[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__max,type,
lattices_ord_arg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__max,type,
lattic501386751176901750rg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
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__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_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
thf(sy_c_List_OBleast,type,
bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oabort__Bleast,type,
abort_Bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_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_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_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_OF,type,
comm_monoid_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset,type,
comm_m9189036328036947845d_mset:
!>[A: $tType] : ( ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ocr__multiset,type,
cr_multiset:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Ofilter__mset,type,
filter_mset:
!>[A: $tType] : ( ( A > $o ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Ofold__mset,type,
fold_mset:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( multiset @ A ) > B ) ).
thf(sy_c_Multiset_Oimage__mset,type,
image_mset:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( multiset @ A ) > ( multiset @ B ) ) ).
thf(sy_c_Multiset_Ointer__mset,type,
inter_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset,type,
linord6283353356039996273ltiset:
!>[A: $tType] : ( ( multiset @ A ) > ( list @ A ) ) ).
thf(sy_c_Multiset_Oms__strict,type,
ms_strict: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Multiset_Oms__weak,type,
ms_weak: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Multiset_Omset,type,
mset:
!>[A: $tType] : ( ( list @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Omset__set,type,
mset_set:
!>[B: $tType] : ( ( set @ B ) > ( multiset @ B ) ) ).
thf(sy_c_Multiset_Omult,type,
mult:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omult1,type,
mult1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omulteqp__code,type,
multeqp_code:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Omultiset_OAbs__multiset,type,
abs_multiset:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Omultiset_Ocount,type,
count:
!>[A: $tType] : ( ( multiset @ A ) > A > nat ) ).
thf(sy_c_Multiset_Omultp,type,
multp:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Omultp__code,type,
multp_code:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Opcr__multiset,type,
pcr_multiset:
!>[C: $tType,B: $tType] : ( ( C > B > $o ) > ( C > nat ) > ( multiset @ B ) > $o ) ).
thf(sy_c_Multiset_Opw__leq,type,
pw_leq: ( multiset @ ( product_prod @ nat @ nat ) ) > ( multiset @ ( product_prod @ nat @ nat ) ) > $o ).
thf(sy_c_Multiset_Orepeat__mset,type,
repeat_mset:
!>[A: $tType] : ( nat > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oreplicate__mset,type,
replicate_mset:
!>[A: $tType] : ( nat > A > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
thf(sy_c_Multiset_Osubset__eq__mset__impl,type,
subset_eq_mset_impl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( option @ $o ) ) ).
thf(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
subset751672762298770561pl_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Multiset_Osubset__mset,type,
subset_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Osubseteq__mset,type,
subseteq_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Ounion__mset,type,
union_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: ( list @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Onat__of__num,type,
nat_of_num: num > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
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_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_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Order__Relation_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder_Omono,type,
mono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder_Ostrict__mono,type,
strict_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__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_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( A > ( set @ A ) > A ) ).
thf(sy_c_Partial__Function_Oflat__ord,type,
partial_flat_ord:
!>[A: $tType] : ( A > A > A > $o ) ).
thf(sy_c_Partial__Function_Ofun__lub,type,
partial_fun_lub:
!>[C: $tType,B: $tType,A: $tType] : ( ( ( set @ C ) > B ) > ( set @ ( A > C ) ) > A > B ) ).
thf(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
thf(sy_c_Partial__Function_Opartial__function__definitions,type,
partia7178651479351089652itions:
!>[A: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > A ) > $o ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Predicate_OSeq,type,
seq2:
!>[A: $tType] : ( ( product_unit > ( seq @ A ) ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Obind,type,
bind2:
!>[A: $tType,B: $tType] : ( ( pred @ A ) > ( A > ( pred @ B ) ) > ( pred @ B ) ) ).
thf(sy_c_Predicate_Oiterate__upto,type,
iterate_upto:
!>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( pred @ A ) ) ).
thf(sy_c_Predicate_Oiterate__upto__rel,type,
iterate_upto_rel:
!>[A: $tType] : ( ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > $o ) ).
thf(sy_c_Predicate_Opred_OPred,type,
pred2:
!>[A: $tType] : ( ( A > $o ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Oseq_OEmpty,type,
empty:
!>[A: $tType] : ( seq @ A ) ).
thf(sy_c_Predicate_Oseq_OInsert,type,
insert2:
!>[A: $tType] : ( A > ( pred @ A ) > ( seq @ A ) ) ).
thf(sy_c_Predicate_Oseq_Ocase__seq,type,
case_seq:
!>[B: $tType,A: $tType] : ( B > ( A > ( pred @ A ) > B ) > ( ( pred @ A ) > ( seq @ A ) > B ) > ( seq @ A ) > B ) ).
thf(sy_c_Predicate_Oset__of__pred,type,
set_of_pred:
!>[A: $tType] : ( ( pred @ A ) > ( set @ A ) ) ).
thf(sy_c_Predicate_Oset__of__seq,type,
set_of_seq:
!>[A: $tType] : ( ( seq @ A ) > ( set @ A ) ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc5280177257484947105e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Random_Oinc__shift,type,
inc_shift: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Oiterate,type,
iterate:
!>[B: $tType,A: $tType] : ( code_natural > ( B > A > ( product_prod @ B @ A ) ) > B > A > ( product_prod @ B @ A ) ) ).
thf(sy_c_Random_Oiterate__rel,type,
iterate_rel:
!>[B: $tType,A: $tType] : ( ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > $o ) ).
thf(sy_c_Random_Olog,type,
log: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Olog__rel,type,
log_rel: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ code_natural ) > $o ).
thf(sy_c_Random_Ominus__shift,type,
minus_shift: code_natural > code_natural > code_natural > code_natural ).
thf(sy_c_Random_Onext,type,
next: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Opick,type,
pick:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > code_natural > A ) ).
thf(sy_c_Random_Orange,type,
range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( ( list @ A ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random_Osplit__seed,type,
split_seed: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random__Pred_Obind,type,
random_bind:
!>[A: $tType,B: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Oempty,type,
random_empty:
!>[A: $tType] : ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Oiterate__upto,type,
random_iterate_upto:
!>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Ounion,type,
random_union:
!>[A: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oratrel,type,
ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Ref__Time_Oalloc,type,
ref_alloc:
!>[A: $tType] : ( A > ( heap_ext @ product_unit ) > ( product_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) ) ) ).
thf(sy_c_Ref__Time_Ochange,type,
ref_change:
!>[A: $tType] : ( ( A > A ) > ( ref @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Ref__Time_Oget,type,
ref_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > A ) ).
thf(sy_c_Ref__Time_Olookup,type,
ref_lookup:
!>[A: $tType] : ( ( ref @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Ref__Time_Opresent,type,
ref_present:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > $o ) ).
thf(sy_c_Ref__Time_Oref,type,
ref_ref:
!>[A: $tType] : ( A > ( heap_Time_Heap @ ( ref @ A ) ) ) ).
thf(sy_c_Ref__Time_Oset,type,
ref_set:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Ref__Time_Oupdate,type,
ref_update:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_Time_Heap @ product_unit ) ) ).
thf(sy_c_Refine__Imp__Hol_Oassert_H,type,
refine_Imp_assert: $o > ( heap_Time_Heap @ product_unit ) ).
thf(sy_c_Refine__Imp__Hol_Orefines,type,
refine_Imp_refines:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
thf(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Relation_ORange,type,
range2:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Relation_ORangep,type,
rangep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > $o ) ).
thf(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oantisymp,type,
antisymp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oasymp,type,
asymp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
thf(sy_c_Relation_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > A > $o ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oirreflp,type,
irreflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oreflp,type,
reflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
thf(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
thf(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Relation_Osingle__valuedp,type,
single_valuedp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Obind,type,
bind3:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert3:
!>[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_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).
thf(sy_c_Set__Interval_Oord__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_Osyntax__fo__nomatch,type,
syntax7388354845996824322omatch:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oleft__total,type,
left_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oleft__unique,type,
left_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_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_Transitive__Closure_Otranclp,type,
transitive_tranclp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_VEBT__List__Assn_OlistI__assn,type,
vEBT_List_listI_assn:
!>[A: $tType,B: $tType] : ( ( set @ nat ) > ( A > B > assn ) > ( list @ A ) > ( list @ B ) > assn ) ).
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_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Wellfounded_OwfP,type,
wfP:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Wfrec_Ocut,type,
cut:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ ( product_prod @ A @ A ) ) > A > A > B ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Zorn_OChains,type,
chains:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Zorn_Ochain__subset,type,
chain_subset:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_Zorn_Ochains,type,
chains2:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ).
thf(sy_c_Zorn_Oinit__seg__of,type,
init_seg_of:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
thf(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Omaxchain,type,
pred_maxchain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Osuc,type,
pred_suc:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Zorn_Opred__on_Osuc__Union__closedp,type,
pred_s7749564232668923593losedp:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A,type,
a: c > d > assn ).
thf(sy_v_F,type,
f: assn ).
thf(sy_v_I,type,
i: set @ nat ).
thf(sy_v_P,type,
p: assn ).
thf(sy_v_Q,type,
q: assn ).
thf(sy_v_i,type,
i2: nat ).
thf(sy_v_x,type,
x: c ).
thf(sy_v_xi,type,
xi: d ).
thf(sy_v_xs,type,
xs: list @ c ).
thf(sy_v_xsi,type,
xsi: list @ d ).
% Relevant facts (8005)
thf(fact_0_entailsI,axiom,
! [P: assn,Q: assn] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> ( rep_assn @ Q @ H ) )
=> ( entails @ P @ Q ) ) ).
% entailsI
thf(fact_1__092_060open_062P_A_092_060Longrightarrow_062_092_060_094sub_062A_Afalse_092_060close_062,axiom,
entails @ p @ ( bot_bot @ assn ) ).
% \<open>P \<Longrightarrow>\<^sub>A false\<close>
thf(fact_2_assms_I3_J,axiom,
member @ nat @ i2 @ i ).
% assms(3)
thf(fact_3_ent__iffI,axiom,
! [A3: assn,B2: assn] :
( ( entails @ A3 @ B2 )
=> ( ( entails @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% ent_iffI
thf(fact_4_ent__refl,axiom,
! [P: assn] : ( entails @ P @ P ) ).
% ent_refl
thf(fact_5_ent__trans,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ P @ Q )
=> ( ( entails @ Q @ R )
=> ( entails @ P @ R ) ) ) ).
% ent_trans
thf(fact_6_is__entails,axiom,
! [P: assn,Q: assn] :
( ( entails @ P @ Q )
=> ( entails @ P @ Q ) ) ).
% is_entails
thf(fact_7_ent__false,axiom,
! [P: assn] : ( entails @ ( bot_bot @ assn ) @ P ) ).
% ent_false
thf(fact_8_ent__fwd,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Q: assn] :
( ( rep_assn @ P @ H2 )
=> ( ( entails @ P @ Q )
=> ( rep_assn @ Q @ H2 ) ) ) ).
% ent_fwd
thf(fact_9_entailsD,axiom,
! [P: assn,Q: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( entails @ P @ Q )
=> ( ( rep_assn @ P @ H2 )
=> ( rep_assn @ Q @ H2 ) ) ) ).
% entailsD
thf(fact_10_entails__def,axiom,
( entails
= ( ^ [P2: assn,Q2: assn] :
! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H3 )
=> ( rep_assn @ Q2 @ H3 ) ) ) ) ).
% entails_def
thf(fact_11_list__assn__mono,axiom,
! [A: $tType,B: $tType,P: A > B > assn,P3: A > B > assn,L: list @ A,L2: list @ B] :
( ! [X: A,X2: B] : ( entails @ ( P @ X @ X2 ) @ ( P3 @ X @ X2 ) )
=> ( entails @ ( vEBT_List_list_assn @ A @ B @ P @ L @ L2 ) @ ( vEBT_List_list_assn @ A @ B @ P3 @ L @ L2 ) ) ) ).
% list_assn_mono
thf(fact_12_ent__false__iff,axiom,
! [P: assn] :
( ( entails @ P @ ( bot_bot @ assn ) )
= ( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H3 ) ) ) ).
% ent_false_iff
thf(fact_13_assms_I4_J,axiom,
entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ c @ d @ i @ a @ ( list_update @ c @ xs @ i2 @ x ) @ ( list_update @ d @ xsi @ i2 @ xi ) ) @ f ) @ q ).
% assms(4)
thf(fact_14_ent__imp__entt,axiom,
! [P: assn,Q: assn] :
( ( entails @ P @ Q )
=> ( entailst @ P @ Q ) ) ).
% ent_imp_entt
thf(fact_15_Rep__assn__inject,axiom,
! [X3: assn,Y: assn] :
( ( ( rep_assn @ X3 )
= ( rep_assn @ Y ) )
= ( X3 = Y ) ) ).
% Rep_assn_inject
thf(fact_16_entt__refl,axiom,
! [A3: assn] : ( entailst @ A3 @ A3 ) ).
% entt_refl
thf(fact_17_star__false__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( bot_bot @ assn ) @ P )
= ( bot_bot @ assn ) ) ).
% star_false_left
thf(fact_18_star__false__right,axiom,
! [P: assn] :
( ( times_times @ assn @ P @ ( bot_bot @ assn ) )
= ( bot_bot @ assn ) ) ).
% star_false_right
thf(fact_19_assn__aci_I10_J,axiom,
! [A4: assn,B3: assn,C2: assn] :
( ( times_times @ assn @ ( times_times @ assn @ A4 @ B3 ) @ C2 )
= ( times_times @ assn @ ( times_times @ assn @ A4 @ C2 ) @ B3 ) ) ).
% assn_aci(10)
thf(fact_20_star__aci_I3_J,axiom,
! [A4: assn,B3: assn,C2: assn] :
( ( times_times @ assn @ A4 @ ( times_times @ assn @ B3 @ C2 ) )
= ( times_times @ assn @ B3 @ ( times_times @ assn @ A4 @ C2 ) ) ) ).
% star_aci(3)
thf(fact_21_star__aci_I2_J,axiom,
( ( times_times @ assn )
= ( ^ [A5: assn,B4: assn] : ( times_times @ assn @ B4 @ A5 ) ) ) ).
% star_aci(2)
thf(fact_22_mod__false,axiom,
! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ ( bot_bot @ assn ) @ H2 ) ).
% mod_false
thf(fact_23_mod__starD,axiom,
! [A3: assn,B2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A3 @ B2 ) @ H2 )
=> ? [H1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),H22: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ A3 @ H1 )
& ( rep_assn @ B2 @ H22 ) ) ) ).
% mod_starD
thf(fact_24_mod__starE,axiom,
! [A4: assn,B3: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A4 @ B3 ) @ H2 )
=> ~ ( ? [X_1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : ( rep_assn @ A4 @ X_1 )
=> ! [H_2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ B3 @ H_2 ) ) ) ).
% mod_starE
thf(fact_25_entt__trans,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( entailst @ A3 @ B2 )
=> ( ( entailst @ B2 @ C3 )
=> ( entailst @ A3 @ C3 ) ) ) ).
% entt_trans
thf(fact_26_star__assoc,axiom,
! [A4: assn,B3: assn,C2: assn] :
( ( times_times @ assn @ ( times_times @ assn @ A4 @ B3 ) @ C2 )
= ( times_times @ assn @ A4 @ ( times_times @ assn @ B3 @ C2 ) ) ) ).
% star_assoc
thf(fact_27_entt__fr__drop,axiom,
! [F: assn,F2: assn,A3: assn] :
( ( entailst @ F @ F2 )
=> ( entailst @ ( times_times @ assn @ F @ A3 ) @ F2 ) ) ).
% entt_fr_drop
thf(fact_28_entt__fr__refl,axiom,
! [F: assn,F2: assn,A3: assn] :
( ( entailst @ F @ F2 )
=> ( entailst @ ( times_times @ assn @ F @ A3 ) @ ( times_times @ assn @ F2 @ A3 ) ) ) ).
% entt_fr_refl
thf(fact_29_entt__frame__fwd,axiom,
! [P: assn,Q: assn,A3: assn,F: assn,B2: assn] :
( ( entailst @ P @ Q )
=> ( ( entailst @ A3 @ ( times_times @ assn @ P @ F ) )
=> ( ( entailst @ ( times_times @ assn @ Q @ F ) @ B2 )
=> ( entailst @ A3 @ B2 ) ) ) ) ).
% entt_frame_fwd
thf(fact_30_entt__star__mono,axiom,
! [A3: assn,B2: assn,C3: assn,D2: assn] :
( ( entailst @ A3 @ B2 )
=> ( ( entailst @ C3 @ D2 )
=> ( entailst @ ( times_times @ assn @ A3 @ C3 ) @ ( times_times @ assn @ B2 @ D2 ) ) ) ) ).
% entt_star_mono
thf(fact_31_assn__times__comm,axiom,
( ( times_times @ assn )
= ( ^ [P2: assn,Q2: assn] : ( times_times @ assn @ Q2 @ P2 ) ) ) ).
% assn_times_comm
thf(fact_32_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_33_mod__frame__fwd,axiom,
! [Ps: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P: assn,R: assn,F: assn] :
( ( rep_assn @ Ps @ H2 )
=> ( ( entails @ P @ R )
=> ( ( entails @ Ps @ ( times_times @ assn @ P @ F ) )
=> ( rep_assn @ ( times_times @ assn @ R @ F ) @ H2 ) ) ) ) ).
% mod_frame_fwd
thf(fact_34_ent__frame__fwd,axiom,
! [P: assn,R: assn,Ps: assn,F: assn,Q: assn] :
( ( entails @ P @ R )
=> ( ( entails @ Ps @ ( times_times @ assn @ P @ F ) )
=> ( ( entails @ ( times_times @ assn @ R @ F ) @ Q )
=> ( entails @ Ps @ Q ) ) ) ) ).
% ent_frame_fwd
thf(fact_35_ent__star__mono,axiom,
! [P: assn,P3: assn,Q: assn,Q3: assn] :
( ( entails @ P @ P3 )
=> ( ( entails @ Q @ Q3 )
=> ( entails @ ( times_times @ assn @ P @ Q ) @ ( times_times @ assn @ P3 @ Q3 ) ) ) ) ).
% ent_star_mono
thf(fact_36_fr__rot__rhs,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( entails @ A3 @ ( times_times @ assn @ B2 @ C3 ) )
=> ( entails @ A3 @ ( times_times @ assn @ C3 @ B2 ) ) ) ).
% fr_rot_rhs
thf(fact_37_fr__refl,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( entails @ A3 @ B2 )
=> ( entails @ ( times_times @ assn @ A3 @ C3 ) @ ( times_times @ assn @ B2 @ C3 ) ) ) ).
% fr_refl
thf(fact_38_fr__rot,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( entails @ ( times_times @ assn @ A3 @ B2 ) @ C3 )
=> ( entails @ ( times_times @ assn @ B2 @ A3 ) @ C3 ) ) ).
% fr_rot
thf(fact_39_assms_I1_J,axiom,
entails @ p @ ( times_times @ assn @ ( times_times @ assn @ ( a @ x @ xi ) @ ( vEBT_List_listI_assn @ c @ d @ ( minus_minus @ ( set @ nat ) @ i @ ( insert3 @ nat @ i2 @ ( bot_bot @ ( set @ nat ) ) ) ) @ a @ xs @ xsi ) ) @ f ) ).
% assms(1)
thf(fact_40_list__update__overwrite,axiom,
! [A: $tType,Xs: list @ A,I: nat,X3: A,Y: A] :
( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I @ Y )
= ( list_update @ A @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_41_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_42_False,axiom,
( ( size_size @ ( list @ c ) @ xs )
!= ( size_size @ ( list @ d ) @ xsi ) ) ).
% False
thf(fact_43_assms_I2_J,axiom,
ord_less @ nat @ i2 @ ( size_size @ ( list @ c ) @ xs ) ).
% assms(2)
thf(fact_44_frame__inference__init,axiom,
! [P: assn,Q: assn,F: assn] :
( ( fI_QUERY @ P @ Q @ F )
=> ( entails @ P @ ( times_times @ assn @ Q @ F ) ) ) ).
% frame_inference_init
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G: A > B] :
( ! [X: A] :
( ( F3 @ X )
= ( G @ X ) )
=> ( F3 = G ) ) ).
% ext
thf(fact_49_FI__QUERY__def,axiom,
( fI_QUERY
= ( ^ [P2: assn,Q2: assn,F4: assn] : ( entails @ P2 @ ( times_times @ assn @ Q2 @ F4 ) ) ) ) ).
% FI_QUERY_def
thf(fact_50_sngr__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: ref @ A,X3: A,Y: A] :
( ( times_times @ assn @ ( sngr_assn @ A @ P4 @ X3 ) @ ( sngr_assn @ A @ P4 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% sngr_same_false
thf(fact_51_ent__wand__frameI,axiom,
! [Q: assn,R: assn,F: assn,S: assn,P: assn,X5: assn] :
( ( entails @ ( times_times @ assn @ ( wand_assn @ Q @ R ) @ F ) @ S )
=> ( ( entails @ P @ ( times_times @ assn @ F @ X5 ) )
=> ( ( entails @ ( times_times @ assn @ Q @ X5 ) @ R )
=> ( entails @ P @ S ) ) ) ) ).
% ent_wand_frameI
thf(fact_52_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_53_ent__mp,axiom,
! [P: assn,Q: assn] : ( entails @ ( times_times @ assn @ P @ ( wand_assn @ P @ Q ) ) @ Q ) ).
% ent_mp
thf(fact_54_snga__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: array @ A,X3: list @ A,Y: list @ A] :
( ( times_times @ assn @ ( snga_assn @ A @ P4 @ X3 ) @ ( snga_assn @ A @ P4 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% snga_same_false
thf(fact_55_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I: nat,X3: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X3 ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_56_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% order_less_imp_not_less
thf(fact_57_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( Y != X3 ) ) ) ).
% order_less_imp_not_eq2
thf(fact_58_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( X3 != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_59_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
| ( X3 = Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ).
% linorder_less_linear
thf(fact_60_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,P: $o] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ X3 )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_61_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% order_less_not_sym
thf(fact_62_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F3: A > C,C2: C] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ C @ ( F3 @ B3 ) @ C2 )
=> ( ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ C @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ C @ ( F3 @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_63_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F3: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X: B,Y2: B] :
( ( ord_less @ B @ X @ Y2 )
=> ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_64_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% order_less_irrefl
thf(fact_65_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B3: A,F3: A > B,C2: B] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ( F3 @ B3 )
= C2 )
=> ( ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ B @ ( F3 @ A4 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_66_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F3: B > A,B3: B,C2: B] :
( ( A4
= ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X: B,Y2: B] :
( ( ord_less @ B @ X @ Y2 )
=> ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F3 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_67_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% order_less_trans
thf(fact_68_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% order_less_asym'
thf(fact_69_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
= ( ( ord_less @ A @ X3 @ Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_neq_iff
thf(fact_70_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% order_less_asym
thf(fact_71_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
=> ( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_neqE
thf(fact_72_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_73_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_74_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_75_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( A4 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_76_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( A4 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_77_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_78_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( ( ord_less @ A @ Y @ X3 )
| ( X3 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_79_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_80_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B3: A] :
( ! [A6: A,B5: A] :
( ( ord_less @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_81_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P5: A > $o] :
? [X6: A] : ( P5 @ X6 ) )
= ( ^ [P2: A > $o] :
? [N2: A] :
( ( P2 @ N2 )
& ! [M: A] :
( ( ord_less @ A @ M @ N2 )
=> ~ ( P2 @ M ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_82_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_83_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_84_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_85_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_86_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A4: A] :
( ! [X: A] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_87_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_88_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_89_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% order.asym
thf(fact_90_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( X3 != Y ) ) ) ).
% less_imp_neq
thf(fact_91_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X3 @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_92_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% gt_ex
thf(fact_93_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X3 ) ) ).
% lt_ex
thf(fact_94_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] :
( ( A4
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A4 ) ) ) ).
% bot.not_eq_extremum
thf(fact_95_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_96_list__assn__aux__ineq__len,axiom,
! [B: $tType,A: $tType,L: list @ A,Li: list @ B,A3: A > B > assn] :
( ( ( size_size @ ( list @ A ) @ L )
!= ( size_size @ ( list @ B ) @ Li ) )
=> ( ( vEBT_List_list_assn @ A @ B @ A3 @ L @ Li )
= ( bot_bot @ assn ) ) ) ).
% list_assn_aux_ineq_len
thf(fact_97_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_98_list__update__swap,axiom,
! [A: $tType,I: nat,I2: nat,Xs: list @ A,X3: A,X7: A] :
( ( I != I2 )
=> ( ( list_update @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I2 @ X7 )
= ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ X7 ) @ I @ X3 ) ) ) ).
% list_update_swap
thf(fact_99_insert__Diff__single,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( insert3 @ A @ A4 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert3 @ A @ A4 @ A3 ) ) ).
% insert_Diff_single
thf(fact_100_insert__Diff1,axiom,
! [A: $tType,X3: A,B2: set @ A,A3: set @ A] :
( ( member @ A @ X3 @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_101_Diff__insert0,axiom,
! [A: $tType,X3: A,A3: set @ A,B2: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_102_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_103_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_104_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_105_singletonI,axiom,
! [A: $tType,A4: A] : ( member @ A @ A4 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_106_Diff__insert,axiom,
! [A: $tType,A3: set @ A,A4: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_107_insert__Diff,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( insert3 @ A @ A4 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_108_Diff__insert2,axiom,
! [A: $tType,A3: set @ A,A4: A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_109_insert__minus__eq,axiom,
! [A: $tType,X3: A,Y: A,A3: set @ A] :
( ( X3 != Y )
=> ( ( minus_minus @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ ( insert3 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert3 @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% insert_minus_eq
thf(fact_110_Diff__insert__absorb,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_111_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_112_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_113_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_114_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_115_insertCI,axiom,
! [A: $tType,A4: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A4 @ B2 )
=> ( A4 = B3 ) )
=> ( member @ A @ A4 @ ( insert3 @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_116_insert__iff,axiom,
! [A: $tType,A4: A,B3: A,A3: set @ A] :
( ( member @ A @ A4 @ ( insert3 @ A @ B3 @ A3 ) )
= ( ( A4 = B3 )
| ( member @ A @ A4 @ A3 ) ) ) ).
% insert_iff
thf(fact_117_insert__absorb2,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( insert3 @ A @ X3 @ ( insert3 @ A @ X3 @ A3 ) )
= ( insert3 @ A @ X3 @ A3 ) ) ).
% insert_absorb2
thf(fact_118_DiffI,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_119_Diff__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( ( member @ A @ C2 @ A3 )
& ~ ( member @ A @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_120_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ).
% Diff_idemp
thf(fact_121_not__psubset__empty,axiom,
! [A: $tType,A3: set @ A] :
~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_122_psubset__imp__ex__mem,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_123_remove__subset,axiom,
! [A: $tType,X3: A,S: set @ A] :
( ( member @ A @ X3 @ S )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ S ) ) ).
% remove_subset
thf(fact_124_memb__imp__not__empty,axiom,
! [A: $tType,X3: A,S: set @ A] :
( ( member @ A @ X3 @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_125_set__notEmptyE,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X: A] :
~ ( member @ A @ X @ S ) ) ).
% set_notEmptyE
thf(fact_126_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_127_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_128_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_129_equals0D,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A4 @ A3 ) ) ).
% equals0D
thf(fact_130_emptyE,axiom,
! [A: $tType,A4: A] :
~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_131_insertE,axiom,
! [A: $tType,A4: A,B3: A,A3: set @ A] :
( ( member @ A @ A4 @ ( insert3 @ A @ B3 @ A3 ) )
=> ( ( A4 != B3 )
=> ( member @ A @ A4 @ A3 ) ) ) ).
% insertE
thf(fact_132_insertI1,axiom,
! [A: $tType,A4: A,B2: set @ A] : ( member @ A @ A4 @ ( insert3 @ A @ A4 @ B2 ) ) ).
% insertI1
thf(fact_133_insertI2,axiom,
! [A: $tType,A4: A,B2: set @ A,B3: A] :
( ( member @ A @ A4 @ B2 )
=> ( member @ A @ A4 @ ( insert3 @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_134_Set_Oset__insert,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ~ ! [B6: set @ A] :
( ( A3
= ( insert3 @ A @ X3 @ B6 ) )
=> ( member @ A @ X3 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_135_insert__ident,axiom,
! [A: $tType,X3: A,A3: set @ A,B2: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ~ ( member @ A @ X3 @ B2 )
=> ( ( ( insert3 @ A @ X3 @ A3 )
= ( insert3 @ A @ X3 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_136_insert__absorb,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( insert3 @ A @ A4 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_137_insert__eq__iff,axiom,
! [A: $tType,A4: A,A3: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A4 @ A3 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert3 @ A @ A4 @ A3 )
= ( insert3 @ A @ B3 @ B2 ) )
= ( ( ( A4 = B3 )
=> ( A3 = B2 ) )
& ( ( A4 != B3 )
=> ? [C4: set @ A] :
( ( A3
= ( insert3 @ A @ B3 @ C4 ) )
& ~ ( member @ A @ B3 @ C4 )
& ( B2
= ( insert3 @ A @ A4 @ C4 ) )
& ~ ( member @ A @ A4 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_138_insert__commute,axiom,
! [A: $tType,X3: A,Y: A,A3: set @ A] :
( ( insert3 @ A @ X3 @ ( insert3 @ A @ Y @ A3 ) )
= ( insert3 @ A @ Y @ ( insert3 @ A @ X3 @ A3 ) ) ) ).
% insert_commute
thf(fact_139_mk__disjoint__insert,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ? [B6: set @ A] :
( ( A3
= ( insert3 @ A @ A4 @ B6 ) )
& ~ ( member @ A @ A4 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_140_DiffE,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_141_DiffD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_142_DiffD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
=> ~ ( member @ A @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_143_singleton__inject,axiom,
! [A: $tType,A4: A,B3: A] :
( ( ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A4 = B3 ) ) ).
% singleton_inject
thf(fact_144_insert__not__empty,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( insert3 @ A @ A4 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_145_doubleton__eq__iff,axiom,
! [A: $tType,A4: A,B3: A,C2: A,D3: A] :
( ( ( insert3 @ A @ A4 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert3 @ A @ C2 @ ( insert3 @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A4 = C2 )
& ( B3 = D3 ) )
| ( ( A4 = D3 )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_146_singleton__iff,axiom,
! [A: $tType,B3: A,A4: A] :
( ( member @ A @ B3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A4 ) ) ).
% singleton_iff
thf(fact_147_singletonD,axiom,
! [A: $tType,B3: A,A4: A] :
( ( member @ A @ B3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A4 ) ) ).
% singletonD
thf(fact_148_insert__Diff__if,axiom,
! [A: $tType,X3: A,B2: set @ A,A3: set @ A] :
( ( ( member @ A @ X3 @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) )
& ( ~ ( member @ A @ X3 @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ B2 )
= ( insert3 @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_149_set__minus__singleton__eq,axiom,
! [A: $tType,X3: A,X5: set @ A] :
( ~ ( member @ A @ X3 @ X5 )
=> ( ( minus_minus @ ( set @ A ) @ X5 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X5 ) ) ).
% set_minus_singleton_eq
thf(fact_150_listI__assn__reinsert,axiom,
! [B: $tType,A: $tType,P: assn,A3: A > B > assn,Xs: list @ A,I: nat,Xsi: list @ B,I3: set @ nat,F: assn,Q: assn] :
( ( entails @ P @ ( times_times @ assn @ ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ I3 @ ( insert3 @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs @ Xsi ) ) @ F ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( member @ nat @ I @ I3 )
=> ( ( entails @ ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ I3 @ A3 @ Xs @ Xsi ) @ F ) @ Q )
=> ( entails @ P @ Q ) ) ) ) ) ).
% listI_assn_reinsert
thf(fact_151_the__elem__eq,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ).
% the_elem_eq
thf(fact_152_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B7: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B7 @ X4 ) ) ) ) ) ).
% minus_apply
thf(fact_153_listI__assn__extract,axiom,
! [A: $tType,B: $tType,I: nat,I3: set @ nat,Xs: list @ A,A3: A > B > assn,Xsi: list @ B] :
( ( member @ nat @ I @ I3 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I3 @ A3 @ Xs @ Xsi )
= ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ ( minus_minus @ ( set @ nat ) @ I3 @ ( insert3 @ nat @ I @ ( bot_bot @ ( set @ nat ) ) ) ) @ A3 @ Xs @ Xsi ) ) ) ) ) ).
% listI_assn_extract
thf(fact_154_is__singletonI,axiom,
! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_155_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D2: A,Q: A > $o] :
( ! [X: A,K: A] :
( ( P @ X )
= ( P @ ( minus_minus @ A @ X @ ( times_times @ A @ K @ D2 ) ) ) )
=> ( ! [X: A,K: A] :
( ( Q @ X )
= ( Q @ ( minus_minus @ A @ X @ ( times_times @ A @ K @ D2 ) ) ) )
=> ! [X8: A,K2: A] :
( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K2 @ D2 ) ) )
| ( Q @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K2 @ D2 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_156_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D2: A,Q: A > $o] :
( ! [X: A,K: A] :
( ( P @ X )
= ( P @ ( minus_minus @ A @ X @ ( times_times @ A @ K @ D2 ) ) ) )
=> ( ! [X: A,K: A] :
( ( Q @ X )
= ( Q @ ( minus_minus @ A @ X @ ( times_times @ A @ K @ D2 ) ) ) )
=> ! [X8: A,K2: A] :
( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K2 @ D2 ) ) )
& ( Q @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K2 @ D2 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_157_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_158_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ A4 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_159_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( times_times @ A @ ( minus_minus @ A @ B3 @ C2 ) @ A4 )
= ( minus_minus @ A @ ( times_times @ A @ B3 @ A4 ) @ ( times_times @ A @ C2 @ A4 ) ) ) ) ).
% left_diff_distrib'
thf(fact_160_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ A4 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_161_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,D3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ D3 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_162_nth__list__update__neq,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A,X3: A] :
( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_163_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_164_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ I )
= X3 ) ) ).
% nth_list_update_eq
thf(fact_165_nth__update__invalid,axiom,
! [A: $tType,I: nat,L: list @ A,J: nat,X3: A] :
( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ J @ X3 ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_update_invalid
thf(fact_166_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% psubset_trans
thf(fact_167_psubsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_168_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z4: list @ A] : Y4 = Z4 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I4 )
= ( nth @ A @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_169_Skolem__list__nth,axiom,
! [A: $tType,K3: nat,P: nat > A > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K3 )
=> ? [X9: A] : ( P @ I4 @ X9 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K3 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K3 )
=> ( P @ I4 @ ( nth @ A @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_170_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I5 )
= ( nth @ A @ Ys @ I5 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_171_obtain__list__from__elements,axiom,
! [A: $tType,N: nat,P: A > nat > $o] :
( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ N )
=> ? [Li2: A] : ( P @ Li2 @ I5 ) )
=> ~ ! [L3: list @ A] :
( ( ( size_size @ ( list @ A ) @ L3 )
= N )
=> ~ ! [I6: nat] :
( ( ord_less @ nat @ I6 @ N )
=> ( P @ ( nth @ A @ L3 @ I6 ) @ I6 ) ) ) ) ).
% obtain_list_from_elements
thf(fact_172_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( A7
= ( insert3 @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_173_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A,Y2: A] :
( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( X = Y2 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_174_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J )
= X3 ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_175_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X3 )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X3 ) ) ) ).
% list_update_same_conv
thf(fact_176_nth__list__update_H,axiom,
! [A: $tType,I: nat,J: nat,L: list @ A,X3: A] :
( ( ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X3 ) @ J )
= X3 ) )
& ( ~ ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X3 ) @ J )
= ( nth @ A @ L @ J ) ) ) ) ).
% nth_list_update'
thf(fact_177_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
=> ( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_178_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ( ( Q @ X )
= ( Q3 @ X ) ) )
=> ? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ Z3 @ X8 )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P3 @ X8 )
& ( Q3 @ X8 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_179_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ( ( Q @ X )
= ( Q3 @ X ) ) )
=> ? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ Z3 @ X8 )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P3 @ X8 )
| ( Q3 @ X8 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_180_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ Z3 @ X8 )
=> ( X8 != T2 ) ) ) ).
% pinf(3)
thf(fact_181_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ Z3 @ X8 )
=> ( X8 != T2 ) ) ) ).
% pinf(4)
thf(fact_182_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ Z3 @ X8 )
=> ~ ( ord_less @ A @ X8 @ T2 ) ) ) ).
% pinf(5)
thf(fact_183_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ Z3 @ X8 )
=> ( ord_less @ A @ T2 @ X8 ) ) ) ).
% pinf(7)
thf(fact_184_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F: D] :
? [Z3: C] :
! [X8: C] :
( ( ord_less @ C @ Z3 @ X8 )
=> ( F = F ) ) ) ).
% pinf(11)
thf(fact_185_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ( ( Q @ X )
= ( Q3 @ X ) ) )
=> ? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z3 )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
= ( ( P3 @ X8 )
& ( Q3 @ X8 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_186_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ( ( Q @ X )
= ( Q3 @ X ) ) )
=> ? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z3 )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
= ( ( P3 @ X8 )
| ( Q3 @ X8 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_187_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z3 )
=> ( X8 != T2 ) ) ) ).
% minf(3)
thf(fact_188_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z3 )
=> ( X8 != T2 ) ) ) ).
% minf(4)
thf(fact_189_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z3 )
=> ( ord_less @ A @ X8 @ T2 ) ) ) ).
% minf(5)
thf(fact_190_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z3 )
=> ~ ( ord_less @ A @ T2 @ X8 ) ) ) ).
% minf(7)
thf(fact_191_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F: D] :
? [Z3: C] :
! [X8: C] :
( ( ord_less @ C @ X8 @ Z3 )
=> ( F = F ) ) ) ).
% minf(11)
thf(fact_192_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( times_times @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) )
= ( times_times @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_193_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ B4 @ A5 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_194_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( times_times @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_195_listI__assn__cong,axiom,
! [A: $tType,B: $tType,I3: set @ nat,I7: set @ nat,Xs: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B,A3: A > B > assn,A8: A > B > assn] :
( ( I3 = I7 )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Xs4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Xsi )
= ( size_size @ ( list @ B ) @ Xsi2 ) )
=> ( ! [I5: nat] :
( ( member @ nat @ I5 @ I3 )
=> ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Xsi ) )
=> ( ( ( nth @ A @ Xs @ I5 )
= ( nth @ A @ Xs4 @ I5 ) )
& ( ( nth @ B @ Xsi @ I5 )
= ( nth @ B @ Xsi2 @ I5 ) )
& ( ( A3 @ ( nth @ A @ Xs @ I5 ) @ ( nth @ B @ Xsi @ I5 ) )
= ( A8 @ ( nth @ A @ Xs4 @ I5 ) @ ( nth @ B @ Xsi2 @ I5 ) ) ) ) ) ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I3 @ A3 @ Xs @ Xsi )
= ( vEBT_List_listI_assn @ A @ B @ I7 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).
% listI_assn_cong
thf(fact_196_listI__assn__weak__cong,axiom,
! [A: $tType,B: $tType,I3: set @ nat,I7: set @ nat,A3: A > B > assn,A8: A > B > assn,Xs: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B] :
( ( I3 = I7 )
=> ( ( A3 = A8 )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Xs4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Xsi )
= ( size_size @ ( list @ B ) @ Xsi2 ) )
=> ( ! [I5: nat] :
( ( member @ nat @ I5 @ I3 )
=> ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Xsi ) )
=> ( ( ( nth @ A @ Xs @ I5 )
= ( nth @ A @ Xs4 @ I5 ) )
& ( ( nth @ B @ Xsi @ I5 )
= ( nth @ B @ Xsi2 @ I5 ) ) ) ) ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ I3 @ A3 @ Xs @ Xsi )
= ( vEBT_List_listI_assn @ A @ B @ I7 @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).
% listI_assn_weak_cong
thf(fact_197_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B7: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B7 @ X4 ) ) ) ) ) ).
% fun_diff_def
thf(fact_198_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C2 ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_199_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( A4 = B3 )
= ( C2 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_200_is__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( is_singleton @ A @ A3 )
=> ~ ! [X: A] :
( A3
!= ( insert3 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_201_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
? [X4: A] :
( A7
= ( insert3 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_202_listI__assn__insert,axiom,
! [A: $tType,B: $tType,I: nat,I3: set @ nat,Xs: list @ A,A3: A > B > assn,Xsi: list @ B] :
( ~ ( member @ nat @ I @ I3 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( insert3 @ nat @ I @ I3 ) @ A3 @ Xs @ Xsi )
= ( times_times @ assn @ ( A3 @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Xsi @ I ) ) @ ( vEBT_List_listI_assn @ A @ B @ I3 @ A3 @ Xs @ Xsi ) ) ) ) ) ).
% listI_assn_insert
thf(fact_203_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B3 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_204_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A4 ) @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_205_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A4 @ B3 )
= ( ord_less @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_206_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_207_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_208_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X4: A,A7: set @ A] : ( minus_minus @ ( set @ A ) @ A7 @ ( insert3 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_209_insert__swap__set__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( insert3 @ A @ ( nth @ A @ L @ I ) @ ( set2 @ A @ ( list_update @ A @ L @ I @ X3 ) ) )
= ( insert3 @ A @ X3 @ ( set2 @ A @ L ) ) ) ) ).
% insert_swap_set_eq
thf(fact_210_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_211_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A7: set @ A] :
( A7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_212_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_213_list__ex__length,axiom,
! [A: $tType] :
( ( list_ex @ A )
= ( ^ [P2: A > $o,Xs3: list @ A] :
? [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P2 @ ( nth @ A @ Xs3 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_214_mod__h__bot__iff_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: ref @ A,X3: A,H2: heap_ext @ product_unit] :
~ ( rep_assn @ ( sngr_assn @ A @ P4 @ X3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(3)
thf(fact_215_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( ( heap @ B )
=> ! [Q4: array @ B,Y: list @ B,H2: heap_ext @ product_unit] :
~ ( rep_assn @ ( snga_assn @ B @ Q4 @ Y ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(4)
thf(fact_216_ent__pure__post__iff,axiom,
! [P: assn,Q: assn,B3: $o] :
( ( entails @ P @ ( times_times @ assn @ Q @ ( pure_assn @ B3 ) ) )
= ( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H3 )
=> B3 )
& ( entails @ P @ Q ) ) ) ).
% ent_pure_post_iff
thf(fact_217_pure__assn__eq__conv,axiom,
! [P: $o,Q: $o] :
( ( ( pure_assn @ P )
= ( pure_assn @ Q ) )
= ( P = Q ) ) ).
% pure_assn_eq_conv
thf(fact_218_member__remove,axiom,
! [A: $tType,X3: A,Y: A,A3: set @ A] :
( ( member @ A @ X3 @ ( remove @ A @ Y @ A3 ) )
= ( ( member @ A @ X3 @ A3 )
& ( X3 != Y ) ) ) ).
% member_remove
thf(fact_219_merge__pure__star,axiom,
! [A4: $o,B3: $o] :
( ( times_times @ assn @ ( pure_assn @ A4 ) @ ( pure_assn @ B3 ) )
= ( pure_assn
@ ( A4
& B3 ) ) ) ).
% merge_pure_star
thf(fact_220_pure__false,axiom,
( ( pure_assn @ $false )
= ( bot_bot @ assn ) ) ).
% pure_false
thf(fact_221_pure__assn__eq__false__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( bot_bot @ assn ) )
= ~ P ) ).
% pure_assn_eq_false_iff
thf(fact_222_mod__pure__star__dist,axiom,
! [P: assn,B3: $o,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ H2 )
= ( ( rep_assn @ P @ H2 )
& B3 ) ) ).
% mod_pure_star_dist
thf(fact_223_ent__pure__pre__iff,axiom,
! [P: assn,B3: $o,Q: assn] :
( ( entails @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ Q )
= ( B3
=> ( entails @ P @ Q ) ) ) ).
% ent_pure_pre_iff
thf(fact_224_mod__h__bot__iff_I5_J,axiom,
! [P: assn,Q: assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(5)
thf(fact_225_mod__h__bot__iff_I1_J,axiom,
! [B3: $o,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( pure_assn @ B3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= B3 ) ).
% mod_h_bot_iff(1)
thf(fact_226_list__ex__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,F3: A > $o,G: A > $o] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ( F3 @ X )
= ( G @ X ) ) )
=> ( ( list_ex @ A @ F3 @ Xs )
= ( list_ex @ A @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_227_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_228_one__assn__raw_Ocases,axiom,
! [X3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( X3
!= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ).
% one_assn_raw.cases
thf(fact_229_bex2I,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ S )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ S )
=> ( P @ A4 @ B3 ) )
=> ? [A6: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ S )
& ( P @ A6 @ B5 ) ) ) ) ).
% bex2I
thf(fact_230_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X3: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F5: A > B,A6: A,B5: A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F5 @ ( product_Pair @ A @ A @ A6 @ B5 ) ) ) ).
% pairself.cases
thf(fact_231_list__assn__cong,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xs4: list @ A,Xsi: list @ B,Xsi2: list @ B,A3: A > B > assn,A8: A > B > assn] :
( ( Xs = Xs4 )
=> ( ( Xsi = Xsi2 )
=> ( ! [X: A,Xi: B] :
( ( member @ A @ X @ ( set2 @ A @ Xs4 ) )
=> ( ( member @ B @ Xi @ ( set2 @ B @ Xsi2 ) )
=> ( ( A3 @ X @ Xi )
= ( A8 @ X @ Xi ) ) ) )
=> ( ( vEBT_List_list_assn @ A @ B @ A3 @ Xs @ Xsi )
= ( vEBT_List_list_assn @ A @ B @ A8 @ Xs4 @ Xsi2 ) ) ) ) ) ).
% list_assn_cong
thf(fact_232_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ? [X: nat] :
( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X )
= X3 )
& ! [Y3: nat] :
( ( ( ord_less @ nat @ Y3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y3 )
= X3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% distinct_Ex1
thf(fact_233_mod__h__bot__indep,axiom,
! [P: assn,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_indep
thf(fact_234_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A4: A,I: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A4 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert3 @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I @ A4 ) ) ) ) ).
% distinct_list_update
thf(fact_235_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs3: list @ A] :
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [J2: nat] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( I4 != J2 )
=> ( ( nth @ A @ Xs3 @ I4 )
!= ( nth @ A @ Xs3 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_236_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_237_all__set__conv__nth,axiom,
! [A: $tType,L: list @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ L ) )
=> ( P @ X4 ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) )
=> ( P @ ( nth @ A @ L @ I4 ) ) ) ) ) ).
% all_set_conv_nth
thf(fact_238_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_239_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_240_in__set__conv__nth,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I4 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_241_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X3: A] :
( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I5 ) ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_242_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_243_in__set__upd__eq__aux,axiom,
! [A: $tType,I: nat,L: list @ A,X3: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X3 = Y )
| ! [Y5: A] : ( member @ A @ X3 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y5 ) ) ) ) ) ) ).
% in_set_upd_eq_aux
thf(fact_244_in__set__upd__cases,axiom,
! [A: $tType,X3: A,L: list @ A,I: nat,Y: A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
=> ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( X3 != Y ) )
=> ( member @ A @ X3 @ ( set2 @ A @ L ) ) ) ) ).
% in_set_upd_cases
thf(fact_245_in__set__upd__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X3: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X3 = Y )
| ( ( member @ A @ X3 @ ( set2 @ A @ L ) )
& ! [Y5: A] : ( member @ A @ X3 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y5 ) ) ) ) ) ) ) ).
% in_set_upd_eq
thf(fact_246_set__update__memI,axiom,
! [A: $tType,N: nat,Xs: list @ A,X3: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ X3 @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X3 ) ) ) ) ).
% set_update_memI
thf(fact_247_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N: nat,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N @ X3 ) )
= ( insert3 @ A @ X3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert3 @ A @ ( nth @ A @ Xs @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_248_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_249_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A9: A,B8: B] :
( ( ( product_Pair @ A @ B @ A4 @ B3 )
= ( product_Pair @ A @ B @ A9 @ B8 ) )
= ( ( A4 = A9 )
& ( B3 = B8 ) ) ) ).
% old.prod.inject
thf(fact_250_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_251_set__remove1__eq,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( set2 @ A @ ( remove1 @ A @ X3 @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_remove1_eq
thf(fact_252_is__empty__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( is_empty @ A @ ( set2 @ A @ Xs ) )
= ( null @ A @ Xs ) ) ).
% is_empty_set
thf(fact_253_set__removeAll,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( set2 @ A @ ( removeAll @ A @ X3 @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_removeAll
thf(fact_254_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
= ( ? [Y5: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N2 ) @ Y5 ) @ R2 )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N2 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_255_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A4: B,B3: C] :
( ( uncurry @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A4 @ B3 ) )
= ( F3 @ A4 @ B3 ) ) ).
% uncurry_apply
thf(fact_256_List_Oset__insert,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( set2 @ A @ ( insert @ A @ X3 @ Xs ) )
= ( insert3 @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% List.set_insert
thf(fact_257_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N2 ) @ ( nth @ B @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_258_ent__pure__post__iff__sng,axiom,
! [P: assn,B3: $o] :
( ( entails @ P @ ( pure_assn @ B3 ) )
= ( ! [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H3 )
=> B3 )
& ( entails @ P @ ( one_one @ assn ) ) ) ) ).
% ent_pure_post_iff_sng
thf(fact_259_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( one_one @ A ) @ A4 )
= A4 ) ) ).
% mult_1
thf(fact_260_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( one_one @ A ) )
= A4 ) ) ).
% mult.right_neutral
thf(fact_261_in__set__remove1,axiom,
! [A: $tType,A4: A,B3: A,Xs: list @ A] :
( ( A4 != B3 )
=> ( ( member @ A @ A4 @ ( set2 @ A @ ( remove1 @ A @ B3 @ Xs ) ) )
= ( member @ A @ A4 @ ( set2 @ A @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_262_removeAll__id,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( removeAll @ A @ X3 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_263_pure__true,axiom,
( ( pure_assn @ $true )
= ( one_one @ assn ) ) ).
% pure_true
thf(fact_264_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( one_one @ assn ) )
= P ) ).
% pure_assn_eq_emp_iff
thf(fact_265_entt__emp,axiom,
! [A3: assn] : ( entailst @ A3 @ ( one_one @ assn ) ) ).
% entt_emp
thf(fact_266_in__set__insert,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X3 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_267_assn__basic__inequalities_I3_J,axiom,
( ( bot_bot @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(3)
thf(fact_268_distinct__insert,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( distinct @ A @ ( insert @ A @ X3 @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_insert
thf(fact_269_ent__pure__pre__iff__sng,axiom,
! [B3: $o,Q: assn] :
( ( entails @ ( pure_assn @ B3 ) @ Q )
= ( B3
=> ( entails @ ( one_one @ assn ) @ Q ) ) ) ).
% ent_pure_pre_iff_sng
thf(fact_270_remove1__commute,axiom,
! [A: $tType,X3: A,Y: A,Zs: list @ A] :
( ( remove1 @ A @ X3 @ ( remove1 @ A @ Y @ Zs ) )
= ( remove1 @ A @ Y @ ( remove1 @ A @ X3 @ Zs ) ) ) ).
% remove1_commute
thf(fact_271_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X3: A] :
( ( ( one_one @ A )
= X3 )
= ( X3
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_272_times__assn__raw_Ocases,axiom,
! [X3: 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 ) ) )] :
~ ! [P6: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q5: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H: heap_ext @ product_unit,As: set @ nat] :
( X3
!= ( 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 ) ) ) @ P6 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Q5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ).
% times_assn_raw.cases
thf(fact_273_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_274_sngr__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R3: ref @ A,X: A,H: heap_ext @ product_unit,As: set @ nat] :
( X3
!= ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R3 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ) ).
% sngr_assn_raw.cases
thf(fact_275_snga__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R3: array @ A,X: list @ A,H: heap_ext @ product_unit,As: set @ nat] :
( X3
!= ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R3 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ) ).
% snga_assn_raw.cases
thf(fact_276_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_277_distinct__remove1__removeAll,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X3 @ Xs )
= ( removeAll @ A @ X3 @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_278_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X3: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
~ ! [B5: $o,H: A,As: set @ B] :
( X3
!= ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B5 @ ( product_Pair @ A @ ( set @ B ) @ H @ As ) ) ) ).
% pure_assn_raw.cases
thf(fact_279_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( one_one @ A ) )
= A4 ) ) ).
% mult.comm_neutral
thf(fact_280_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( one_one @ A ) @ A4 )
= A4 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_281_assn__one__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ P )
= P ) ).
% assn_one_left
thf(fact_282_norm__assertion__simps_I1_J,axiom,
! [A4: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ A4 )
= A4 ) ).
% norm_assertion_simps(1)
thf(fact_283_norm__assertion__simps_I2_J,axiom,
! [A4: assn] :
( ( times_times @ assn @ A4 @ ( one_one @ assn ) )
= A4 ) ).
% norm_assertion_simps(2)
thf(fact_284_remove1__idem,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X3 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_285_notin__set__remove1,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ~ ( member @ A @ X3 @ ( set2 @ A @ ( remove1 @ A @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_286_distinct__remove1,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( remove1 @ A @ X3 @ Xs ) ) ) ).
% distinct_remove1
thf(fact_287_distinct__removeAll,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( removeAll @ A @ X3 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_288_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_289_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_290_prod__induct7,axiom,
! [G2: $tType,F6: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G2 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G2 ) ) ) ) )] :
( ! [A6: A,B5: B,C5: C,D4: D,E2: E,F5: F6,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G2 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G2 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F6 @ G2 ) @ E2 @ ( product_Pair @ F6 @ G2 @ F5 @ G3 ) ) ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct7
thf(fact_291_prod__induct6,axiom,
! [F6: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) )] :
( ! [A6: A,B5: B,C5: C,D4: D,E2: E,F5: F6] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ F6 ) @ D4 @ ( product_Pair @ E @ F6 @ E2 @ F5 ) ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct6
thf(fact_292_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A6: A,B5: B,C5: C,D4: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C5 @ ( product_Pair @ D @ E @ D4 @ E2 ) ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct5
thf(fact_293_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A6: A,B5: B,C5: C,D4: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D4 ) ) ) )
=> ( P @ X3 ) ) ).
% prod_induct4
thf(fact_294_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A6: A,B5: B,C5: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_295_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F6: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G2 ) ) ) ) )] :
~ ! [A6: A,B5: B,C5: C,D4: D,E2: E,F5: F6,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G2 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G2 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F6 @ G2 ) @ E2 @ ( product_Pair @ F6 @ G2 @ F5 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_296_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F6: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) )] :
~ ! [A6: A,B5: B,C5: C,D4: D,E2: E,F5: F6] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E @ F6 ) @ D4 @ ( product_Pair @ E @ F6 @ E2 @ F5 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_297_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A6: A,B5: B,C5: C,D4: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C5 @ ( product_Pair @ D @ E @ D4 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_298_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B5: B,C5: C,D4: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C5 @ D4 ) ) ) ) ).
% prod_cases4
thf(fact_299_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B5: B,C5: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C5 ) ) ) ).
% prod_cases3
thf(fact_300_Pair__inject,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A9: A,B8: B] :
( ( ( product_Pair @ A @ B @ A4 @ B3 )
= ( product_Pair @ A @ B @ A9 @ B8 ) )
=> ~ ( ( A4 = A9 )
=> ( B3 != B8 ) ) ) ).
% Pair_inject
thf(fact_301_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_302_surj__pair,axiom,
! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
? [X: A,Y2: B] :
( P4
= ( product_Pair @ A @ B @ X @ Y2 ) ) ).
% surj_pair
thf(fact_303_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_304_remove__code_I1_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( remove @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( removeAll @ A @ X3 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_305_mod__emp__simp,axiom,
! [H2: heap_ext @ product_unit] : ( rep_assn @ ( one_one @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_emp_simp
thf(fact_306_length__removeAll__less,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_307_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A4: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( F1 @ A4 @ B3 ) ) ).
% old.prod.rec
thf(fact_308_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_309_remove__code_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( remove @ A @ X3 @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( insert @ A @ X3 @ Xs ) ) ) ).
% remove_code(2)
thf(fact_310_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A4: B,B3: C] :
( ( produc5280177257484947105e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A4 @ B3 ) )
= ( C2 @ A4 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_311_curry__uncurry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C] :
( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F3 ) )
= F3 ) ).
% curry_uncurry_id
thf(fact_312_uncurry__curry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F3 ) )
= F3 ) ).
% uncurry_curry_id
thf(fact_313_pure__assn__raw_Oelims_I3_J,axiom,
! [B: $tType,A: $tType,X3: $o,Xa: product_prod @ A @ ( set @ B )] :
( ~ ( pure_assn_raw @ A @ B @ X3 @ Xa )
=> ~ ! [H: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As ) )
=> ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X3 ) ) ) ).
% pure_assn_raw.elims(3)
thf(fact_314_pure__assn__raw_Oelims_I2_J,axiom,
! [B: $tType,A: $tType,X3: $o,Xa: product_prod @ A @ ( set @ B )] :
( ( pure_assn_raw @ A @ B @ X3 @ Xa )
=> ~ ! [H: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As ) )
=> ~ ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X3 ) ) ) ).
% pure_assn_raw.elims(2)
thf(fact_315_pure__assn__raw_Oelims_I1_J,axiom,
! [A: $tType,B: $tType,X3: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
( ( ( pure_assn_raw @ A @ B @ X3 @ Xa )
= Y )
=> ~ ! [H: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As ) )
=> ( Y
= ( ~ ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X3 ) ) ) ) ) ).
% pure_assn_raw.elims(1)
thf(fact_316_pure__assn__raw_Osimps,axiom,
! [A: $tType,B: $tType,B3: $o,H2: A,As2: set @ B] :
( ( pure_assn_raw @ A @ B @ B3 @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As2 ) )
= ( ( As2
= ( bot_bot @ ( set @ B ) ) )
& B3 ) ) ).
% pure_assn_raw.simps
thf(fact_317_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_318_curryI,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A4: A,B3: B] :
( ( F3 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( product_curry @ A @ B @ $o @ F3 @ A4 @ B3 ) ) ).
% curryI
thf(fact_319_zip__eq__zip__same__len,axiom,
! [A: $tType,B: $tType,A4: list @ A,B3: list @ B,A9: list @ A,B8: list @ B] :
( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ( ( size_size @ ( list @ A ) @ A9 )
= ( size_size @ ( list @ B ) @ B8 ) )
=> ( ( ( zip @ A @ B @ A4 @ B3 )
= ( zip @ A @ B @ A9 @ B8 ) )
= ( ( A4 = A9 )
& ( B3 = B8 ) ) ) ) ) ).
% zip_eq_zip_same_len
thf(fact_320_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F7: ( product_prod @ B @ C ) > A,A5: B,B4: C] : ( F7 @ ( product_Pair @ B @ C @ A5 @ B4 ) ) ) ) ).
% curry_conv
thf(fact_321_zip__inj,axiom,
! [A: $tType,B: $tType,A4: list @ A,B3: list @ B,A9: list @ A,B8: list @ B] :
( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ( ( size_size @ ( list @ A ) @ A9 )
= ( size_size @ ( list @ B ) @ B8 ) )
=> ( ( ( zip @ A @ B @ A4 @ B3 )
= ( zip @ A @ B @ A9 @ B8 ) )
=> ( ( A4 = A9 )
& ( B3 = B8 ) ) ) ) ) ).
% zip_inj
thf(fact_322_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_323_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_324_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_325_curryD,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A4: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F3 @ A4 @ B3 )
=> ( F3 @ ( product_Pair @ A @ B @ A4 @ B3 ) ) ) ).
% curryD
thf(fact_326_curryE,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A4: A,B3: B] :
( ( product_curry @ A @ B @ $o @ F3 @ A4 @ B3 )
=> ( F3 @ ( product_Pair @ A @ B @ A4 @ B3 ) ) ) ).
% curryE
thf(fact_327_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_328_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_329_in__set__zipE,axiom,
! [A: $tType,B: $tType,X3: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ~ ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ~ ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_330_zip__same,axiom,
! [A: $tType,A4: A,B3: A,Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
= ( ( member @ A @ A4 @ ( set2 @ A @ Xs ) )
& ( A4 = B3 ) ) ) ).
% zip_same
thf(fact_331_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,X3: A,Ys: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X3 ) @ ( list_update @ B @ Ys @ I @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ ( product_Pair @ A @ B @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_332_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ~ ! [Y2: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_333_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ Ys ) )
=> ~ ! [X: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_334_length__remove1,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X3 @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X3 @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_remove1
thf(fact_335_insert__code_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( insert3 @ A @ X3 @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( removeAll @ A @ X3 @ Xs ) ) ) ).
% insert_code(2)
thf(fact_336_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_337_one__assn__raw_Osimps,axiom,
! [H2: heap_ext @ product_unit,As2: set @ nat] :
( ( one_assn_raw @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
= ( As2
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% one_assn_raw.simps
thf(fact_338_one__assn__raw_Oelims_I1_J,axiom,
! [X3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( one_assn_raw @ X3 )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( X3
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( Y
= ( As
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% one_assn_raw.elims(1)
thf(fact_339_one__assn__raw_Oelims_I2_J,axiom,
! [X3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( one_assn_raw @ X3 )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( X3
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( As
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(2)
thf(fact_340_one__assn__raw_Oelims_I3_J,axiom,
! [X3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( one_assn_raw @ X3 )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( X3
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( As
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(3)
thf(fact_341_psubset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X3: A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ B2 ) )
= ( ( ( member @ A @ X3 @ B2 )
=> ( ord_less @ ( set @ A ) @ A3 @ B2 ) )
& ( ~ ( member @ A @ X3 @ B2 )
=> ( ( ( member @ A @ X3 @ A3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_342_lexn__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexn @ A @ R2 @ N ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= N )
& ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ) ).
% lexn_length
thf(fact_343_pairwise__alt,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R4: A > A > $o,S2: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert3 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( R4 @ X4 @ Y5 ) ) ) ) ) ).
% pairwise_alt
thf(fact_344_preciseD_H,axiom,
! [B: $tType,A: $tType,R: A > B > assn,A4: A,P4: B,F: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),A9: A,F2: assn] :
( ( precise @ A @ B @ R )
=> ( ( rep_assn @ ( times_times @ assn @ ( R @ A4 @ P4 ) @ F ) @ H2 )
=> ( ( rep_assn @ ( times_times @ assn @ ( R @ A9 @ P4 ) @ F2 ) @ H2 )
=> ( A4 = A9 ) ) ) ) ).
% preciseD'
thf(fact_345_and__extract__pure__left__iff,axiom,
! [B3: $o,Q: assn] :
( ( inf_inf @ assn @ ( pure_assn @ B3 ) @ Q )
= ( times_times @ assn @ ( inf_inf @ assn @ ( one_one @ assn ) @ Q ) @ ( pure_assn @ B3 ) ) ) ).
% and_extract_pure_left_iff
thf(fact_346_and__extract__pure__right__iff,axiom,
! [P: assn,B3: $o] :
( ( inf_inf @ assn @ P @ ( pure_assn @ B3 ) )
= ( times_times @ assn @ ( inf_inf @ assn @ ( one_one @ assn ) @ P ) @ ( pure_assn @ B3 ) ) ) ).
% and_extract_pure_right_iff
thf(fact_347_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_348_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_349_subsetI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% subsetI
thf(fact_350_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_351_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A] :
( ( inf_inf @ A @ A4 @ A4 )
= A4 ) ) ).
% inf.idem
thf(fact_352_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ X3 )
= X3 ) ) ).
% inf_idem
thf(fact_353_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( inf_inf @ A @ A4 @ ( inf_inf @ A @ A4 @ B3 ) )
= ( inf_inf @ A @ A4 @ B3 ) ) ) ).
% inf.left_idem
thf(fact_354_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y ) )
= ( inf_inf @ A @ X3 @ Y ) ) ) ).
% inf_left_idem
thf(fact_355_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A4 @ B3 ) @ B3 )
= ( inf_inf @ A @ A4 @ B3 ) ) ) ).
% inf.right_idem
thf(fact_356_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y )
= ( inf_inf @ A @ X3 @ Y ) ) ) ).
% inf_right_idem
thf(fact_357_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F7: A > B,G4: A > B,X4: A] : ( inf_inf @ B @ ( F7 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% inf_apply
thf(fact_358_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) )
= ( ( ord_less_eq @ A @ X3 @ Y )
& ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_359_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ ( inf_inf @ A @ B3 @ C2 ) )
= ( ( ord_less_eq @ A @ A4 @ B3 )
& ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_360_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_361_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_362_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_363_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_364_insert__subset,axiom,
! [A: $tType,X3: A,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ B2 )
= ( ( member @ A @ X3 @ B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_365_psubsetI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_366_merge__pure__and,axiom,
! [A4: $o,B3: $o] :
( ( inf_inf @ assn @ ( pure_assn @ A4 ) @ ( pure_assn @ B3 ) )
= ( pure_assn
@ ( A4
& B3 ) ) ) ).
% merge_pure_and
thf(fact_367_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A4: A,A3: set @ A] :
( ( ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert3 @ A @ A4 @ A3 ) )
= ( ( A4 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_368_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A4: A,A3: set @ A,B3: A] :
( ( ( insert3 @ A @ A4 @ A3 )
= ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A4 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_369_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_370_and__extract__pure__right__ctx__iff,axiom,
! [P: assn,Q: assn,B3: $o] :
( ( inf_inf @ assn @ P @ ( times_times @ assn @ Q @ ( pure_assn @ B3 ) ) )
= ( times_times @ assn @ ( inf_inf @ assn @ P @ Q ) @ ( pure_assn @ B3 ) ) ) ).
% and_extract_pure_right_ctx_iff
thf(fact_371_and__extract__pure__left__ctx__iff,axiom,
! [P: assn,B3: $o,Q: assn] :
( ( inf_inf @ assn @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ Q )
= ( times_times @ assn @ ( inf_inf @ assn @ P @ Q ) @ ( pure_assn @ B3 ) ) ) ).
% and_extract_pure_left_ctx_iff
thf(fact_372_mod__h__bot__iff_I6_J,axiom,
! [P: assn,Q: assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( inf_inf @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(6)
thf(fact_373_in__mono,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_374_subsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_375_equalityE,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_376_pairwiseD,axiom,
! [A: $tType,R: A > A > $o,S: set @ A,X3: A,Y: A] :
( ( pairwise @ A @ R @ S )
=> ( ( member @ A @ X3 @ S )
=> ( ( member @ A @ Y @ S )
=> ( ( X3 != Y )
=> ( R @ X3 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_377_pairwiseI,axiom,
! [A: $tType,S: set @ A,R: A > A > $o] :
( ! [X: A,Y2: A] :
( ( member @ A @ X @ S )
=> ( ( member @ A @ Y2 @ S )
=> ( ( X != Y2 )
=> ( R @ X @ Y2 ) ) ) )
=> ( pairwise @ A @ R @ S ) ) ).
% pairwiseI
thf(fact_378_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ A @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_379_equalityD1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_380_equalityD2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_381_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A7 )
=> ( member @ A @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_382_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_383_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_384_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R4: A > A > $o,S2: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ S2 )
=> ( ( X4 != Y5 )
=> ( R4 @ X4 @ Y5 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_385_subset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_386_pairwise__mono,axiom,
! [A: $tType,P: A > A > $o,A3: set @ A,Q: A > A > $o,B2: set @ A] :
( ( pairwise @ A @ P @ A3 )
=> ( ! [X: A,Y2: A] :
( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( pairwise @ A @ Q @ B2 ) ) ) ) ).
% pairwise_mono
thf(fact_387_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y4: set @ A,Z4: set @ A] : Y4 = Z4 )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_388_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S: set @ A,T4: set @ A] :
( ( pairwise @ A @ P @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T4 @ S )
=> ( pairwise @ A @ P @ T4 ) ) ) ).
% pairwise_subset
thf(fact_389_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_390_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( A4 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ( C2 = D3 )
=> ( ord_less_eq @ A @ A4 @ D3 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_391_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( P @ X4 ) ) ) ) ).
% subset_Collect_conv
thf(fact_392_assn__aci_I4_J,axiom,
! [X3: assn,Y: assn] :
( ( inf_inf @ assn @ X3 @ ( inf_inf @ assn @ X3 @ Y ) )
= ( inf_inf @ assn @ X3 @ Y ) ) ).
% assn_aci(4)
thf(fact_393_assn__aci_I3_J,axiom,
! [X3: assn,Y: assn,Z2: assn] :
( ( inf_inf @ assn @ X3 @ ( inf_inf @ assn @ Y @ Z2 ) )
= ( inf_inf @ assn @ Y @ ( inf_inf @ assn @ X3 @ Z2 ) ) ) ).
% assn_aci(3)
thf(fact_394_assn__aci_I1_J,axiom,
( ( inf_inf @ assn )
= ( ^ [X4: assn,Y5: assn] : ( inf_inf @ assn @ Y5 @ X4 ) ) ) ).
% assn_aci(1)
thf(fact_395_norm__assertion__simps_I31_J,axiom,
! [X3: assn] :
( ( inf_inf @ assn @ X3 @ X3 )
= X3 ) ).
% norm_assertion_simps(31)
thf(fact_396_norm__assertion__simps_I14_J,axiom,
! [X3: assn,Y: assn,Z2: assn] :
( ( inf_inf @ assn @ ( inf_inf @ assn @ X3 @ Y ) @ Z2 )
= ( inf_inf @ assn @ X3 @ ( inf_inf @ assn @ Y @ Z2 ) ) ) ).
% norm_assertion_simps(14)
thf(fact_397_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ~ ( ord_less_eq @ A @ A4 @ B3 ) )
= ( ( ord_less_eq @ A @ B3 @ A4 )
& ( B3 != A4 ) ) ) ) ).
% nle_le
thf(fact_398_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_399_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z4: A] : Y4 = Z4 )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_400_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_401_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_402_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X3 )
=> ( X3 = Y ) ) ) ) ).
% order_antisym
thf(fact_403_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% order.trans
thf(fact_404_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_405_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A4: A,B3: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_406_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z4: A] : Y4 = Z4 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_407_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_408_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_409_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ).
% antisym
thf(fact_410_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_411_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_412_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B] :
( ! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% le_funI
thf(fact_413_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F7: A > B,G4: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F7 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_414_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z4: A] : Y4 = Z4 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_415_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F3: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X: B,Y2: B] :
( ( ord_less_eq @ B @ X @ Y2 )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_416_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F3: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C @ ( F3 @ B3 ) @ C2 )
=> ( ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ C @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A4 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_417_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( X3 = Y )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% order_eq_refl
thf(fact_418_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
| ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linorder_linear
thf(fact_419_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F3: B > A,B3: B,C2: B] :
( ( A4
= ( F3 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X: B,Y2: B] :
( ( ord_less_eq @ B @ X @ Y2 )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F3 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_420_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B3: A,F3: A > B,C2: B] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( F3 @ B3 )
= C2 )
=> ( ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A4 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_421_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linorder_le_cases
thf(fact_422_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
= ( X3 = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_423_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y ) )
= ( inf_inf @ A @ X3 @ Y ) ) ) ).
% inf_sup_aci(4)
thf(fact_424_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X3 @ Z2 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_425_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y ) @ Z2 )
= ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_426_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X4: A,Y5: A] : ( inf_inf @ A @ Y5 @ X4 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_427_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_428_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ X3 ) ) ).
% inf_sup_ord(1)
thf(fact_429_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ X3 ) ) ).
% inf_le1
thf(fact_430_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_431_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A4 @ B3 ) )
=> ~ ( ( ord_less_eq @ A @ X3 @ A4 )
=> ~ ( ord_less_eq @ A @ X3 @ B3 ) ) ) ) ).
% le_infE
thf(fact_432_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ X3 @ A4 )
=> ( ( ord_less_eq @ A @ X3 @ B3 )
=> ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A4 @ B3 ) ) ) ) ) ).
% le_infI
thf(fact_433_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,C2: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ D3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ ( inf_inf @ A @ C2 @ D3 ) ) ) ) ) ).
% inf_mono
thf(fact_434_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,X3: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ X3 ) ) ) ).
% le_infI1
thf(fact_435_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,X3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ X3 ) ) ) ).
% le_infI2
thf(fact_436_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A4 @ B3 ) @ C2 )
= ( inf_inf @ A @ A4 @ ( inf_inf @ A @ B3 @ C2 ) ) ) ) ).
% inf.assoc
thf(fact_437_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y ) @ Z2 )
= ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ).
% inf_assoc
thf(fact_438_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( A4
= ( inf_inf @ A @ A4 @ B3 ) ) ) ) ).
% inf.orderE
thf(fact_439_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( inf_inf @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% inf.orderI
thf(fact_440_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F3: A > A > A,X3: A,Y: A] :
( ! [X: A,Y2: A] : ( ord_less_eq @ A @ ( F3 @ X @ Y2 ) @ X )
=> ( ! [X: A,Y2: A] : ( ord_less_eq @ A @ ( F3 @ X @ Y2 ) @ Y2 )
=> ( ! [X: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ X @ Z3 )
=> ( ord_less_eq @ A @ X @ ( F3 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X3 @ Y )
= ( F3 @ X3 @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_441_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y5: A] :
( ( inf_inf @ A @ X4 @ Y5 )
= X4 ) ) ) ) ).
% le_iff_inf
thf(fact_442_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( inf_inf @ A @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb1
thf(fact_443_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( inf_inf @ A @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb2
thf(fact_444_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A5: A,B4: A] : ( inf_inf @ A @ B4 @ A5 ) ) ) ) ).
% inf.commute
thf(fact_445_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( inf_inf @ A @ X3 @ Y )
= X3 ) ) ) ).
% inf_absorb1
thf(fact_446_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( inf_inf @ A @ X3 @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_447_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X4: A,Y5: A] : ( inf_inf @ A @ Y5 @ X4 ) ) ) ) ).
% inf_commute
thf(fact_448_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ ( inf_inf @ A @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A4 @ B3 )
=> ~ ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% inf.boundedE
thf(fact_449_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ A4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ ( inf_inf @ A @ B3 @ C2 ) ) ) ) ) ).
% inf.boundedI
thf(fact_450_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ) ).
% inf_greatest
thf(fact_451_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( A5
= ( inf_inf @ A @ A5 @ B4 ) ) ) ) ) ).
% inf.order_iff
thf(fact_452_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ A4 ) ) ).
% inf.cobounded1
thf(fact_453_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ B3 ) ) ).
% inf.cobounded2
thf(fact_454_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( inf_inf @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_455_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( inf_inf @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_456_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% inf.coboundedI1
thf(fact_457_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% inf.coboundedI2
thf(fact_458_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( inf_inf @ A @ B3 @ ( inf_inf @ A @ A4 @ C2 ) )
= ( inf_inf @ A @ A4 @ ( inf_inf @ A @ B3 @ C2 ) ) ) ) ).
% inf.left_commute
thf(fact_459_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X3 @ Z2 ) ) ) ) ).
% inf_left_commute
thf(fact_460_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F7: A > B,G4: A > B,X4: A] : ( inf_inf @ B @ ( F7 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% inf_fun_def
thf(fact_461_listrel1__mono,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel1 @ A @ S3 ) ) ) ).
% listrel1_mono
thf(fact_462_listrel__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( listrel @ A @ B @ R2 ) @ ( listrel @ A @ B @ S3 ) ) ) ).
% listrel_mono
thf(fact_463_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_464_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ A4 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_465_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( inf_inf @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_466_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ ( inf_inf @ A @ B3 @ C2 ) )
=> ~ ( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_467_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( inf_inf @ A @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb4
thf(fact_468_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( inf_inf @ A @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb3
thf(fact_469_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,X3: A,A4: A] :
( ( ord_less @ A @ B3 @ X3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A4 @ B3 ) @ X3 ) ) ) ).
% less_infI2
thf(fact_470_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,X3: A,B3: A] :
( ( ord_less @ A @ A4 @ X3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A4 @ B3 ) @ X3 ) ) ) ).
% less_infI1
thf(fact_471_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ Z3 @ X8 )
=> ~ ( ord_less_eq @ A @ X8 @ T2 ) ) ) ).
% pinf(6)
thf(fact_472_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ Z3 @ X8 )
=> ( ord_less_eq @ A @ T2 @ X8 ) ) ) ).
% pinf(8)
thf(fact_473_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z3 )
=> ( ord_less_eq @ A @ X8 @ T2 ) ) ) ).
% minf(6)
thf(fact_474_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X8: A] :
( ( ord_less @ A @ X8 @ Z3 )
=> ~ ( ord_less_eq @ A @ T2 @ X8 ) ) ) ).
% minf(8)
thf(fact_475_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_476_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ).
% linorder_le_less_linear
thf(fact_477_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F3: A > C,C2: C] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C @ ( F3 @ B3 ) @ C2 )
=> ( ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ C @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ C @ ( F3 @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_478_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F3: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X: B,Y2: B] :
( ( ord_less_eq @ B @ X @ Y2 )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_479_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F3: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ C @ ( F3 @ B3 ) @ C2 )
=> ( ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ C @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ C @ ( F3 @ A4 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_480_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F3: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X: B,Y2: B] :
( ( ord_less @ B @ X @ Y2 )
=> ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_481_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% order_less_le_trans
thf(fact_482_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% order_le_less_trans
thf(fact_483_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( A4 != B3 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% order_neq_le_trans
thf(fact_484_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% order_le_neq_trans
thf(fact_485_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% order_less_imp_le
thf(fact_486_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linorder_not_less
thf(fact_487_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X3 @ Y ) )
= ( ord_less @ A @ Y @ X3 ) ) ) ).
% linorder_not_le
thf(fact_488_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ) ).
% order_less_le
thf(fact_489_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ) ).
% order_le_less
thf(fact_490_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_491_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_492_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ~ ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_493_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_494_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_495_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_496_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less @ A @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_497_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X3 @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_498_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X3 )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_499_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ~ ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_500_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_501_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_502_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_503_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_504_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ~ ( ord_less_eq @ A @ Y @ X3 )
=> ( ord_less @ A @ X3 @ Y ) ) ) ).
% not_le_imp_less
thf(fact_505_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ~ ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_506_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z2: A] :
( ! [X: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_507_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y: A] :
( ! [X: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ord_less_eq @ A @ Y @ X ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_508_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% antisym_conv2
thf(fact_509_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ X3 @ Y )
= ( X3 = Y ) ) ) ) ).
% antisym_conv1
thf(fact_510_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ~ ( ord_less @ A @ A4 @ B3 ) )
= ( ~ ( ord_less_eq @ A @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% nless_le
thf(fact_511_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% leI
thf(fact_512_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y ) ) ) ).
% leD
thf(fact_513_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
=> ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_514_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
= ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_515_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A4 ) ) ).
% bot.extremum
thf(fact_516_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ! [X: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).
% subrelI
thf(fact_517_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
= ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_518_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B3 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_519_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A4 ) @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_520_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,D3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ D3 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_521_mod__and__dist,axiom,
! [P: assn,Q: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( inf_inf @ assn @ P @ Q ) @ H2 )
= ( ( rep_assn @ P @ H2 )
& ( rep_assn @ Q @ H2 ) ) ) ).
% mod_and_dist
thf(fact_522_ent__conjI,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( entails @ A3 @ B2 )
=> ( ( entails @ A3 @ C3 )
=> ( entails @ A3 @ ( inf_inf @ assn @ B2 @ C3 ) ) ) ) ).
% ent_conjI
thf(fact_523_ent__conjE1,axiom,
! [A3: assn,C3: assn,B2: assn] :
( ( entails @ A3 @ C3 )
=> ( entails @ ( inf_inf @ assn @ A3 @ B2 ) @ C3 ) ) ).
% ent_conjE1
thf(fact_524_ent__conjE2,axiom,
! [B2: assn,C3: assn,A3: assn] :
( ( entails @ B2 @ C3 )
=> ( entails @ ( inf_inf @ assn @ A3 @ B2 ) @ C3 ) ) ).
% ent_conjE2
thf(fact_525_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert3 @ A @ B3 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_526_subset__insertI,axiom,
! [A: $tType,B2: set @ A,A4: A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( insert3 @ A @ A4 @ B2 ) ) ).
% subset_insertI
thf(fact_527_subset__insert,axiom,
! [A: $tType,X3: A,A3: set @ A,B2: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% subset_insert
thf(fact_528_insert__mono,axiom,
! [A: $tType,C3: set @ A,D2: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ A4 @ C3 ) @ ( insert3 @ A @ A4 @ D2 ) ) ) ).
% insert_mono
thf(fact_529_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X4 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_530_double__diff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_531_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ A3 ) ).
% Diff_subset
thf(fact_532_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,D2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ D2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_533_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
| ( A7 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_534_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_535_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_536_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_537_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_538_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( A7 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_539_psubsetE,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% psubsetE
thf(fact_540_norm__assertion__simps_I9_J,axiom,
! [X3: assn] :
( ( inf_inf @ assn @ ( bot_bot @ assn ) @ X3 )
= ( bot_bot @ assn ) ) ).
% norm_assertion_simps(9)
thf(fact_541_norm__assertion__simps_I10_J,axiom,
! [X3: assn] :
( ( inf_inf @ assn @ X3 @ ( bot_bot @ assn ) )
= ( bot_bot @ assn ) ) ).
% norm_assertion_simps(10)
thf(fact_542_precise__def,axiom,
! [B: $tType,A: $tType] :
( ( precise @ A @ B )
= ( ^ [R4: A > B > assn] :
! [A5: A,A10: A,H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P7: B,F4: assn,F8: assn] :
( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R4 @ A5 @ P7 ) @ F4 ) @ ( times_times @ assn @ ( R4 @ A10 @ P7 ) @ F8 ) ) @ H3 )
=> ( A5 = A10 ) ) ) ) ).
% precise_def
thf(fact_543_preciseI,axiom,
! [B: $tType,A: $tType,R: A > B > assn] :
( ! [A6: A,A11: A,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P8: B,F9: assn,F10: assn] :
( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R @ A6 @ P8 ) @ F9 ) @ ( times_times @ assn @ ( R @ A11 @ P8 ) @ F10 ) ) @ H )
=> ( A6 = A11 ) )
=> ( precise @ A @ B @ R ) ) ).
% preciseI
thf(fact_544_preciseD,axiom,
! [B: $tType,A: $tType,R: A > B > assn,A4: A,P4: B,F: assn,A9: A,F2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( precise @ A @ B @ R )
=> ( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R @ A4 @ P4 ) @ F ) @ ( times_times @ assn @ ( R @ A9 @ P4 ) @ F2 ) ) @ H2 )
=> ( A4 = A9 ) ) ) ).
% preciseD
thf(fact_545_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_546_pairwise__insert,axiom,
! [A: $tType,R2: A > A > $o,X3: A,S3: set @ A] :
( ( pairwise @ A @ R2 @ ( insert3 @ A @ X3 @ S3 ) )
= ( ! [Y5: A] :
( ( ( member @ A @ Y5 @ S3 )
& ( Y5 != X3 ) )
=> ( ( R2 @ X3 @ Y5 )
& ( R2 @ Y5 @ X3 ) ) )
& ( pairwise @ A @ R2 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_547_prec__frame,axiom,
! [B: $tType,A: $tType,P: A > B > assn,R1: assn,R22: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X3: A,P4: B,F12: assn,Y: A,F22: assn] :
( ( precise @ A @ B @ P )
=> ( ( rep_assn @ ( inf_inf @ assn @ R1 @ R22 ) @ H2 )
=> ( ( entails @ R1 @ ( times_times @ assn @ ( P @ X3 @ P4 ) @ F12 ) )
=> ( ( entails @ R22 @ ( times_times @ assn @ ( P @ Y @ P4 ) @ F22 ) )
=> ( X3 = Y ) ) ) ) ) ).
% prec_frame
thf(fact_548_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_549_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_550_subset__minus__empty,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_551_subset__Diff__insert,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X3: A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B2 @ ( insert3 @ A @ X3 @ C3 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B2 @ C3 ) )
& ~ ( member @ A @ X3 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_552_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A,X3: A,I: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X3 ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_553_set__remove1__subset,axiom,
! [A: $tType,X3: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X3 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_remove1_subset
thf(fact_554_subset__code_I2_J,axiom,
! [B: $tType,A3: set @ B,Ys: list @ B] :
( ( ord_less_eq @ ( set @ B ) @ A3 @ ( coset @ B @ Ys ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Ys ) )
=> ~ ( member @ B @ X4 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_555_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A3: A] : ( pairwise @ A @ P @ ( insert3 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_556_prec__frame__expl,axiom,
! [A: $tType,P: A > assn,F12: assn,F22: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),R1: assn,R22: assn,X3: A,Y: A] :
( ! [X: A,Y2: A] :
( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( P @ X ) @ F12 ) @ ( times_times @ assn @ ( P @ Y2 ) @ F22 ) ) @ H2 )
=> ( X = Y2 ) )
=> ( ( rep_assn @ ( inf_inf @ assn @ R1 @ R22 ) @ H2 )
=> ( ( entails @ R1 @ ( times_times @ assn @ ( P @ X3 ) @ F12 ) )
=> ( ( entails @ R22 @ ( times_times @ assn @ ( P @ Y ) @ F22 ) )
=> ( X3 = Y ) ) ) ) ) ).
% prec_frame_expl
thf(fact_557_prec__frame_H,axiom,
! [A: $tType,P: A > assn,X3: A,F12: assn,Y: A,F22: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),R1: assn,R22: assn] :
( ( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( P @ X3 ) @ F12 ) @ ( times_times @ assn @ ( P @ Y ) @ F22 ) ) @ H2 )
=> ( X3 = Y ) )
=> ( ( rep_assn @ ( inf_inf @ assn @ R1 @ R22 ) @ H2 )
=> ( ( entails @ R1 @ ( times_times @ assn @ ( P @ X3 ) @ F12 ) )
=> ( ( entails @ R22 @ ( times_times @ assn @ ( P @ Y ) @ F22 ) )
=> ( X3 = Y ) ) ) ) ) ).
% prec_frame'
thf(fact_558_subset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X3: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ B2 ) )
= ( ( ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) )
& ( ~ ( member @ A @ X3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_559_Diff__single__insert,axiom,
! [A: $tType,A3: set @ A,X3: A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_560_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I: nat,X3: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I @ X3 ) ) @ ( insert3 @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_561_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_right
thf(fact_562_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_left
thf(fact_563_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ( minus_minus @ A @ X3 @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% diff_shunt_var
thf(fact_564_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_565_less__imp__diff__less,axiom,
! [J: nat,K3: nat,N: nat] :
( ( ord_less @ nat @ J @ K3 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K3 ) ) ).
% less_imp_diff_less
thf(fact_566_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( ord_less @ nat @ M2 @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_567_insert__subsetI,axiom,
! [A: $tType,X3: A,A3: set @ A,X5: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ X5 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_568_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S: list @ A,I3: ( set @ A ) > B > $o,Sigma_0: B,F3: B > A > B] :
( ( distinct @ A @ S )
=> ( ( I3 @ ( set2 @ A @ S ) @ Sigma_0 )
=> ( ! [X: A,It: set @ A,Sigma: B] :
( ( member @ A @ X @ It )
=> ( ( ord_less_eq @ ( set @ A ) @ It @ ( set2 @ A @ S ) )
=> ( ( I3 @ It @ Sigma )
=> ( I3 @ ( minus_minus @ ( set @ A ) @ It @ ( insert3 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( F3 @ Sigma @ X ) ) ) ) )
=> ( I3 @ ( bot_bot @ ( set @ A ) ) @ ( foldl @ B @ A @ F3 @ Sigma_0 @ S ) ) ) ) ) ).
% distinct_foldl_invar
thf(fact_569_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X: A] :
~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_570_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A4: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( P @ A4 )
=> ( ~ ( P @ B3 )
=> ? [C5: A] :
( ( ord_less_eq @ A @ A4 @ C5 )
& ( ord_less_eq @ A @ C5 @ B3 )
& ! [X8: A] :
( ( ( ord_less_eq @ A @ A4 @ X8 )
& ( ord_less @ A @ X8 @ C5 ) )
=> ( P @ X8 ) )
& ! [D5: A] :
( ! [X: A] :
( ( ( ord_less_eq @ A @ A4 @ X )
& ( ord_less @ A @ X @ D5 ) )
=> ( P @ X ) )
=> ( ord_less_eq @ A @ D5 @ C5 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_571_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B8: B,A9: B] :
( ( ~ ( ord_less_eq @ B @ B8 @ A9 ) )
= ( ord_less @ B @ A9 @ B8 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_572_Int__subset__iff,axiom,
! [A: $tType,C3: set @ A,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C3 @ A3 )
& ( ord_less_eq @ ( set @ A ) @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_573_Int__insert__left__if0,axiom,
! [A: $tType,A4: A,C3: set @ A,B2: set @ A] :
( ~ ( member @ A @ A4 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert3 @ A @ A4 @ B2 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_574_Int__insert__left__if1,axiom,
! [A: $tType,A4: A,C3: set @ A,B2: set @ A] :
( ( member @ A @ A4 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert3 @ A @ A4 @ B2 ) @ C3 )
= ( insert3 @ A @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_575_insert__inter__insert,axiom,
! [A: $tType,A4: A,A3: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert3 @ A @ A4 @ A3 ) @ ( insert3 @ A @ A4 @ B2 ) )
= ( insert3 @ A @ A4 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_576_Int__insert__right__if0,axiom,
! [A: $tType,A4: A,A3: set @ A,B2: set @ A] :
( ~ ( member @ A @ A4 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_577_Int__insert__right__if1,axiom,
! [A: $tType,A4: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ B2 ) )
= ( insert3 @ A @ A4 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_578_insert__disjoint_I1_J,axiom,
! [A: $tType,A4: A,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert3 @ A @ A4 @ A3 ) @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A4 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_579_insert__disjoint_I2_J,axiom,
! [A: $tType,A4: A,A3: set @ A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert3 @ A @ A4 @ A3 ) @ B2 ) )
= ( ~ ( member @ A @ A4 @ B2 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_580_disjoint__insert_I1_J,axiom,
! [A: $tType,B2: set @ A,A4: A,A3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B2 @ ( insert3 @ A @ A4 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A4 @ B2 )
& ( ( inf_inf @ ( set @ A ) @ B2 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_581_disjoint__insert_I2_J,axiom,
! [A: $tType,A3: set @ A,B3: A,B2: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( insert3 @ A @ B3 @ B2 ) ) )
= ( ~ ( member @ A @ B3 @ A3 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_582_Diff__disjoint,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_583_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X3: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X3 )
= Xs ) ) ).
% list_update_beyond
thf(fact_584_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F7: A > B,G4: A > B] :
( ( ord_less_eq @ ( A > B ) @ F7 @ G4 )
& ~ ( ord_less_eq @ ( A > B ) @ G4 @ F7 ) ) ) ) ) ).
% less_fun_def
thf(fact_585_foldl__A1__eq,axiom,
! [A: $tType,F3: A > A > A,N: A,I: A,Ww: list @ A] :
( ! [E2: A] :
( ( F3 @ N @ E2 )
= E2 )
=> ( ! [E2: A] :
( ( F3 @ E2 @ N )
= E2 )
=> ( ! [A6: A,B5: A,C5: A] :
( ( F3 @ A6 @ ( F3 @ B5 @ C5 ) )
= ( F3 @ ( F3 @ A6 @ B5 ) @ C5 ) )
=> ( ( foldl @ A @ A @ F3 @ I @ Ww )
= ( F3 @ I @ ( foldl @ A @ A @ F3 @ N @ Ww ) ) ) ) ) ) ).
% foldl_A1_eq
thf(fact_586_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_587_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_588_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_589_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less @ nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_590_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_591_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I: nat,J: nat] :
( ! [I5: nat,J3: nat] :
( ( ord_less @ nat @ I5 @ J3 )
=> ( ord_less @ nat @ ( F3 @ I5 ) @ ( F3 @ J3 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_592_disjoint__iff__not__equal,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ B2 )
=> ( X4 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_593_Int__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_594_Int__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_595_disjoint__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ~ ( member @ A @ X4 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_596_Int__emptyI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ~ ( member @ A @ X @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_597_disjointI,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ~ ( member @ A @ X @ B3 ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjointI
thf(fact_598_inter__eq__subsetI,axiom,
! [A: $tType,S: set @ A,S4: set @ A,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ S4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ S4 )
= ( inf_inf @ ( set @ A ) @ B2 @ S4 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ S )
= ( inf_inf @ ( set @ A ) @ B2 @ S ) ) ) ) ).
% inter_eq_subsetI
thf(fact_599_Int__Collect__mono,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B2 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_600_Int__greatest,axiom,
! [A: $tType,C3: set @ A,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ C3 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_601_Int__absorb2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= A3 ) ) ).
% Int_absorb2
thf(fact_602_Int__absorb1,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_603_Int__lower2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_604_Int__lower1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ A3 ) ).
% Int_lower1
thf(fact_605_Int__mono,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B2: set @ A,D2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( inf_inf @ ( set @ A ) @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_606_Int__insert__left,axiom,
! [A: $tType,A4: A,C3: set @ A,B2: set @ A] :
( ( ( member @ A @ A4 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert3 @ A @ A4 @ B2 ) @ C3 )
= ( insert3 @ A @ A4 @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) ) )
& ( ~ ( member @ A @ A4 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert3 @ A @ A4 @ B2 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_607_Int__insert__right,axiom,
! [A: $tType,A4: A,A3: set @ A,B2: set @ A] :
( ( ( member @ A @ A4 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ B2 ) )
= ( insert3 @ A @ A4 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) )
& ( ~ ( member @ A @ A4 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_608_Int__Diff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Int_Diff
thf(fact_609_Diff__Int2,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_610_Diff__Diff__Int,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_611_Diff__Int__distrib,axiom,
! [A: $tType,C3: set @ A,A3: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ C3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C3 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C3 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_612_Diff__Int__distrib2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_613_exists__leI,axiom,
! [N: nat,P: nat > $o] :
( ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ N )
=> ~ ( P @ N3 ) )
=> ( P @ N ) )
=> ? [N4: nat] :
( ( ord_less_eq @ nat @ N4 @ N )
& ( P @ N4 ) ) ) ).
% exists_leI
thf(fact_614_foldl__cong,axiom,
! [A: $tType,B: $tType,A4: A,B3: A,L: list @ B,K3: list @ B,F3: A > B > A,G: A > B > A] :
( ( A4 = B3 )
=> ( ( L = K3 )
=> ( ! [A6: A,X: B] :
( ( member @ B @ X @ ( set2 @ B @ L ) )
=> ( ( F3 @ A6 @ X )
= ( G @ A6 @ X ) ) )
=> ( ( foldl @ A @ B @ F3 @ A4 @ L )
= ( foldl @ A @ B @ G @ B3 @ K3 ) ) ) ) ) ).
% foldl_cong
thf(fact_615_less__eq__assn__def,axiom,
( ( ord_less_eq @ assn )
= ( ^ [A5: assn,B4: assn] :
( A5
= ( inf_inf @ assn @ A5 @ B4 ) ) ) ) ).
% less_eq_assn_def
thf(fact_616_less__diff__iff,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K3 @ M2 )
=> ( ( ord_less_eq @ nat @ K3 @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ K3 ) @ ( minus_minus @ nat @ N @ K3 ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_617_diff__less__mono,axiom,
! [A4: nat,B3: nat,C2: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( ( ord_less_eq @ nat @ C2 @ A4 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A4 @ C2 ) @ ( minus_minus @ nat @ B3 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_618_disjoint__mono,axiom,
! [A: $tType,A4: set @ A,A9: set @ A,B3: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ A9 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ B8 )
=> ( ( ( inf_inf @ ( set @ A ) @ A9 @ B8 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% disjoint_mono
thf(fact_619_disjoint__alt__simp2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B2 )
!= A3 )
= ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp2
thf(fact_620_disjoint__alt__simp1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B2 )
= A3 )
= ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp1
thf(fact_621_Int__Diff__disjoint,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_622_Diff__triv,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ B2 )
= A3 ) ) ).
% Diff_triv
thf(fact_623_foldl__absorb1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X3: A,Zs: list @ A] :
( ( times_times @ A @ X3 @ ( foldl @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) @ Zs ) )
= ( foldl @ A @ A @ ( times_times @ A ) @ X3 @ Zs ) ) ) ).
% foldl_absorb1
thf(fact_624_length__removeAll__less__eq,axiom,
! [A: $tType,X3: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_625_disjoint__alt__simp3,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ A3 )
= ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp3
thf(fact_626_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% verit_comp_simplify1(1)
thf(fact_627_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A4: A] :
? [B5: A] :
( ( ord_less @ A @ A4 @ B5 )
| ( ord_less @ A @ B5 @ A4 ) ) ) ).
% ex_gt_or_lt
thf(fact_628_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P: A > $o,A4: A] :
( ! [X: A] :
( ! [Y3: A] :
( ( ord_less @ B @ ( F3 @ Y3 ) @ ( F3 @ X ) )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A4 ) ) ) ).
% measure_induct
thf(fact_629_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P: A > $o,A4: A] :
( ! [X: A] :
( ! [Y3: A] :
( ( ord_less @ B @ ( F3 @ Y3 ) @ ( F3 @ X ) )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A4 ) ) ) ).
% measure_induct_rule
thf(fact_630_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S3: B,R: set @ ( product_prod @ A @ B ),S5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R )
=> ( ( S5 = S3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S5 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_631_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_632_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less @ nat @ M2 @ N )
| ( ord_less @ nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_633_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_634_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_635_less__not__refl3,axiom,
! [S3: nat,T2: nat] :
( ( ord_less @ nat @ S3 @ T2 )
=> ( S3 != T2 ) ) ).
% less_not_refl3
thf(fact_636_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_637_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N5 )
=> ( P @ M3 ) )
=> ( P @ N5 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_638_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N5 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_639_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less @ nat @ X3 @ Y )
=> ( ord_less @ nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_640_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X3: A] :
( ! [X: A] :
( ~ ( P @ X )
=> ? [Y3: A] :
( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X ) )
& ~ ( P @ Y3 ) ) )
=> ( P @ X3 ) ) ).
% infinite_descent_measure
thf(fact_641_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X3: A,Y: A] :
( ( ( size_size @ A @ X3 )
!= ( size_size @ A @ Y ) )
=> ( X3 != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_642_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_643_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_644_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_645_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_646_rev__update,axiom,
! [A: $tType,K3: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ K3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K3 @ Y ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K3 ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% rev_update
thf(fact_647_butlast__list__update,axiom,
! [A: $tType,K3: nat,Xs: list @ A,X3: A] :
( ( ( K3
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K3 @ X3 ) )
= ( butlast @ A @ Xs ) ) )
& ( ( K3
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K3 @ X3 ) )
= ( list_update @ A @ ( butlast @ A @ Xs ) @ K3 @ X3 ) ) ) ) ).
% butlast_list_update
thf(fact_648_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_649_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_650_star__assnI,axiom,
! [P: assn,H2: heap_ext @ product_unit,As2: set @ nat,Q: assn,As3: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As3 ) )
=> ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As3 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( sup_sup @ ( set @ nat ) @ As2 @ As3 ) ) ) ) ) ) ).
% star_assnI
thf(fact_651_mod__star__conv,axiom,
! [A3: assn,B2: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A3 @ B2 ) @ H2 )
= ( ? [Hr: heap_ext @ product_unit,As1: set @ nat,As22: set @ nat] :
( ( H2
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( rep_assn @ A3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As1 ) )
& ( rep_assn @ B2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As22 ) ) ) ) ) ).
% mod_star_conv
thf(fact_652_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_653_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A] :
( ( sup_sup @ A @ A4 @ A4 )
= A4 ) ) ).
% sup.idem
thf(fact_654_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ X3 )
= X3 ) ) ).
% sup_idem
thf(fact_655_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( sup_sup @ A @ A4 @ ( sup_sup @ A @ A4 @ B3 ) )
= ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_656_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y ) )
= ( sup_sup @ A @ X3 @ Y ) ) ) ).
% sup_left_idem
thf(fact_657_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ B3 )
= ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_658_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F7: A > B,G4: A > B,X4: A] : ( sup_sup @ B @ ( F7 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% sup_apply
thf(fact_659_IntI,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% IntI
thf(fact_660_Int__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
= ( ( member @ A @ C2 @ A3 )
& ( member @ A @ C2 @ B2 ) ) ) ).
% Int_iff
thf(fact_661_UnCI,axiom,
! [A: $tType,C2: A,B2: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ A3 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_662_Un__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ( member @ A @ C2 @ A3 )
| ( member @ A @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_663_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A] :
( ( ord_min @ A @ A4 @ A4 )
= A4 ) ) ).
% min.idem
thf(fact_664_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_min @ A @ A4 @ ( ord_min @ A @ A4 @ B3 ) )
= ( ord_min @ A @ A4 @ B3 ) ) ) ).
% min.left_idem
thf(fact_665_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_min @ A @ ( ord_min @ A @ A4 @ B3 ) @ B3 )
= ( ord_min @ A @ A4 @ B3 ) ) ) ).
% min.right_idem
thf(fact_666_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_667_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_668_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X3 @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_669_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 )
= ( ( ord_less_eq @ A @ B3 @ A4 )
& ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% sup.bounded_iff
thf(fact_670_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A4: A] :
( ( sup_sup @ A @ A4 @ ( bot_bot @ A ) )
= A4 ) ) ).
% sup_bot.right_neutral
thf(fact_671_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A4: A,B3: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A4 @ B3 ) )
= ( ( A4
= ( bot_bot @ A ) )
& ( B3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_672_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A4: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A4 )
= A4 ) ) ).
% sup_bot.left_neutral
thf(fact_673_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A4: A,B3: A] :
( ( ( sup_sup @ A @ A4 @ B3 )
= ( bot_bot @ A ) )
= ( ( A4
= ( bot_bot @ A ) )
& ( B3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_674_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A,Y: A] :
( ( ( sup_sup @ A @ X3 @ Y )
= ( bot_bot @ A ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_675_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X3 @ Y ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_676_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% sup_bot_right
thf(fact_677_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X3 )
= X3 ) ) ).
% sup_bot_left
thf(fact_678_inf__sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ X3 @ Y ) )
= X3 ) ) ).
% inf_sup_absorb
thf(fact_679_sup__inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] :
( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ X3 @ Y ) )
= X3 ) ) ).
% sup_inf_absorb
thf(fact_680_Un__empty,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_681_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_682_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_683_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_684_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_685_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_min @ A @ A4 @ B3 )
= A4 ) ) ) ).
% min.absorb1
thf(fact_686_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_min @ A @ A4 @ B3 )
= B3 ) ) ) ).
% min.absorb2
thf(fact_687_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ ( ord_min @ A @ B3 @ C2 ) )
= ( ( ord_less_eq @ A @ A4 @ B3 )
& ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% min.bounded_iff
thf(fact_688_Un__subset__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_689_Un__insert__right,axiom,
! [A: $tType,A3: set @ A,A4: A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ B2 ) )
= ( insert3 @ A @ A4 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_690_Un__insert__left,axiom,
! [A: $tType,A4: A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert3 @ A @ A4 @ B2 ) @ C3 )
= ( insert3 @ A @ A4 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_691_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ Z2 @ ( ord_min @ A @ X3 @ Y ) )
= ( ( ord_less @ A @ Z2 @ X3 )
& ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_692_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_min @ A @ A4 @ B3 )
= B3 ) ) ) ).
% min.absorb4
thf(fact_693_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_min @ A @ A4 @ B3 )
= A4 ) ) ) ).
% min.absorb3
thf(fact_694_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_695_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_696_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( ord_min @ A @ A4 @ B3 ) @ A4 )
= ( ord_less @ A @ B3 @ A4 ) ) ) ).
% min_less_self_conv(1)
thf(fact_697_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( ord_min @ A @ A4 @ B3 ) @ B3 )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% min_less_self_conv(2)
thf(fact_698_min__simps_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_min @ A @ A4 @ B3 )
= A4 ) ) ) ).
% min_simps(1)
thf(fact_699_min__simps_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_min @ A @ A4 @ B3 )
= B3 ) ) ) ).
% min_simps(2)
thf(fact_700_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_min @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_701_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_702_Int__Un__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S: set @ A] :
( ( sup_sup @ ( set @ A ) @ T4 @ ( inf_inf @ ( set @ A ) @ S @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_703_Int__Un__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ S @ ( inf_inf @ ( set @ A ) @ S @ T4 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_704_Int__Un__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_705_Int__Un__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T4 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_706_Un__Int__eq_I4_J,axiom,
! [A: $tType,T4: set @ A,S: set @ A] :
( ( inf_inf @ ( set @ A ) @ T4 @ ( sup_sup @ ( set @ A ) @ S @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_707_Un__Int__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ S @ ( sup_sup @ ( set @ A ) @ S @ T4 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_708_Un__Int__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_709_Un__Int__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T4 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_710_Un__Diff__cancel2,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) @ A3 )
= ( sup_sup @ ( set @ A ) @ B2 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_711_Un__Diff__cancel,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_712_set__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rev @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rev
thf(fact_713_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_714_distinct__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rev @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rev
thf(fact_715_set__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rotate1 @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate1
thf(fact_716_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_717_distinct1__rotate,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rotate1 @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct1_rotate
thf(fact_718_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_719_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_720_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_721_IntE,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ~ ( member @ A @ C2 @ B2 ) ) ) ).
% IntE
thf(fact_722_IntD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% IntD1
thf(fact_723_IntD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
=> ( member @ A @ C2 @ B2 ) ) ).
% IntD2
thf(fact_724_Int__assoc,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Int_assoc
thf(fact_725_Int__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_726_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% Int_commute
thf(fact_727_Un__Int__crazy,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) ) @ ( inf_inf @ ( set @ A ) @ C3 @ A3 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) @ ( sup_sup @ ( set @ A ) @ C3 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_728_Int__Un__distrib,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_729_Un__Int__distrib,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_730_Int__Un__distrib2,axiom,
! [A: $tType,B2: set @ A,C3: set @ A,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) @ A3 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ A3 ) @ ( inf_inf @ ( set @ A ) @ C3 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_731_Int__left__absorb,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ).
% Int_left_absorb
thf(fact_732_Un__Int__distrib2,axiom,
! [A: $tType,B2: set @ A,C3: set @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) @ A3 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B2 @ A3 ) @ ( sup_sup @ ( set @ A ) @ C3 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_733_Int__left__commute,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_734_less__assn__def,axiom,
( ( ord_less @ assn )
= ( ^ [A5: assn,B4: assn] :
( ( ord_less_eq @ assn @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% less_assn_def
thf(fact_735_UnE,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
=> ( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_736_UnI1,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_737_UnI2,axiom,
! [A: $tType,C2: A,B2: set @ A,A3: set @ A] :
( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_738_bex__Un,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
& ( P @ X4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( P @ X4 ) )
| ? [X4: A] :
( ( member @ A @ X4 @ B2 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_739_ball__Un,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( P @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_740_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_assoc
thf(fact_741_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_742_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% Un_commute
thf(fact_743_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ).
% Un_left_absorb
thf(fact_744_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_745_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_746_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y ) )
= ( sup_sup @ A @ X3 @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_747_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_748_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y ) @ Z2 )
= ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_749_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X4: A,Y5: A] : ( sup_sup @ A @ Y5 @ X4 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_750_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A4 @ B3 ) @ C2 )
= ( ord_min @ A @ A4 @ ( ord_min @ A @ B3 @ C2 ) ) ) ) ).
% min.assoc
thf(fact_751_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( ord_min @ A @ B4 @ A5 ) ) ) ) ).
% min.commute
thf(fact_752_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_min @ A @ B3 @ ( ord_min @ A @ A4 @ C2 ) )
= ( ord_min @ A @ A4 @ ( ord_min @ A @ B3 @ C2 ) ) ) ) ).
% min.left_commute
thf(fact_753_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ C2 )
= ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_754_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y ) @ Z2 )
= ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% sup_assoc
thf(fact_755_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B4: A] : ( sup_sup @ A @ B4 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_756_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X4: A,Y5: A] : ( sup_sup @ A @ Y5 @ X4 ) ) ) ) ).
% sup_commute
thf(fact_757_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A4 @ C2 ) )
= ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_758_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% sup_left_commute
thf(fact_759_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F7: A > B,G4: A > B,X4: A] : ( sup_sup @ B @ ( F7 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% sup_fun_def
thf(fact_760_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_761_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_762_sorted__wrt__mono__rel,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o,Q: A > A > $o] :
( ! [X: A,Y2: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) ) ) )
=> ( ( sorted_wrt @ A @ P @ Xs )
=> ( sorted_wrt @ A @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_763_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ A5 @ B4 ) ) ) ) ).
% min_def
thf(fact_764_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_min @ A @ X3 @ Y )
= X3 ) ) ) ).
% min_absorb1
thf(fact_765_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_min @ A @ X3 @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_766_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,C2: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ D3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A4 @ B3 ) @ ( ord_min @ A @ C2 @ D3 ) ) ) ) ) ).
% min.mono
thf(fact_767_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( A4
= ( ord_min @ A @ A4 @ B3 ) ) ) ) ).
% min.orderE
thf(fact_768_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( ord_min @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% min.orderI
thf(fact_769_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ ( ord_min @ A @ B3 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A4 @ B3 )
=> ~ ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% min.boundedE
thf(fact_770_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ A4 @ C2 )
=> ( ord_less_eq @ A @ A4 @ ( ord_min @ A @ B3 @ C2 ) ) ) ) ) ).
% min.boundedI
thf(fact_771_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( A5
= ( ord_min @ A @ A5 @ B4 ) ) ) ) ) ).
% min.order_iff
thf(fact_772_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A4 @ B3 ) @ A4 ) ) ).
% min.cobounded1
thf(fact_773_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A4 @ B3 ) @ B3 ) ) ).
% min.cobounded2
thf(fact_774_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_min @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% min.absorb_iff1
thf(fact_775_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_min @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% min.absorb_iff2
thf(fact_776_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% min.coboundedI1
thf(fact_777_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% min.coboundedI2
thf(fact_778_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X3 @ Z2 )
| ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% min_le_iff_disj
thf(fact_779_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% min.strict_coboundedI2
thf(fact_780_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ A4 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% min.strict_coboundedI1
thf(fact_781_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( ord_min @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_782_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ ( ord_min @ A @ B3 @ C2 ) )
=> ~ ( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% min.strict_boundedE
thf(fact_783_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 )
= ( ( ord_less @ A @ X3 @ Z2 )
| ( ord_less @ A @ Y @ Z2 ) ) ) ) ).
% min_less_iff_disj
thf(fact_784_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 )
= ( ord_min @ A @ ( minus_minus @ A @ X3 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% min_diff_distrib_left
thf(fact_785_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X3: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_786_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_787_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A,X3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ X3 )
=> ~ ( ( ord_less_eq @ A @ A4 @ X3 )
=> ~ ( ord_less_eq @ A @ B3 @ X3 ) ) ) ) ).
% le_supE
thf(fact_788_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,X3: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ X3 )
=> ( ( ord_less_eq @ A @ B3 @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ X3 ) ) ) ) ).
% le_supI
thf(fact_789_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% sup_ge1
thf(fact_790_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X3: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X3 @ Y ) ) ) ).
% sup_ge2
thf(fact_791_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ X3 @ A4 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% le_supI1
thf(fact_792_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,B3: A,A4: A] :
( ( ord_less_eq @ A @ X3 @ B3 )
=> ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% le_supI2
thf(fact_793_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A4: A,D3: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ( ord_less_eq @ A @ D3 @ B3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D3 ) @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ) ).
% sup.mono
thf(fact_794_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,C2: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ D3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A4 @ B3 ) @ ( sup_sup @ A @ C2 @ D3 ) ) ) ) ) ).
% sup_mono
thf(fact_795_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X3: A,Z2: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X3 ) ) ) ) ).
% sup_least
thf(fact_796_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y5: A] :
( ( sup_sup @ A @ X4 @ Y5 )
= Y5 ) ) ) ) ).
% le_iff_sup
thf(fact_797_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4
= ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.orderE
thf(fact_798_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( sup_sup @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% sup.orderI
thf(fact_799_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F3: A > A > A,X3: A,Y: A] :
( ! [X: A,Y2: A] : ( ord_less_eq @ A @ X @ ( F3 @ X @ Y2 ) )
=> ( ! [X: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F3 @ X @ Y2 ) )
=> ( ! [X: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ Z3 @ X )
=> ( ord_less_eq @ A @ ( F3 @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup @ A @ X3 @ Y )
= ( F3 @ X3 @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_800_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( sup_sup @ A @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb1
thf(fact_801_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( sup_sup @ A @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb2
thf(fact_802_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( sup_sup @ A @ X3 @ Y )
= X3 ) ) ) ).
% sup_absorb1
thf(fact_803_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( sup_sup @ A @ X3 @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_804_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A4 )
=> ~ ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% sup.boundedE
thf(fact_805_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 ) ) ) ) ).
% sup.boundedI
thf(fact_806_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B4 ) ) ) ) ) ).
% sup.order_iff
thf(fact_807_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ A4 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.cobounded1
thf(fact_808_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.cobounded2
thf(fact_809_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( sup_sup @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_810_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( sup_sup @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_811_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.coboundedI1
thf(fact_812_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.coboundedI2
thf(fact_813_inf__min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% inf_min
thf(fact_814_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_815_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ A4 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_816_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( A5
= ( sup_sup @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_817_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 )
=> ~ ( ( ord_less @ A @ B3 @ A4 )
=> ~ ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_818_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( sup_sup @ A @ A4 @ B3 )
= B3 ) ) ) ).
% sup.absorb4
thf(fact_819_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( sup_sup @ A @ A4 @ B3 )
= A4 ) ) ) ).
% sup.absorb3
thf(fact_820_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,B3: A,A4: A] :
( ( ord_less @ A @ X3 @ B3 )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% less_supI2
thf(fact_821_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,A4: A,B3: A] :
( ( ord_less @ A @ X3 @ A4 )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% less_supI1
thf(fact_822_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% boolean_algebra.disj_zero_right
thf(fact_823_distrib__imp1,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ! [X: A,Y2: A,Z3: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y2 @ Z3 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y2 ) @ ( inf_inf @ A @ X @ Z3 ) ) )
=> ( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y ) @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ) ).
% distrib_imp1
thf(fact_824_distrib__imp2,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ! [X: A,Y2: A,Z3: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y2 @ Z3 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y2 ) @ ( sup_sup @ A @ X @ Z3 ) ) )
=> ( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y ) @ ( inf_inf @ A @ X3 @ Z2 ) ) ) ) ) ).
% distrib_imp2
thf(fact_825_inf__sup__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y ) @ ( inf_inf @ A @ X3 @ Z2 ) ) ) ) ).
% inf_sup_distrib1
thf(fact_826_inf__sup__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z2: A,X3: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X3 )
= ( sup_sup @ A @ ( inf_inf @ A @ Y @ X3 ) @ ( inf_inf @ A @ Z2 @ X3 ) ) ) ) ).
% inf_sup_distrib2
thf(fact_827_sup__inf__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y ) @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% sup_inf_distrib1
thf(fact_828_sup__inf__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z2: A,X3: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z2 ) @ X3 )
= ( inf_inf @ A @ ( sup_sup @ A @ Y @ X3 ) @ ( sup_sup @ A @ Z2 @ X3 ) ) ) ) ).
% sup_inf_distrib2
thf(fact_829_Un__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_830_Un__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Un_empty_right
thf(fact_831_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_832_Un__mono,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B2: set @ A,D2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_833_Un__least,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C3 ) ) ) ).
% Un_least
thf(fact_834_Un__upper1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ).
% Un_upper1
thf(fact_835_Un__upper2,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ).
% Un_upper2
thf(fact_836_Un__absorb1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_837_Un__absorb2,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= A3 ) ) ).
% Un_absorb2
thf(fact_838_subset__UnE,axiom,
! [A: $tType,C3: set @ A,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
=> ~ ! [A12: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A12 @ A3 )
=> ! [B9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B9 @ B2 )
=> ( C3
!= ( sup_sup @ ( set @ A ) @ A12 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_839_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( sup_sup @ ( set @ A ) @ A7 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_840_Un__Int__assoc__eq,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C3 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_841_set__diff__diff__left,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( minus_minus @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% set_diff_diff_left
thf(fact_842_Un__Diff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ C3 ) @ ( minus_minus @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_Diff
thf(fact_843_Diff__Un,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Diff_Un
thf(fact_844_Diff__Int,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B2 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Diff_Int
thf(fact_845_Int__Diff__Un,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_846_Un__Diff__Int,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_847_in__set__butlastD,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_848_distinct__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( butlast @ A @ Xs ) ) ) ).
% distinct_butlast
thf(fact_849_butlast__update_H,axiom,
! [A: $tType,L: list @ A,I: nat,X3: A] :
( ( list_update @ A @ ( butlast @ A @ L ) @ I @ X3 )
= ( butlast @ A @ ( list_update @ A @ L @ I @ X3 ) ) ) ).
% butlast_update'
thf(fact_850_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I4: nat,J2: nat] :
( ( ord_less_eq @ nat @ I4 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J2 ) @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_851_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_852_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_853_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_854_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_855_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y ) @ ( inf_inf @ A @ X3 @ Z2 ) ) @ ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% distrib_inf_le
thf(fact_856_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y @ Z2 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y ) @ ( sup_sup @ A @ X3 @ Z2 ) ) ) ) ).
% distrib_sup_le
thf(fact_857_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A4: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A4 @ Xs ) ) ) ) ).
% sorted_remove1
thf(fact_858_insert__is__Un,axiom,
! [A: $tType] :
( ( insert3 @ A )
= ( ^ [A5: A] : ( sup_sup @ ( set @ A ) @ ( insert3 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_859_Un__singleton__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X3: A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_860_singleton__Un__iff,axiom,
! [A: $tType,X3: A,A3: set @ A,B2: set @ A] :
( ( ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_861_Diff__partition,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_862_Diff__subset__conv,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_863_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_864_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_865_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_866_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P2: A > A > $o,Xs3: list @ A] :
! [I4: nat,J2: nat] :
( ( ord_less @ nat @ I4 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P2 @ ( nth @ A @ Xs3 @ I4 ) @ ( nth @ A @ Xs3 @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_867_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_868_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_869_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat,J2: nat] :
( ( ord_less_eq @ nat @ I4 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_870_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_871_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat,J2: nat] :
( ( ord_less @ nat @ I4 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_872_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_873_times__assn__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As2: set @ nat] :
( ( times_assn_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ 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 ) @ H2 @ As1 ) )
& ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As22 ) ) ) ) ) ).
% times_assn_raw.simps
thf(fact_874_times__assn__raw_Oelims_I1_J,axiom,
! [X3: ( 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 @ X3 @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ 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 ) ) )
& ( X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As22 ) ) ) ) ) ) ) ).
% times_assn_raw.elims(1)
thf(fact_875_times__assn__raw_Oelims_I2_J,axiom,
! [X3: ( 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 @ X3 @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ~ ? [As12: set @ nat,As23: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As23 ) ) ) ) ) ).
% times_assn_raw.elims(2)
thf(fact_876_times__assn__raw_Oelims_I3_J,axiom,
! [X3: ( 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 @ X3 @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ? [As13: set @ nat,As24: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As24 ) ) ) ) ) ).
% times_assn_raw.elims(3)
thf(fact_877_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K3: A,F3: A > nat,B3: nat] :
( ( P @ K3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less @ nat @ ( F3 @ Y2 ) @ B3 ) )
=> ? [X: A] :
( ( P @ X )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ ( F3 @ Y3 ) @ ( F3 @ X ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_878_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( P @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ! [I6: nat] :
( ( ord_less @ nat @ K @ I6 )
=> ( P @ I6 ) )
=> ( P @ K ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_879_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_880_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_881_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I4 ) ) @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_882_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C2 @ A4 ) )
= ( B3 = C2 ) ) ) ).
% add_right_cancel
thf(fact_883_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add_left_cancel
thf(fact_884_mod__or__dist,axiom,
! [P: assn,Q: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( sup_sup @ assn @ P @ Q ) @ H2 )
= ( ( rep_assn @ P @ H2 )
| ( rep_assn @ Q @ H2 ) ) ) ).
% mod_or_dist
thf(fact_885_merge__pure__or,axiom,
! [A4: $o,B3: $o] :
( ( sup_sup @ assn @ ( pure_assn @ A4 ) @ ( pure_assn @ B3 ) )
= ( pure_assn
@ ( A4
| B3 ) ) ) ).
% merge_pure_or
thf(fact_886_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_887_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_888_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_889_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_890_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ).
% add_diff_cancel
thf(fact_891_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ).
% diff_add_cancel
thf(fact_892_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% add_diff_cancel_left
thf(fact_893_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ A4 )
= B3 ) ) ).
% add_diff_cancel_left'
thf(fact_894_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ A4 @ B3 ) ) ) ).
% add_diff_cancel_right
thf(fact_895_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ).
% add_diff_cancel_right'
thf(fact_896_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_897_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_898_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_899_nat__add__left__cancel__less,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K3 @ M2 ) @ ( plus_plus @ nat @ K3 @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_900_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( plus_plus @ A @ B3 @ ( minus_minus @ A @ A4 @ B3 ) )
= A4 ) ) ) ).
% le_add_diff_inverse
thf(fact_901_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ) ).
% le_add_diff_inverse2
thf(fact_902_min__Suc__gt_I1_J,axiom,
! [A4: nat,B3: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( ( ord_min @ nat @ ( suc @ A4 ) @ B3 )
= ( suc @ A4 ) ) ) ).
% min_Suc_gt(1)
thf(fact_903_min__Suc__gt_I2_J,axiom,
! [A4: nat,B3: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( ( ord_min @ nat @ B3 @ ( suc @ A4 ) )
= ( suc @ A4 ) ) ) ).
% min_Suc_gt(2)
thf(fact_904_mod__h__bot__iff_I7_J,axiom,
! [P: assn,Q: assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( sup_sup @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
| ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(7)
thf(fact_905_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_906_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ~ ! [Q6: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M2 @ Q6 ) ) ) ) ).
% less_natE
thf(fact_907_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_908_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_909_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M: nat,N2: nat] :
? [K4: nat] :
( N2
= ( suc @ ( plus_plus @ nat @ M @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_910_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ? [K: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M2 @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_911_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C2 @ A4 ) )
=> ( B3 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_912_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C2 ) )
=> ( B3 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_913_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) )
= ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_914_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_915_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= ( plus_plus @ A @ C2 @ A4 ) )
= ( B3 = C2 ) ) ) ).
% add.right_cancel
thf(fact_916_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( plus_plus @ A @ A4 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% add.left_cancel
thf(fact_917_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_918_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: A,K3: A,B3: A,A4: A] :
( ( B2
= ( plus_plus @ A @ K3 @ B3 ) )
=> ( ( plus_plus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K3 @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_919_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K3: A,A4: A,B3: A] :
( ( A3
= ( plus_plus @ A @ K3 @ A4 ) )
=> ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ K3 @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_920_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K3: A,L: A] :
( ( ( I = J )
& ( K3 = L ) )
=> ( ( plus_plus @ A @ I @ K3 )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_921_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_imp_le_right
thf(fact_922_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_imp_le_left
thf(fact_923_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
? [C6: A] :
( B4
= ( plus_plus @ A @ A5 @ C6 ) ) ) ) ) ).
% le_iff_add
thf(fact_924_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_925_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ~ ! [C5: A] :
( B3
!= ( plus_plus @ A @ A4 @ C5 ) ) ) ) ).
% less_eqE
thf(fact_926_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_left_mono
thf(fact_927_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_928_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K3: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K3 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_929_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K3: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K3 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_930_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K3: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K3 = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_931_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K3: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K3 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_932_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K3: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K3 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_933_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K3: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K3 = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_934_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_935_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_strict_left_mono
thf(fact_936_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_937_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_imp_less_left
thf(fact_938_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_imp_less_right
thf(fact_939_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_940_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_941_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_942_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_943_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_944_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A4: A,E3: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ E3 ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_945_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K3: A,A4: A,B3: A] :
( ( A3
= ( plus_plus @ A @ K3 @ A4 ) )
=> ( ( minus_minus @ A @ A3 @ B3 )
= ( plus_plus @ A @ K3 @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_946_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( minus_minus @ A @ A4 @ B3 )
= C2 )
= ( A4
= ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% diff_eq_eq
thf(fact_947_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( A4
= ( minus_minus @ A @ C2 @ B3 ) )
= ( ( plus_plus @ A @ A4 @ B3 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_948_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( plus_plus @ A @ A4 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_949_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( minus_minus @ A @ A4 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C2 ) @ B3 ) ) ) ).
% diff_diff_eq2
thf(fact_950_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ C2 ) @ B3 ) ) ) ).
% diff_add_eq
thf(fact_951_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C2 ) @ B3 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_952_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ( plus_plus @ A @ C2 @ B3 )
= A4 )
=> ( C2
= ( minus_minus @ A @ A4 @ B3 ) ) ) ) ).
% add_implies_diff
thf(fact_953_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( minus_minus @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_954_star__or__dist1,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( times_times @ assn @ ( sup_sup @ assn @ A3 @ B2 ) @ C3 )
= ( sup_sup @ assn @ ( times_times @ assn @ A3 @ C3 ) @ ( times_times @ assn @ B2 @ C3 ) ) ) ).
% star_or_dist1
thf(fact_955_star__or__dist2,axiom,
! [C3: assn,A3: assn,B2: assn] :
( ( times_times @ assn @ C3 @ ( sup_sup @ assn @ A3 @ B2 ) )
= ( sup_sup @ assn @ ( times_times @ assn @ C3 @ A3 ) @ ( times_times @ assn @ C3 @ B2 ) ) ) ).
% star_or_dist2
thf(fact_956_Nat_OlessE,axiom,
! [I: nat,K3: nat] :
( ( ord_less @ nat @ I @ K3 )
=> ( ( K3
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( K3
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_957_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_958_Suc__lessE,axiom,
! [I: nat,K3: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K3 )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( K3
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_959_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_960_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_961_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_962_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_963_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_964_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M2 @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_965_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_966_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_967_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_968_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_969_less__trans__Suc,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ K3 )
=> ( ord_less @ nat @ ( suc @ I ) @ K3 ) ) ) ).
% less_trans_Suc
thf(fact_970_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I5: nat] : ( P @ I5 @ ( suc @ I5 ) )
=> ( ! [I5: nat,J3: nat,K: nat] :
( ( ord_less @ nat @ I5 @ J3 )
=> ( ( ord_less @ nat @ J3 @ K )
=> ( ( P @ I5 @ J3 )
=> ( ( P @ J3 @ K )
=> ( P @ I5 @ K ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_971_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I5: nat] :
( ( J
= ( suc @ I5 ) )
=> ( P @ I5 ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ J )
=> ( ( P @ ( suc @ I5 ) )
=> ( P @ I5 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_972_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_973_add__lessD1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K3 )
=> ( ord_less @ nat @ I @ K3 ) ) ).
% add_lessD1
thf(fact_974_add__less__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K3 @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K3 ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_975_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_976_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_977_add__less__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K3 ) @ ( plus_plus @ nat @ J @ K3 ) ) ) ).
% add_less_mono1
thf(fact_978_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_979_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_980_less__add__eq__less,axiom,
! [K3: nat,L: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ K3 @ L )
=> ( ( ( plus_plus @ nat @ M2 @ L )
= ( plus_plus @ nat @ K3 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_981_ent__disjI2__direct,axiom,
! [B2: assn,A3: assn] : ( entails @ B2 @ ( sup_sup @ assn @ A3 @ B2 ) ) ).
% ent_disjI2_direct
thf(fact_982_ent__disjI1__direct,axiom,
! [A3: assn,B2: assn] : ( entails @ A3 @ ( sup_sup @ assn @ A3 @ B2 ) ) ).
% ent_disjI1_direct
thf(fact_983_ent__disjI2_H,axiom,
! [A3: assn,C3: assn,B2: assn] :
( ( entails @ A3 @ C3 )
=> ( entails @ A3 @ ( sup_sup @ assn @ B2 @ C3 ) ) ) ).
% ent_disjI2'
thf(fact_984_ent__disjI1_H,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( entails @ A3 @ B2 )
=> ( entails @ A3 @ ( sup_sup @ assn @ B2 @ C3 ) ) ) ).
% ent_disjI1'
thf(fact_985_ent__disjI2,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ ( sup_sup @ assn @ P @ Q ) @ R )
=> ( entails @ Q @ R ) ) ).
% ent_disjI2
thf(fact_986_ent__disjI1,axiom,
! [P: assn,Q: assn,R: assn] :
( ( entails @ ( sup_sup @ assn @ P @ Q ) @ R )
=> ( entails @ P @ R ) ) ).
% ent_disjI1
thf(fact_987_ent__disjE,axiom,
! [A3: assn,C3: assn,B2: assn] :
( ( entails @ A3 @ C3 )
=> ( ( entails @ B2 @ C3 )
=> ( entails @ ( sup_sup @ assn @ A3 @ B2 ) @ C3 ) ) ) ).
% ent_disjE
thf(fact_988_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X3 @ ( ord_min @ A @ Y @ Z2 ) )
= ( ord_min @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( plus_plus @ A @ X3 @ Z2 ) ) ) ) ).
% min_add_distrib_right
thf(fact_989_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X3 @ Y ) @ Z2 )
= ( ord_min @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% min_add_distrib_left
thf(fact_990_entt__disjE,axiom,
! [A3: assn,M5: assn,B2: assn] :
( ( entailst @ A3 @ M5 )
=> ( ( entailst @ B2 @ M5 )
=> ( entailst @ ( sup_sup @ assn @ A3 @ B2 ) @ M5 ) ) ) ).
% entt_disjE
thf(fact_991_entt__disjD1,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( entailst @ ( sup_sup @ assn @ A3 @ B2 ) @ C3 )
=> ( entailst @ A3 @ C3 ) ) ).
% entt_disjD1
thf(fact_992_entt__disjD2,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( entailst @ ( sup_sup @ assn @ A3 @ B2 ) @ C3 )
=> ( entailst @ B2 @ C3 ) ) ).
% entt_disjD2
thf(fact_993_entt__disjI1_H,axiom,
! [A3: assn,B2: assn,C3: assn] :
( ( entailst @ A3 @ B2 )
=> ( entailst @ A3 @ ( sup_sup @ assn @ B2 @ C3 ) ) ) ).
% entt_disjI1'
thf(fact_994_entt__disjI2_H,axiom,
! [A3: assn,C3: assn,B2: assn] :
( ( entailst @ A3 @ C3 )
=> ( entailst @ A3 @ ( sup_sup @ assn @ B2 @ C3 ) ) ) ).
% entt_disjI2'
thf(fact_995_entt__disjI1__direct,axiom,
! [A3: assn,B2: assn] : ( entailst @ A3 @ ( sup_sup @ assn @ A3 @ B2 ) ) ).
% entt_disjI1_direct
thf(fact_996_entt__disjI2__direct,axiom,
! [B2: assn,A3: assn] : ( entailst @ B2 @ ( sup_sup @ assn @ A3 @ B2 ) ) ).
% entt_disjI2_direct
thf(fact_997_norm__assertion__simps_I6_J,axiom,
! [X3: assn] :
( ( sup_sup @ assn @ X3 @ ( bot_bot @ assn ) )
= X3 ) ).
% norm_assertion_simps(6)
thf(fact_998_norm__assertion__simps_I5_J,axiom,
! [X3: assn] :
( ( sup_sup @ assn @ ( bot_bot @ assn ) @ X3 )
= X3 ) ).
% norm_assertion_simps(5)
thf(fact_999_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K3: A,F3: A > nat,N: nat] :
( ( P @ K3 )
=> ( ! [X: A] :
( ( P @ X )
=> ? [Y3: A] :
( ( P @ Y3 )
& ~ ( ord_less_eq @ nat @ ( F3 @ Y3 ) @ ( F3 @ X ) ) ) )
=> ? [Y2: A] :
( ( P @ Y2 )
& ~ ( ord_less @ nat @ ( F3 @ Y2 ) @ ( plus_plus @ nat @ ( F3 @ K3 ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_1000_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K3: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K3 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1001_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K3: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K3 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K3 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1002_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_1003_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_1004_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( ord_less_eq @ A @ A4 @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% diff_le_eq
thf(fact_1005_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( minus_minus @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_1006_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K3: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ N )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K3 ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_1007_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ A4 )
= B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1008_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K3: A,N: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K3 ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K3 ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K3 ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K3 ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1009_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C2 ) @ A4 ) ) ) ) ).
% le_add_diff
thf(fact_1010_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B3 @ A4 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A4 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1011_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B3 @ A4 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B3 ) @ A4 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1012_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B3 ) @ A4 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B3 @ A4 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1013_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C2 ) @ A4 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1014_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B3 @ C2 ) @ A4 )
= ( plus_plus @ A @ ( minus_minus @ A @ B3 @ A4 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1015_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B3 @ A4 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A4 ) @ B3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1016_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ A4 @ ( minus_minus @ A @ B3 @ A4 ) )
= B3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1017_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( minus_minus @ A @ B3 @ A4 )
= C2 )
= ( B3
= ( plus_plus @ A @ C2 @ A4 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1018_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A] : ( ord_less @ A @ A4 @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_1019_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_1020_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( ord_less @ A @ A4 @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% diff_less_eq
thf(fact_1021_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ A4 @ ( minus_minus @ A @ C2 @ B3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_1022_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,B3: A] :
( ~ ( ord_less @ A @ A4 @ B3 )
=> ( ( plus_plus @ A @ B3 @ ( minus_minus @ A @ A4 @ B3 ) )
= A4 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1023_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X3: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ A @ X3 @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_1024_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,E3: A,C2: A,B3: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A4 ) @ E3 ) @ D3 ) ) ) ) ).
% eq_add_iff2
thf(fact_1025_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,E3: A,C2: A,B3: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ E3 ) @ C2 )
= D3 ) ) ) ).
% eq_add_iff1
thf(fact_1026_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N6: nat] :
( ! [N5: nat] : ( ord_less @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
=> ( ( ord_less @ nat @ N @ N6 )
=> ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ N6 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1027_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,M2: nat] :
( ! [N5: nat] : ( ord_less @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
=> ( ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ M2 ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1028_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1029_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1030_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N5: nat] :
( ( ord_less_eq @ nat @ I @ N5 )
=> ( ( ord_less @ nat @ N5 @ J )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1031_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N5: nat] :
( ( ord_less_eq @ nat @ I @ N5 )
=> ( ( ord_less @ nat @ N5 @ J )
=> ( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1032_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1033_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1034_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1035_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1036_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1037_nat__in__between__eq_I1_J,axiom,
! [A4: nat,B3: nat] :
( ( ( ord_less @ nat @ A4 @ B3 )
& ( ord_less_eq @ nat @ B3 @ ( suc @ A4 ) ) )
= ( B3
= ( suc @ A4 ) ) ) ).
% nat_in_between_eq(1)
thf(fact_1038_nat__in__between__eq_I2_J,axiom,
! [A4: nat,B3: nat] :
( ( ( ord_less_eq @ nat @ A4 @ B3 )
& ( ord_less @ nat @ B3 @ ( suc @ A4 ) ) )
= ( B3 = A4 ) ) ).
% nat_in_between_eq(2)
thf(fact_1039_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( ( suc @ ( minus_minus @ nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1040_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1041_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M2: nat,K3: nat] :
( ! [M6: nat,N5: nat] :
( ( ord_less @ nat @ M6 @ N5 )
=> ( ord_less @ nat @ ( F3 @ M6 ) @ ( F3 @ N5 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F3 @ M2 ) @ K3 ) @ ( F3 @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1042_Suc__mult__less__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K3 ) @ M2 ) @ ( times_times @ nat @ ( suc @ K3 ) @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1043_less__diff__conv,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K3 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I @ K3 ) @ J ) ) ).
% less_diff_conv
thf(fact_1044_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less @ nat @ M2 @ N )
=> ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1045_mlex__bound,axiom,
! [A4: nat,A3: nat,B3: nat,N7: nat] :
( ( ord_less @ nat @ A4 @ A3 )
=> ( ( ord_less @ nat @ B3 @ N7 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A4 @ N7 ) @ B3 ) @ ( times_times @ nat @ A3 @ N7 ) ) ) ) ).
% mlex_bound
thf(fact_1046_mlex__fst__decrI,axiom,
! [A4: nat,A9: nat,B3: nat,N7: nat,B8: nat] :
( ( ord_less @ nat @ A4 @ A9 )
=> ( ( ord_less @ nat @ B3 @ N7 )
=> ( ( ord_less @ nat @ B8 @ N7 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A4 @ N7 ) @ B3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A9 @ N7 ) @ B8 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_1047_mlex__snd__decrI,axiom,
! [A4: nat,A9: nat,B3: nat,B8: nat,N7: nat] :
( ( A4 = A9 )
=> ( ( ord_less @ nat @ B3 @ B8 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A4 @ N7 ) @ B3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A9 @ N7 ) @ B8 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_1048_mlex__leI,axiom,
! [A4: nat,A9: nat,B3: nat,B8: nat,N7: nat] :
( ( ord_less_eq @ nat @ A4 @ A9 )
=> ( ( ord_less_eq @ nat @ B3 @ B8 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A4 @ N7 ) @ B3 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A9 @ N7 ) @ B8 ) ) ) ) ).
% mlex_leI
thf(fact_1049_le__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E3: A,C2: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A4 ) @ E3 ) @ D3 ) ) ) ) ).
% le_add_iff2
thf(fact_1050_le__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E3: A,C2: A,B3: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ E3 ) @ C2 ) @ D3 ) ) ) ).
% le_add_iff1
thf(fact_1051_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E3: A,C2: A,B3: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B3 @ A4 ) @ E3 ) @ D3 ) ) ) ) ).
% less_add_iff2
thf(fact_1052_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,E3: A,C2: A,B3: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B3 @ E3 ) @ D3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ E3 ) @ C2 ) @ D3 ) ) ) ).
% less_add_iff1
thf(fact_1053_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A] :
( ( minus_minus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X3 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_1054_less__diff__conv2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K3 @ J )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K3 ) @ I )
= ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K3 ) ) ) ) ).
% less_diff_conv2
thf(fact_1055_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_1056_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ ( suc @ I4 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1057_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_1058_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1059_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1060_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_1061_sorted__in__between,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: nat,J: nat,L: list @ A,X3: 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 ) @ X3 )
=> ( ( ord_less @ A @ X3 @ ( nth @ A @ L @ J ) )
=> ~ ! [K: nat] :
( ( ord_less_eq @ nat @ I @ K )
=> ( ( ord_less @ nat @ K @ J )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ K ) @ X3 )
=> ~ ( ord_less @ A @ X3 @ ( nth @ A @ L @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
thf(fact_1062_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W2: A,Y: A,X3: A,Z2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y ) @ ( times_times @ A @ X3 @ Z2 ) )
= ( plus_plus @ A @ ( times_times @ A @ W2 @ Z2 ) @ ( times_times @ A @ X3 @ Y ) ) )
= ( ( W2 = X3 )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
thf(fact_1063_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( A4 != B3 )
& ( C2 != D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A4 @ D3 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_1064_nth__step__trancl,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A ),N: nat,M2: nat] :
( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ ( suc @ N5 ) ) @ ( nth @ A @ Xs @ N5 ) ) @ R ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ ( nth @ A @ Xs @ M2 ) ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% nth_step_trancl
thf(fact_1065_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X4: A] : ( minus_minus @ A @ ( plus_plus @ A @ X4 @ X4 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_1066_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_1067_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_1068_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% add.right_neutral
thf(fact_1069_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A4 @ A4 ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_1070_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B3: A,A4: A] :
( ( ( plus_plus @ A @ B3 @ A4 )
= A4 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_1071_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A4: A,B3: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= A4 )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_1072_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( plus_plus @ A @ B3 @ A4 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_1073_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( plus_plus @ A @ A4 @ B3 ) )
= ( B3
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_1074_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Y: A] :
( ( ( plus_plus @ A @ X3 @ Y )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1075_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X3 @ Y ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1076_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A4 )
= A4 ) ) ).
% add_0
thf(fact_1077_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ( times_times @ A @ A4 @ C2 )
= ( times_times @ A @ B3 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4 = B3 ) ) ) ) ).
% mult_cancel_right
thf(fact_1078_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ( times_times @ A @ C2 @ A4 )
= ( times_times @ A @ C2 @ B3 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4 = B3 ) ) ) ) ).
% mult_cancel_left
thf(fact_1079_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A4: A,B3: A] :
( ( ( times_times @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
= ( ( A4
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_1080_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_1081_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_1082_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ A4 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1083_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% diff_zero
thf(fact_1084_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_1085_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% diff_0_right
thf(fact_1086_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ A4 @ A4 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_1087_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_1088_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_1089_bot__nat__0_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A4 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1090_trancl__single,axiom,
! [A: $tType,A4: A,B3: A] :
( ( transitive_trancl @ A @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) )
= ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trancl_single
thf(fact_1091_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B3 @ A4 ) @ B3 )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_1092_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_1093_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( plus_plus @ A @ A4 @ B3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% le_add_same_cancel1
thf(fact_1094_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( plus_plus @ A @ B3 @ A4 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% le_add_same_cancel2
thf(fact_1095_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1096_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ A4 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1097_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_1098_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% zero_comp_diff_simps(1)
thf(fact_1099_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ A4 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1100_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1101_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( plus_plus @ A @ B3 @ A4 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel2
thf(fact_1102_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( plus_plus @ A @ A4 @ B3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel1
thf(fact_1103_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ B3 )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_1104_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B3 @ A4 ) @ B3 )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_1105_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B3 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_1106_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A4 @ B3 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ).
% zero_comp_diff_simps(2)
thf(fact_1107_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B3: A] :
( ( C2
= ( times_times @ A @ C2 @ B3 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_1108_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A4: A] :
( ( ( times_times @ A @ C2 @ A4 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_1109_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B3: A] :
( ( C2
= ( times_times @ A @ B3 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_1110_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A4: A,C2: A] :
( ( ( times_times @ A @ A4 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A4
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_1111_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( plus_plus @ A @ A4 @ B3 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_1112_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_1113_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_1114_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1115_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_1116_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1117_nat__mult__less__cancel__disj,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K3 @ M2 ) @ ( times_times @ nat @ K3 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
& ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1118_mult__less__cancel2,axiom,
! [M2: nat,K3: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M2 @ K3 ) @ ( times_times @ nat @ N @ K3 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
& ( ord_less @ nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1119_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1120_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_1121_slice__complete,axiom,
! [A: $tType,Xs: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs )
= Xs ) ).
% slice_complete
thf(fact_1122_Suc__pred,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1123_nat__mult__le__cancel__disj,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ M2 ) @ ( times_times @ nat @ K3 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1124_mult__le__cancel2,axiom,
! [M2: nat,K3: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M2 @ K3 ) @ ( times_times @ nat @ N @ K3 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1125_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1126_assn__aci_I8_J,axiom,
! [X3: assn,Y: assn] :
( ( sup_sup @ assn @ X3 @ ( sup_sup @ assn @ X3 @ Y ) )
= ( sup_sup @ assn @ X3 @ Y ) ) ).
% assn_aci(8)
thf(fact_1127_assn__aci_I7_J,axiom,
! [X3: assn,Y: assn,Z2: assn] :
( ( sup_sup @ assn @ X3 @ ( sup_sup @ assn @ Y @ Z2 ) )
= ( sup_sup @ assn @ Y @ ( sup_sup @ assn @ X3 @ Z2 ) ) ) ).
% assn_aci(7)
thf(fact_1128_assn__aci_I5_J,axiom,
( ( sup_sup @ assn )
= ( ^ [X4: assn,Y5: assn] : ( sup_sup @ assn @ Y5 @ X4 ) ) ) ).
% assn_aci(5)
thf(fact_1129_norm__assertion__simps_I32_J,axiom,
! [X3: assn] :
( ( sup_sup @ assn @ X3 @ X3 )
= X3 ) ).
% norm_assertion_simps(32)
thf(fact_1130_norm__assertion__simps_I15_J,axiom,
! [X3: assn,Y: assn,Z2: assn] :
( ( sup_sup @ assn @ ( sup_sup @ assn @ X3 @ Y ) @ Z2 )
= ( sup_sup @ assn @ X3 @ ( sup_sup @ assn @ Y @ Z2 ) ) ) ).
% norm_assertion_simps(15)
thf(fact_1131_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X3: A] :
( ( ( zero_zero @ A )
= X3 )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_1132_trancl__mono__mp,axiom,
! [A: $tType,U2: set @ ( product_prod @ A @ A ),V: set @ ( product_prod @ A @ A ),X3: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V )
=> ( ( member @ ( product_prod @ A @ A ) @ X3 @ ( transitive_trancl @ A @ U2 ) )
=> ( member @ ( product_prod @ A @ A ) @ X3 @ ( transitive_trancl @ A @ V ) ) ) ) ).
% trancl_mono_mp
thf(fact_1133_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_1134_trancl__sub__insert__trancl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X3: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R ) @ ( transitive_trancl @ A @ ( insert3 @ ( product_prod @ A @ A ) @ X3 @ R ) ) ) ).
% trancl_sub_insert_trancl
thf(fact_1135_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ).
% zero_le
thf(fact_1136_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_1137_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_1138_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_1139_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_1140_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_1141_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A4 )
= A4 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_1142_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% add.comm_neutral
thf(fact_1143_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A4 )
= A4 ) ) ).
% add.group_left_neutral
thf(fact_1144_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A4 @ C2 )
= ( times_times @ A @ B3 @ C2 ) )
= ( A4 = B3 ) ) ) ) ).
% mult_right_cancel
thf(fact_1145_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A4 )
= ( times_times @ A @ C2 @ B3 ) )
= ( A4 = B3 ) ) ) ) ).
% mult_left_cancel
thf(fact_1146_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ B3 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_1147_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A4: A,B3: A] :
( ( ( times_times @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
=> ( ( A4
= ( zero_zero @ A ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_1148_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A4: A,B3: A] :
( ( ( times_times @ A @ A4 @ B3 )
!= ( zero_zero @ A ) )
=> ( ( A4
!= ( zero_zero @ A ) )
& ( B3
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_1149_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_1150_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y4: A,Z4: A] : Y4 = Z4 )
= ( ^ [A5: A,B4: A] :
( ( minus_minus @ A @ A5 @ B4 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1151_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X3: A] :
( ! [X: A] :
( ( ( V @ X )
= ( zero_zero @ nat ) )
=> ( P @ X ) )
=> ( ! [X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X ) )
=> ( ~ ( P @ X )
=> ? [Y3: A] :
( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X ) )
& ~ ( P @ Y3 ) ) ) )
=> ( P @ X3 ) ) ) ).
% infinite_descent0_measure
thf(fact_1152_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N5 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1153_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_1154_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_1155_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_1156_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_1157_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_1158_bot__nat__0_Oextremum__strict,axiom,
! [A4: nat] :
~ ( ord_less @ nat @ A4 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_1159_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A4: A,B3: A,C2: A,D3: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A4 = B3 )
& ( C2 != D3 ) )
=> ( ( plus_plus @ A @ A4 @ ( times_times @ A @ R2 @ C2 ) )
!= ( plus_plus @ A @ B3 @ ( times_times @ A @ R2 @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1160_nat__mult__eq__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ( ( times_times @ nat @ K3 @ M2 )
= ( times_times @ nat @ K3 @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1161_nat__mult__less__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K3 @ M2 ) @ ( times_times @ nat @ K3 @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1162_lexn_Osimps_I1_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_1163_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X3 @ Y )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1164_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X3 @ Y )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1165_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_1166_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1167_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_1168_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% add_decreasing2
thf(fact_1169_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_1170_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% add_decreasing
thf(fact_1171_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1172_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1173_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ A4 ) ) ) ).
% zero_le_square
thf(fact_1174_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% split_mult_pos_le
thf(fact_1175_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1176_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1177_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1178_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1179_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1180_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_1181_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_1182_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1183_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1184_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1185_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B3 @ A4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1186_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1187_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1188_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_1189_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_1190_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_1191_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1192_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X3 @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_1193_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_1194_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% add_pos_pos
thf(fact_1195_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ! [C5: A] :
( ( B3
= ( plus_plus @ A @ A4 @ C5 ) )
=> ( C5
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1196_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_1197_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_1198_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ ( times_times @ A @ A4 @ A4 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_1199_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_1200_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_1201_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_1202_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_1203_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B3 @ A4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_1204_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1205_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_1206_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B3 @ A4 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_1207_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1208_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1209_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1210_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1211_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A4 @ B3 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1212_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1213_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1214_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A4 @ B3 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1215_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1216_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_1217_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_1218_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_1219_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1220_nat__mult__le__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ M2 ) @ ( times_times @ nat @ K3 @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1221_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1222_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1223_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1224_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_1225_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ? [J2: nat] :
( ( M2
= ( suc @ J2 ) )
& ( ord_less @ nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1226_nat__compl__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N5: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N5 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N5 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct
thf(fact_1227_nat__compl__induct_H,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N5: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N5 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N5 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct'
thf(fact_1228_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
& ! [I6: nat] :
( ( ord_less @ nat @ I6 @ K )
=> ~ ( P @ I6 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_1229_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ( plus_plus @ nat @ I @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1230_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_1231_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1232_mult__less__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ord_less @ nat @ ( times_times @ nat @ I @ K3 ) @ ( times_times @ nat @ J @ K3 ) ) ) ) ).
% mult_less_mono1
thf(fact_1233_mult__less__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ord_less @ nat @ ( times_times @ nat @ K3 @ I ) @ ( times_times @ nat @ K3 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1234_nat__geq__1__eq__neqz,axiom,
! [X3: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X3 )
= ( X3
!= ( zero_zero @ nat ) ) ) ).
% nat_geq_1_eq_neqz
thf(fact_1235_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_1236_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_1237_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_1238_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_1239_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1240_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1241_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1242_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1243_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1244_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1245_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1246_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1247_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1248_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1249_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1250_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1251_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1252_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1253_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1254_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1255_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X3: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1256_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A4: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ A4 ) ) ) ) ).
% mult_left_le
thf(fact_1257_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1258_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Y ) @ X3 ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1259_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X3 ) @ X3 ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1260_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X3: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1261_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_1262_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less @ nat @ K @ N )
& ! [I6: nat] :
( ( ord_less_eq @ nat @ I6 @ K )
=> ~ ( P @ I6 ) )
& ( P @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1263_length__pos__if__in__set,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1264_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1265_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1266_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1267_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_1268_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1269_nat__diff__split,axiom,
! [P: nat > $o,A4: nat,B3: nat] :
( ( P @ ( minus_minus @ nat @ A4 @ B3 ) )
= ( ( ( ord_less @ nat @ A4 @ B3 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D6: nat] :
( ( A4
= ( plus_plus @ nat @ B3 @ D6 ) )
=> ( P @ D6 ) ) ) ) ).
% nat_diff_split
thf(fact_1270_nat__diff__split__asm,axiom,
! [P: nat > $o,A4: nat,B3: nat] :
( ( P @ ( minus_minus @ nat @ A4 @ B3 ) )
= ( ~ ( ( ( ord_less @ nat @ A4 @ B3 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D6: nat] :
( ( A4
= ( plus_plus @ nat @ B3 @ D6 ) )
& ~ ( P @ D6 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1271_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1272_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A4: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A4 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1273_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1274_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1275_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1276_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A4: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A4 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1277_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ B3 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B3 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1278_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A4 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1279_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X3: A,A4: A,Y: A,U: A,V2: A] :
( ( ord_less_eq @ A @ X3 @ A4 )
=> ( ( ord_less_eq @ A @ Y @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X3 ) @ ( times_times @ A @ V2 @ Y ) ) @ A4 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1280_nz__le__conv__less,axiom,
! [K3: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ( ord_less_eq @ nat @ K3 @ M2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ K3 @ ( suc @ ( zero_zero @ nat ) ) ) @ M2 ) ) ) ).
% nz_le_conv_less
thf(fact_1281_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( N
= ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_1282_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( minus_minus @ nat @ ( suc @ M2 ) @ N )
= ( minus_minus @ nat @ M2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1283_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X3: A,A4: A,Y: A,U: A,V2: A] :
( ( ord_less @ A @ X3 @ A4 )
=> ( ( ord_less @ A @ Y @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X3 ) @ ( times_times @ A @ V2 @ Y ) ) @ A4 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1284_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ! [Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X3 ) @ Y ) ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_1285_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X3
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1286_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1287_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1288_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ X3 ) @ ( times_times @ A @ Z2 @ Y ) )
= ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1289_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ Y @ E2 ) ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% field_le_epsilon
thf(fact_1290_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q4: A,R2: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q4 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_1291_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X3: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X3 @ X3 ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X3
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1292_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) )
= ( ord_less @ A @ X3 @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_1293_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X8: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X8 ) ) ).
% linordered_field_no_lb
thf(fact_1294_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X8: A] :
? [X_1: A] : ( ord_less @ A @ X8 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_1295_converse__trancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ B3 ) @ R2 )
=> ( P @ Y2 ) )
=> ( ! [Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ Z3 )
=> ( P @ Y2 ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% converse_trancl_induct
thf(fact_1296_trancl__trans__induct,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),P: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [X: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 )
=> ( P @ X @ Y2 ) )
=> ( ! [X: A,Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ X @ Y2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P @ Y2 @ Z3 )
=> ( P @ X @ Z3 ) ) ) ) )
=> ( P @ X3 @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_1297_trancl__into__trancl2,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_1298_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_1299_irrefl__trancl__rD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( X3 != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_1300_converse__tranclE,axiom,
! [A: $tType,X3: A,Z2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_1301_r__r__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% r_r_into_trancl
thf(fact_1302_trancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y2 ) @ R2 )
=> ( P @ Y2 ) )
=> ( ! [Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R2 )
=> ( ( P @ Y2 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% trancl_induct
thf(fact_1303_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ! [A6: A,B5: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B5 ) ) @ R2 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B5 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( P @ A6 @ B5 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_1304_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ R2 )
=> ~ ! [B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B5 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A22 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_1305_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ? [A5: A,B4: A] :
( ( A1 = A5 )
& ( A22 = B4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R2 ) )
| ? [A5: A,B4: A,C6: A] :
( ( A1 = A5 )
& ( A22 = C6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ ( transitive_trancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C6 ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_1306_trancl_Or__into__trancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_1307_tranclE,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ~ ! [C5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C5 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C5 @ B3 ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_1308_trancl__trans,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_1309_nonempty__has__size,axiom,
! [A: $tType,S: multiset @ A] :
( ( S
!= ( zero_zero @ ( multiset @ A ) ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( multiset @ A ) @ S ) ) ) ).
% nonempty_has_size
thf(fact_1310_bijective__Empty,axiom,
! [B: $tType,A: $tType] : ( bijective @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% bijective_Empty
thf(fact_1311_total__on__singleton,axiom,
! [A: $tType,X3: A] : ( total_on @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_1312_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X3: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_1313_Field__insert,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) )
= ( sup_sup @ ( set @ A ) @ ( insert3 @ A @ A4 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) ) ) ).
% Field_insert
thf(fact_1314_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_1315_times__assn__raw_Opelims_I3_J,axiom,
! [X3: ( 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 @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ 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 ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) )
=> ? [As13: set @ nat,As24: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As24 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
thf(fact_1316_times__assn__raw_Opelims_I2_J,axiom,
! [X3: ( 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 @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ 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 ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) )
=> ~ ? [As12: set @ nat,As23: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As23 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
thf(fact_1317_list_Oinject,axiom,
! [A: $tType,X21: A,X222: list @ A,Y21: A,Y222: list @ A] :
( ( ( cons @ A @ X21 @ X222 )
= ( cons @ A @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_1318_Field__Un,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) )
= ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R2 ) @ ( field2 @ A @ S3 ) ) ) ).
% Field_Un
thf(fact_1319_list__ex__simps_I1_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( list_ex @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( ( P @ X3 )
| ( list_ex @ A @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_1320_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_1321_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X222 ) )
= ( insert3 @ A @ X21 @ ( set2 @ A @ X222 ) ) ) ).
% list.simps(15)
thf(fact_1322_nth__Cons__0,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ ( zero_zero @ nat ) )
= X3 ) ).
% nth_Cons_0
thf(fact_1323_nth__Cons__Suc,axiom,
! [A: $tType,X3: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ ( suc @ N ) )
= ( nth @ A @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1324_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_1325_Field__empty,axiom,
! [A: $tType] :
( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_1326_not__in__set__insert,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X3 @ Xs )
= ( cons @ A @ X3 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1327_Cons__listrel1__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X3 = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1328_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X3: B,Xs: list @ B] :
( ( enumerate @ B @ N @ ( cons @ B @ X3 @ Xs ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X3 ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1329_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F5: nat > A > A,A6: nat,B5: nat,Acc: A] :
( X3
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F5 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B5 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1330_distinct__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( distinct @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ).
% distinct_enumerate
thf(fact_1331_not__Cons__self2,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_1332_list__tail__coinc,axiom,
! [A: $tType,N1: A,R12: list @ A,N22: A,R23: list @ A] :
( ( ( cons @ A @ N1 @ R12 )
= ( cons @ A @ N22 @ R23 ) )
=> ( ( N1 = N22 )
& ( R12 = R23 ) ) ) ).
% list_tail_coinc
thf(fact_1333_union__diff__assoc,axiom,
! [A: $tType,C3: multiset @ A,B2: multiset @ A,A3: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C3 @ B2 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ C3 )
= ( plus_plus @ ( multiset @ A ) @ A3 @ ( minus_minus @ ( multiset @ A ) @ B2 @ C3 ) ) ) ) ).
% union_diff_assoc
thf(fact_1334_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 ) )
=> ~ ! [X: A,Xs5: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs5 ) )
=> ! [Y2: B,Ys4: list @ B] :
( ( Ys
= ( cons @ B @ Y2 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair @ A @ B @ X @ Y2 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs5 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1335_FieldI1,axiom,
! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( member @ A @ I @ ( field2 @ A @ R ) ) ) ).
% FieldI1
thf(fact_1336_FieldI2,axiom,
! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( member @ A @ J @ ( field2 @ A @ R ) ) ) ).
% FieldI2
thf(fact_1337_set__ConsD,axiom,
! [A: $tType,Y: A,X3: A,Xs: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) )
=> ( ( Y = X3 )
| ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1338_list_Oset__cases,axiom,
! [A: $tType,E3: A,A4: list @ A] :
( ( member @ A @ E3 @ ( set2 @ A @ A4 ) )
=> ( ! [Z22: list @ A] :
( A4
!= ( cons @ A @ E3 @ Z22 ) )
=> ~ ! [Z1: A,Z22: list @ A] :
( ( A4
= ( cons @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E3 @ ( set2 @ A @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1339_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1340_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y: A,X222: list @ A,X21: A] :
( ( member @ A @ Y @ ( set2 @ A @ X222 ) )
=> ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1341_distinct__length__2__or__more,axiom,
! [A: $tType,A4: A,B3: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ A4 @ ( cons @ A @ B3 @ Xs ) ) )
= ( ( A4 != B3 )
& ( distinct @ A @ ( cons @ A @ A4 @ Xs ) )
& ( distinct @ A @ ( cons @ A @ B3 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_1342_foldl__Cons,axiom,
! [B: $tType,A: $tType,F3: B > A > B,A4: B,X3: A,Xs: list @ A] :
( ( foldl @ B @ A @ F3 @ A4 @ ( cons @ A @ X3 @ Xs ) )
= ( foldl @ B @ A @ F3 @ ( F3 @ A4 @ X3 ) @ Xs ) ) ).
% foldl_Cons
thf(fact_1343_remove1_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A] :
( ( ( X3 = Y )
=> ( ( remove1 @ A @ X3 @ ( cons @ A @ Y @ Xs ) )
= Xs ) )
& ( ( X3 != Y )
=> ( ( remove1 @ A @ X3 @ ( cons @ A @ Y @ Xs ) )
= ( cons @ A @ Y @ ( remove1 @ A @ X3 @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_1344_total__onI,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ! [X: A,Y2: A] :
( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R2 ) ) ) ) )
=> ( total_on @ A @ A3 @ R2 ) ) ).
% total_onI
thf(fact_1345_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ A7 )
=> ( ( X4 != Y5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_1346_total__on__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% total_on_empty
thf(fact_1347_removeAll_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A] :
( ( ( X3 = Y )
=> ( ( removeAll @ A @ X3 @ ( cons @ A @ Y @ Xs ) )
= ( removeAll @ A @ X3 @ Xs ) ) )
& ( ( X3 != Y )
=> ( ( removeAll @ A @ X3 @ ( cons @ A @ Y @ Xs ) )
= ( cons @ A @ Y @ ( removeAll @ A @ X3 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_1348_zipf_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B > C,A4: A,As2: list @ A,B3: B,Bs: list @ B] :
( ( zipf @ A @ B @ C @ F3 @ ( cons @ A @ A4 @ As2 ) @ ( cons @ B @ B3 @ Bs ) )
= ( cons @ C @ ( F3 @ A4 @ B3 ) @ ( zipf @ A @ B @ C @ F3 @ As2 @ Bs ) ) ) ).
% zipf.simps(2)
thf(fact_1349_null__rec_I1_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
~ ( null @ A @ ( cons @ A @ X3 @ Xs ) ) ).
% null_rec(1)
thf(fact_1350_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y5: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1351_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y5: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1352_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1353_mono__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ R2 ) @ ( field2 @ A @ S3 ) ) ) ).
% mono_Field
thf(fact_1354_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X3: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1355_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Zs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs ) ) )
= ( ( ord_less_eq @ A @ X3 @ Y )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted2
thf(fact_1356_distinct_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X3 @ Xs ) )
= ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1357_list__update__code_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A] :
( ( list_update @ A @ ( cons @ A @ X3 @ Xs ) @ ( zero_zero @ nat ) @ Y )
= ( cons @ A @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1358_list__update__code_I3_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,I: nat,Y: A] :
( ( list_update @ A @ ( cons @ A @ X3 @ Xs ) @ ( suc @ I ) @ Y )
= ( cons @ A @ X3 @ ( list_update @ A @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1359_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X3: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ X3 @ Ys ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I2
thf(fact_1360_merge_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X1: A,X22: A,L12: list @ A,L23: list @ A] :
( ( ( ord_less @ A @ X1 @ X22 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X22 @ L23 ) )
= ( cons @ A @ X1 @ ( merge @ A @ L12 @ ( cons @ A @ X22 @ L23 ) ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X22 )
=> ( ( ( X1 = X22 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X22 @ L23 ) )
= ( cons @ A @ X1 @ ( merge @ A @ L12 @ L23 ) ) ) )
& ( ( X1 != X22 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X22 @ L23 ) )
= ( cons @ A @ X22 @ ( merge @ A @ ( cons @ A @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ) ) ).
% merge.simps(3)
thf(fact_1361_list__assn_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,P: A > C > assn,A4: A,As2: list @ A,C2: C,Cs: list @ C] :
( ( vEBT_List_list_assn @ A @ C @ P @ ( cons @ A @ A4 @ As2 ) @ ( cons @ C @ C2 @ Cs ) )
= ( times_times @ assn @ ( P @ A4 @ C2 ) @ ( vEBT_List_list_assn @ A @ C @ P @ As2 @ Cs ) ) ) ).
% list_assn.simps(2)
thf(fact_1362_list__assn__simps_I2_J,axiom,
! [A: $tType,B: $tType,P: A > B > assn,A4: A,As2: list @ A,C2: B,Cs: list @ B] :
( ( vEBT_List_list_assn @ A @ B @ P @ ( cons @ A @ A4 @ As2 ) @ ( cons @ B @ C2 @ Cs ) )
= ( times_times @ assn @ ( P @ A4 @ C2 ) @ ( vEBT_List_list_assn @ A @ B @ P @ As2 @ Cs ) ) ) ).
% list_assn_simps(2)
thf(fact_1363_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X4: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( member @ A @ X4 @ ( set2 @ A @ Xs3 ) ) @ Xs3 @ ( cons @ A @ X4 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_1364_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X4: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X4 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1365_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X3 @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_1366_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X3 @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ord_less @ A @ X3 @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_1367_bijective__def,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( ! [X4: A,Y5: B,Z6: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z6 ) @ R4 ) )
=> ( Y5 = Z6 ) )
& ! [X4: A,Y5: A,Z6: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z6 ) @ R4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y5 @ Z6 ) @ R4 ) )
=> ( X4 = Y5 ) ) ) ) ) ).
% bijective_def
thf(fact_1368_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [X: A] :
( ( Xs
= ( cons @ A @ X @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) )
=> ~ ! [Zs2: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1369_Cons__listrel1E1,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [Y2: A] :
( ( Ys
= ( cons @ A @ Y2 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 ) )
=> ~ ! [Zs2: list @ A] :
( ( Ys
= ( cons @ A @ X3 @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1370_listrel1I1,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I1
thf(fact_1371_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_1372_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys ) @ Xs ) @ ( listrel @ A @ B @ R2 ) )
=> ~ ! [Y2: B,Ys5: list @ B] :
( ( Xs
= ( cons @ B @ Y2 @ Ys5 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y2 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys5 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_1373_listrel_OCons,axiom,
! [B: $tType,A: $tType,X3: A,Y: B,R2: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_1374_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_1375_nth__Cons_H,axiom,
! [A: $tType,N: nat,X3: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N )
= X3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_1376_nth__equal__first__eq,axiom,
! [A: $tType,X3: A,Xs: list @ A,N: nat] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N )
= X3 )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1377_nth__non__equal__first__eq,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A,N: nat] :
( ( X3 != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X3 @ 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_1378_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X3: A,Xs: list @ A] :
( ( ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( 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 @ X3 @ Xs ) )
= ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% slice_Cons
thf(fact_1379_times__assn__raw_Opelims_I1_J,axiom,
! [X3: ( 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 @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ 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 ) ) )
& ( X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As22 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(1)
thf(fact_1380_length__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_Cons
thf(fact_1381_Cons__lenlex__iff,axiom,
! [A: $tType,M2: A,Ms: list @ A,N: A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M2 @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R2 ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1382_Cons__in__lex,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X3 = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_1383_list__assn_Oelims,axiom,
! [C: $tType,A: $tType,X3: A > C > assn,Xa: list @ A,Xb: list @ C,Y: assn] :
( ( ( vEBT_List_list_assn @ A @ C @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ C ) )
=> ( Y
!= ( one_one @ assn ) ) ) )
=> ( ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [C5: C,Cs2: list @ C] :
( ( Xb
= ( cons @ C @ C5 @ Cs2 ) )
=> ( Y
!= ( times_times @ assn @ ( X3 @ A6 @ C5 ) @ ( vEBT_List_list_assn @ A @ C @ X3 @ As @ Cs2 ) ) ) ) )
=> ( ( ? [V3: A,Va: list @ A] :
( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xb
= ( nil @ C ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V3: C,Va: list @ C] :
( Xb
= ( cons @ C @ V3 @ Va ) )
=> ( Y
!= ( bot_bot @ assn ) ) ) ) ) ) ) ) ).
% list_assn.elims
thf(fact_1384_count__list_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A] :
( ( ( X3 = Y )
=> ( ( count_list @ A @ ( cons @ A @ X3 @ Xs ) @ Y )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) )
& ( ( X3 != Y )
=> ( ( count_list @ A @ ( cons @ A @ X3 @ Xs ) @ Y )
= ( count_list @ A @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_1385_count__notin,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X3 )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_1386_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ M2 ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less_eq @ nat @ N @ M2 ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_1387_in__measures_I2_J,axiom,
! [A: $tType,X3: A,Y: A,F3: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) )
= ( ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X3 )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_1388_nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X3 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1389_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_1390_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_1391_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_1392_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_1393_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_1394_list__update__nonempty,axiom,
! [A: $tType,Xs: list @ A,K3: nat,X3: A] :
( ( ( list_update @ A @ Xs @ K3 @ X3 )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% list_update_nonempty
thf(fact_1395_enumerate__simps_I1_J,axiom,
! [A: $tType,N: nat] :
( ( enumerate @ A @ N @ ( nil @ A ) )
= ( nil @ ( product_prod @ nat @ A ) ) ) ).
% enumerate_simps(1)
thf(fact_1396_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_1397_list__ex__simps_I2_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex @ A @ P @ ( nil @ A ) ) ).
% list_ex_simps(2)
thf(fact_1398_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] :
( ( slice @ A @ Begin @ End @ ( nil @ A ) )
= ( nil @ A ) ) ).
% slice_Nil
thf(fact_1399_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_1400_in__measures_I1_J,axiom,
! [A: $tType,X3: A,Y: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_1401_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ( power_power @ A @ A4 @ M2 )
= ( power_power @ A @ A4 @ N ) )
= ( M2 = N ) ) ) ) ).
% power_inject_exp
thf(fact_1402_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_1403_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_1404_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_1405_singleton__rev__conv,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( ( cons @ A @ X3 @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( ( cons @ A @ X3 @ ( nil @ A ) )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_1406_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_1407_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R2 ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_1408_insert__Nil,axiom,
! [A: $tType,X3: A] :
( ( insert @ A @ X3 @ ( nil @ A ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_1409_list__assn__aux__simps_I1_J,axiom,
! [A: $tType,B: $tType,P: A > B > assn,L2: list @ B] :
( ( vEBT_List_list_assn @ A @ B @ P @ ( nil @ A ) @ L2 )
= ( pure_assn
@ ( L2
= ( nil @ B ) ) ) ) ).
% list_assn_aux_simps(1)
thf(fact_1410_list__assn__aux__simps_I2_J,axiom,
! [B: $tType,A: $tType,P: A > B > assn,L: list @ A] :
( ( vEBT_List_list_assn @ A @ B @ P @ L @ ( nil @ B ) )
= ( pure_assn
@ ( L
= ( nil @ A ) ) ) ) ).
% list_assn_aux_simps(2)
thf(fact_1411_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,X3: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ X3 ) @ ( power_power @ A @ B3 @ Y ) )
= ( ord_less @ nat @ X3 @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_1412_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A4: A,N: nat] :
( ( ( power_power @ A @ A4 @ N )
= ( zero_zero @ A ) )
= ( ( A4
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_1413_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_1414_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ B3 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ M2 ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1415_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,B3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A4 @ N ) @ ( power_power @ A @ B3 @ N ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_1416_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: A,X3: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ X3 ) @ ( power_power @ A @ B3 @ Y ) )
= ( ord_less_eq @ nat @ X3 @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_1417_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_1418_transpose_Ocases,axiom,
! [A: $tType,X3: list @ ( list @ A )] :
( ( X3
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1419_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_1420_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_1421_Nil__notin__lex,axiom,
! [A: $tType,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) @ ( lex @ A @ R2 ) ) ).
% Nil_notin_lex
thf(fact_1422_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_1423_mset__distrib,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,M5: multiset @ A,N7: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A3 @ B2 )
= ( plus_plus @ ( multiset @ A ) @ M5 @ N7 ) )
=> ~ ! [Am: multiset @ A,An: multiset @ A] :
( ( A3
= ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
=> ! [Bm: multiset @ A,Bn: multiset @ A] :
( ( B2
= ( plus_plus @ ( multiset @ A ) @ Bm @ Bn ) )
=> ( ( M5
= ( plus_plus @ ( multiset @ A ) @ Am @ Bm ) )
=> ( N7
!= ( plus_plus @ ( multiset @ A ) @ An @ Bn ) ) ) ) ) ) ).
% mset_distrib
thf(fact_1424_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_1425_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_1426_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A4 @ N ) @ A4 )
= ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) ) ) ).
% power_commutes
thf(fact_1427_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A4 @ B3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A4 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_1428_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X3: A,Y: A,N: nat] :
( ( ( times_times @ A @ X3 @ Y )
= ( times_times @ A @ Y @ X3 ) )
=> ( ( times_times @ A @ ( power_power @ A @ X3 @ N ) @ Y )
= ( times_times @ A @ Y @ ( power_power @ A @ X3 @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_1429_successively_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P6: A > A > $o] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P6 @ ( nil @ A ) ) )
=> ( ! [P6: A > A > $o,X: A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P6 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ~ ! [P6: A > A > $o,X: A,Y2: A,Xs2: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P6 @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1430_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F5: A > B,X: A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F5 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ( ! [F5: A > B,X: A,Y2: A,Zs2: list @ A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F5 @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Zs2 ) ) ) )
=> ~ ! [A6: A > B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A6 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_1431_sorted__wrt_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P6: A > A > $o] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P6 @ ( nil @ A ) ) )
=> ~ ! [P6: A > A > $o,X: A,Ys5: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P6 @ ( cons @ A @ X @ Ys5 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1432_neq__NilE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [X: A,Xs2: list @ A] :
( L
!= ( cons @ A @ X @ Xs2 ) ) ) ).
% neq_NilE
thf(fact_1433_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W22: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [E2: A,W12: list @ A,W23: list @ B] :
( ( P @ W12 @ W23 )
=> ( P @ ( cons @ A @ E2 @ W12 ) @ W23 ) )
=> ( ! [E2: B,W13: list @ A,W24: list @ B] :
( ( P @ W13 @ W24 )
=> ( P @ W13 @ ( cons @ B @ E2 @ W24 ) ) )
=> ( P @ W1 @ W22 ) ) ) ) ).
% list_2pre_induct
thf(fact_1434_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X12: A,X23: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_1435_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 )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys5: list @ B] :
( ( R @ X @ Y2 )
=> ( ( P @ Xs2 @ ( cons @ B @ Y2 @ Ys5 ) )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys5 ) ) ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys5: list @ B] :
( ~ ( R @ X @ Y2 )
=> ( ( P @ ( cons @ A @ X @ Xs2 ) @ Ys5 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys5 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_1436_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1437_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
( ( List
= ( cons @ A @ X21 @ X222 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_1438_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X223: list @ A] :
( Y
!= ( cons @ A @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1439_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A] :
( ! [X: A,Xs2: list @ A] :
( X3
!= ( cons @ A @ X @ Xs2 ) )
=> ( X3
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_1440_remdups__adj_Ocases,axiom,
! [A: $tType,X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ( ! [X: A] :
( X3
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( X3
!= ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1441_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y5: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y5 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1442_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 ) )
=> ( ! [X: A,Xs2: list @ A] : ( P @ ( cons @ A @ X @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys5: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys5 ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys5: list @ B] :
( ( P @ Xs2 @ Ys5 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys5 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1443_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1444_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_1445_sorted__wrt_Osimps_I1_J,axiom,
! [A: $tType,P: A > A > $o] : ( sorted_wrt @ A @ P @ ( nil @ A ) ) ).
% sorted_wrt.simps(1)
thf(fact_1446_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_1447_list__update_Osimps_I1_J,axiom,
! [A: $tType,I: nat,V2: A] :
( ( list_update @ A @ ( nil @ A ) @ I @ V2 )
= ( nil @ A ) ) ).
% list_update.simps(1)
thf(fact_1448_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_1449_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_1450_foldl__Nil,axiom,
! [A: $tType,B: $tType,F3: B > A > B,A4: B] :
( ( foldl @ B @ A @ F3 @ A4 @ ( nil @ A ) )
= A4 ) ).
% foldl_Nil
thf(fact_1451_remove1_Osimps_I1_J,axiom,
! [A: $tType,X3: A] :
( ( remove1 @ A @ X3 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remove1.simps(1)
thf(fact_1452_removeAll_Osimps_I1_J,axiom,
! [A: $tType,X3: A] :
( ( removeAll @ A @ X3 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% removeAll.simps(1)
thf(fact_1453_merge_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L23: list @ A] :
( ( merge @ A @ ( nil @ A ) @ L23 )
= L23 ) ) ).
% merge.simps(1)
thf(fact_1454_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_1455_zipf_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B > C] :
( ( zipf @ A @ B @ C @ F3 @ ( nil @ A ) @ ( nil @ B ) )
= ( nil @ C ) ) ).
% zipf.simps(1)
thf(fact_1456_null__rec_I2_J,axiom,
! [B: $tType] : ( null @ B @ ( nil @ B ) ) ).
% null_rec(2)
thf(fact_1457_eq__Nil__null,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
= ( nil @ A ) )
= ( null @ A @ Xs ) ) ).
% eq_Nil_null
thf(fact_1458_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A4 @ N ) ) ) ) ).
% zero_less_power
thf(fact_1459_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X3: A,Y: A,N: nat] :
( ( ( times_times @ A @ X3 @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ Y @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_1460_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A4: A,N: nat] :
( ( power_power @ A @ A4 @ ( suc @ N ) )
= ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) ) ) ).
% power_Suc
thf(fact_1461_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,N: nat] :
( ( power_power @ A @ A4 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A4 @ N ) @ A4 ) ) ) ).
% power_Suc2
thf(fact_1462_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( power_power @ A @ A4 @ ( plus_plus @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ A4 @ M2 ) @ ( power_power @ A @ A4 @ N ) ) ) ) ).
% power_add
thf(fact_1463_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M2 ) @ ( power_power @ nat @ I @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1464_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 ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys5: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys5 ) )
=> ( ( P @ Xs2 @ Ys5 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys5 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1465_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 ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys5: list @ B,Z3: C,Zs2: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys5 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys5 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys5 ) @ ( cons @ C @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1466_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,Ws: 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 ) @ Ws ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys5: list @ B,Z3: C,Zs2: list @ C,W: D,Ws2: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys5 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys5 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys5 @ Zs2 @ Ws2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys5 ) @ ( cons @ C @ Z3 @ Zs2 ) @ ( cons @ D @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1467_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_1468_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_1469_len__greater__imp__nonempty,axiom,
! [A: $tType,X3: nat,L: list @ A] :
( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ L ) )
=> ( L
!= ( nil @ A ) ) ) ).
% len_greater_imp_nonempty
thf(fact_1470_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X3: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F5: A > B > C] :
( X3
!= ( 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,B5: B,Bs2: list @ B] :
( X3
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ ( cons @ B @ B5 @ Bs2 ) ) ) )
=> ( ! [A6: A > B > C,V3: A,Va: list @ A] :
( X3
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [A6: A > B > C,V3: B,Va: list @ B] :
( X3
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ).
% zipf.cases
thf(fact_1471_merge_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [L22: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L22 ) )
=> ( ! [V3: A,Va: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ A ) ) )
=> ~ ! [X12: A,L1: list @ A,X23: A,L22: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ L1 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ).
% merge.cases
thf(fact_1472_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X3: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [P6: A > B > $o] :
( X3
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [P6: A > B > $o,A6: A,As: list @ A,B5: B,Bs2: list @ B] :
( X3
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ ( cons @ B @ B5 @ Bs2 ) ) ) )
=> ( ! [P6: A > B > $o,V3: A,Va: list @ A] :
( X3
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [P6: A > B > $o,V3: B,Va: list @ B] :
( X3
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ).
% list_all_zip.cases
thf(fact_1473_partition__rev_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
( ! [P6: A > $o,Yes: list @ A,No: list @ A] :
( X3
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P6 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) )
=> ~ ! [P6: A > $o,Yes: list @ A,No: list @ A,X: A,Xs2: list @ A] :
( X3
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P6 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X @ Xs2 ) ) ) ) ) ).
% partition_rev.cases
thf(fact_1474_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R6: A > A > $o,Sl: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R6 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ~ ! [R6: A > A > $o,Sl: list @ A,X: A,Xs2: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R6 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X @ Xs2 ) ) ) ) ) ).
% quicksort_by_rel.cases
thf(fact_1475_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R6: A > A > $o,X: A,Xs2: list @ A,Y2: A,Ys5: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R6 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys5 ) ) ) )
=> ( ! [R6: A > A > $o,Xs2: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R6 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) ) )
=> ~ ! [R6: A > A > $o,V3: A,Va: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R6 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V3 @ Va ) ) ) ) ) ) ).
% mergesort_by_rel_merge.cases
thf(fact_1476_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
( ! [Xs1: list @ A,Xs22: list @ A] :
( X3
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( nil @ A ) ) )
=> ( ! [Xs1: list @ A,Xs22: list @ A,X: A] :
( X3
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ~ ! [Xs1: list @ A,Xs22: list @ A,X12: A,X23: A,Xs2: list @ A] :
( X3
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs2 ) ) ) ) ) ) ).
% mergesort_by_rel_split.cases
thf(fact_1477_list__assn_Ocases,axiom,
! [A: $tType,C: $tType,X3: product_prod @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) )] :
( ! [P6: A > C > assn] :
( X3
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ P6 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( nil @ C ) ) ) )
=> ( ! [P6: A > C > assn,A6: A,As: list @ A,C5: C,Cs2: list @ C] :
( X3
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ P6 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ A6 @ As ) @ ( cons @ C @ C5 @ Cs2 ) ) ) )
=> ( ! [Uu2: A > C > assn,V3: A,Va: list @ A] :
( X3
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ Uu2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ C ) ) ) )
=> ~ ! [Uu2: A > C > assn,V3: C,Va: list @ C] :
( X3
!= ( product_Pair @ ( A > C > assn ) @ ( product_prod @ ( list @ A ) @ ( list @ C ) ) @ Uu2 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( cons @ C @ V3 @ Va ) ) ) ) ) ) ) ).
% list_assn.cases
thf(fact_1478_shuffles_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys5: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys5 ) )
=> ( ! [Xs2: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ~ ! [X: A,Xs2: list @ A,Y2: A,Ys5: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys5 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1479_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X3: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F5: A > B,Bs2: list @ B] :
( X3
!= ( 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] :
( X3
!= ( 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_1480_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys5: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys5 ) )
=> ~ ! [X: A,Xs2: list @ A,Ys5: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Ys5 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_1481_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_1482_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_1483_distinct__singleton,axiom,
! [A: $tType,X3: A] : ( distinct @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_1484_sorted__wrt1,axiom,
! [A: $tType,P: A > A > $o,X3: A] : ( sorted_wrt @ A @ P @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% sorted_wrt1
thf(fact_1485_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_1486_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_1487_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_1488_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) )
=> ( Xs
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_1489_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R2 ) )
=> ( Xs
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_1490_listrel_ONil,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) ) ).
% listrel.Nil
thf(fact_1491_merge_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A,Va2: list @ A] :
( ( merge @ A @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ A ) )
= ( cons @ A @ V2 @ Va2 ) ) ) ).
% merge.simps(2)
thf(fact_1492_list__assn__simps_I1_J,axiom,
! [B: $tType,A: $tType,P: A > B > assn] :
( ( vEBT_List_list_assn @ A @ B @ P @ ( nil @ A ) @ ( nil @ B ) )
= ( one_one @ assn ) ) ).
% list_assn_simps(1)
thf(fact_1493_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_1494_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat,B3: A] :
( ( ord_less @ A @ ( power_power @ A @ A4 @ N ) @ ( power_power @ A @ B3 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_1495_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1496_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ N ) @ ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1497_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_1498_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N7: nat,A4: A] :
( ( ord_less @ nat @ N @ N7 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ N ) @ ( power_power @ A @ A4 @ N7 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_1499_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ A @ ( power_power @ A @ A4 @ M2 ) @ ( power_power @ A @ A4 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_1500_zero__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ).
% zero_power
thf(fact_1501_power__gt__expt,axiom,
! [N: nat,K3: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ K3 @ ( power_power @ nat @ N @ K3 ) ) ) ).
% power_gt_expt
thf(fact_1502_lenlex__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_1503_length__compl__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [E2: A,L3: list @ A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L3 ) )
=> ( P @ Ll ) )
=> ( P @ ( cons @ A @ E2 @ L3 ) ) )
=> ( P @ L ) ) ) ).
% length_compl_induct
thf(fact_1504_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_1505_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_1506_butlast__subset,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( butlast @ A @ Xs ) ) @ A3 ) ) ) ).
% butlast_subset
thf(fact_1507_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_1508_merge_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( merge @ A @ X3 @ Xa )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ( ! [V3: A,Va: list @ A] :
( ( X3
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V3 @ Va ) ) ) )
=> ~ ! [X12: A,L1: list @ A] :
( ( X3
= ( cons @ A @ X12 @ L1 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ~ ( ( ( ord_less @ A @ X12 @ X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L1 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( ( X12 = X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L1 @ L22 ) ) ) )
& ( ( X12 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge.elims
thf(fact_1509_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_1510_list__assn__simps_I3_J,axiom,
! [B: $tType,A: $tType,P: A > B > assn,A4: A,As2: list @ A] :
( ( vEBT_List_list_assn @ A @ B @ P @ ( cons @ A @ A4 @ As2 ) @ ( nil @ B ) )
= ( bot_bot @ assn ) ) ).
% list_assn_simps(3)
thf(fact_1511_list__assn__simps_I4_J,axiom,
! [B: $tType,A: $tType,P: A > B > assn,C2: B,Cs: list @ B] :
( ( vEBT_List_list_assn @ A @ B @ P @ ( nil @ A ) @ ( cons @ B @ C2 @ Cs ) )
= ( bot_bot @ assn ) ) ).
% list_assn_simps(4)
thf(fact_1512_list__assn_Osimps_I3_J,axiom,
! [C: $tType,A: $tType,Uu: A > C > assn,V2: A,Va2: list @ A] :
( ( vEBT_List_list_assn @ A @ C @ Uu @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ C ) )
= ( bot_bot @ assn ) ) ).
% list_assn.simps(3)
thf(fact_1513_list__assn_Osimps_I4_J,axiom,
! [C: $tType,A: $tType,Uu: A > C > assn,V2: C,Va2: list @ C] :
( ( vEBT_List_list_assn @ A @ C @ Uu @ ( nil @ A ) @ ( cons @ C @ V2 @ Va2 ) )
= ( bot_bot @ assn ) ) ).
% list_assn.simps(4)
thf(fact_1514_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1515_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_1516_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N7: nat,A4: A] :
( ( ord_less @ nat @ N @ N7 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ N7 ) @ ( power_power @ A @ A4 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1517_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A4: A,B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ( power_power @ A @ A4 @ N )
= ( power_power @ A @ B3 @ N ) )
= ( A4 = B3 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1518_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat,B3: A] :
( ( ( power_power @ A @ A4 @ N )
= ( power_power @ A @ B3 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A4 = B3 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1519_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A4 @ M2 ) @ ( power_power @ A @ A4 @ N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_1520_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A4 @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_1521_the__elem__set,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= X3 ) ).
% the_elem_set
thf(fact_1522_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A4 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_1523_measures__less,axiom,
! [A: $tType,F3: A > nat,X3: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ).
% measures_less
thf(fact_1524_measures__lesseq,axiom,
! [A: $tType,F3: A > nat,X3: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ Fs ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_1525_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X3 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_1526_lenlex__length,axiom,
! [A: $tType,Ms: list @ A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% lenlex_length
thf(fact_1527_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X4: A,Y5: B,Xs3: list @ A,Ys3: list @ B] :
( ( A1
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y5 @ Ys3 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys3 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_1528_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R2 ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X: A,Y2: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys5: list @ B] :
( ( A22
= ( cons @ B @ Y2 @ Ys5 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys5 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1529_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: A,B3: A,N: nat] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_1530_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P7: A,M: nat] :
( if @ A
@ ( M
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P7 @ ( power_power @ A @ P7 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1531_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A4 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A4 )
= ( power_power @ A @ A4 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1532_subset__mset_Osum__list__update,axiom,
! [A: $tType,K3: nat,Xs: list @ ( multiset @ A ),X3: multiset @ A] :
( ( ord_less @ nat @ K3 @ ( size_size @ ( list @ ( multiset @ A ) ) @ Xs ) )
=> ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( list_update @ ( multiset @ A ) @ Xs @ K3 @ X3 ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ X3 ) @ ( nth @ ( multiset @ A ) @ Xs @ K3 ) ) ) ) ).
% subset_mset.sum_list_update
thf(fact_1533_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_1534_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) )
=> ~ ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I5 @ Xs )
= ( take @ A @ I5 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Ys @ I5 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_1535_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X3 ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1536_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_1537_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_1538_refl__on__singleton,axiom,
! [A: $tType,X3: A] : ( refl_on @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_1539_greaterThan__Suc,axiom,
! [K3: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K3 ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K3 ) @ ( insert3 @ nat @ ( suc @ K3 ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_1540_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K3: A] :
( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K3 ) )
= ( ord_less @ A @ K3 @ I ) ) ) ).
% greaterThan_iff
thf(fact_1541_take__update,axiom,
! [A: $tType,N: nat,L: list @ A,I: nat,X3: A] :
( ( take @ A @ N @ ( list_update @ A @ L @ I @ X3 ) )
= ( list_update @ A @ ( take @ A @ N @ L ) @ I @ X3 ) ) ).
% take_update
thf(fact_1542_take__update__swap,axiom,
! [A: $tType,M2: nat,Xs: list @ A,N: nat,X3: A] :
( ( take @ A @ M2 @ ( list_update @ A @ Xs @ N @ X3 ) )
= ( list_update @ A @ ( take @ A @ M2 @ Xs ) @ N @ X3 ) ) ).
% take_update_swap
thf(fact_1543_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_1544_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ).
% of_nat_less_iff
thf(fact_1545_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_1546_take__Suc__Cons,axiom,
! [A: $tType,N: nat,X3: A,Xs: list @ A] :
( ( take @ A @ ( suc @ N ) @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( take @ A @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_1547_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs3: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_1548_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_1549_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_1550_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_1551_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_1552_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_1553_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_1554_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_1555_subset__mset_Osum__list__eq__0__iff,axiom,
! [A: $tType,Ns: list @ ( multiset @ A )] :
( ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Ns )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ ( set2 @ ( multiset @ A ) @ Ns ) )
=> ( X4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.sum_list_eq_0_iff
thf(fact_1556_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% of_nat_0_less_iff
thf(fact_1557_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B3: nat,W2: nat,X3: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less @ nat @ ( power_power @ nat @ B3 @ W2 ) @ X3 ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1558_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,B3: nat,W2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B3 ) @ W2 ) )
= ( ord_less @ nat @ X3 @ ( power_power @ nat @ B3 @ W2 ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1559_merge__list_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) )] :
( ( X3
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [L3: list @ A] :
( X3
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( X3
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A ),L3: list @ A] :
( X3
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ~ ! [Acc2: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( X3
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc2 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) ) ) ) ) ) ) ).
% merge_list.cases
thf(fact_1560_mergesort__by__rel_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] :
~ ! [R6: A > A > $o,Xs2: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R6 @ Xs2 ) ) ).
% mergesort_by_rel.cases
thf(fact_1561_take__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ! [I5: nat] :
( ( take @ A @ I5 @ Xs )
= ( take @ A @ I5 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_1562_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_1563_take__Nil,axiom,
! [A: $tType,N: nat] :
( ( take @ A @ N @ ( nil @ A ) )
= ( nil @ A ) ) ).
% take_Nil
thf(fact_1564_in__set__takeD,axiom,
! [A: $tType,X3: A,N: nat,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1565_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X3: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X3 ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X3 ) ) ) ) ).
% mult_of_nat_commute
thf(fact_1566_distinct__take,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( take @ A @ I @ Xs ) ) ) ).
% distinct_take
thf(fact_1567_sorted__wrt__take,axiom,
! [A: $tType,F3: A > A > $o,Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ F3 @ Xs )
=> ( sorted_wrt @ A @ F3 @ ( take @ A @ N @ Xs ) ) ) ).
% sorted_wrt_take
thf(fact_1568_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1569_refl__onD,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_1570_refl__onD1,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( member @ A @ X3 @ A3 ) ) ) ).
% refl_onD1
thf(fact_1571_refl__onD2,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( member @ A @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_1572_refl__on__Int,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),B2: set @ A,S3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( refl_on @ A @ B2 @ S3 )
=> ( refl_on @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ).
% refl_on_Int
thf(fact_1573_greaterThan__non__empty,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
( ( set_ord_greaterThan @ A @ X3 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% greaterThan_non_empty
thf(fact_1574_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_1575_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_1576_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_1577_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_1578_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_1579_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% of_nat_less_imp_less
thf(fact_1580_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N: nat,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1581_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_1582_refl__on__Un,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),B2: set @ A,S3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( refl_on @ A @ B2 @ S3 )
=> ( refl_on @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ).
% refl_on_Un
thf(fact_1583_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_1584_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_1585_of__nat__diff,axiom,
! [A: $tType] :
( ( semiring_1_cancel @ A )
=> ! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% of_nat_diff
thf(fact_1586_zip__obtain__same__length,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ ( product_prod @ A @ B ) ) > $o] :
( ! [Zs2: list @ A,Ws2: list @ B,N5: nat] :
( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ B ) @ Ws2 ) )
=> ( ( N5
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( Zs2
= ( take @ A @ N5 @ Xs ) )
=> ( ( Ws2
= ( take @ B @ N5 @ Ys ) )
=> ( P @ ( zip @ A @ B @ Zs2 @ Ws2 ) ) ) ) ) )
=> ( P @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_obtain_same_length
thf(fact_1587_zdiff__int__split,axiom,
! [P: int > $o,X3: nat,Y: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X3 @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X3 )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X3 ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X3 @ Y )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_1588_nth__take__lemma,axiom,
! [A: $tType,K3: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ K3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K3 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ K3 )
=> ( ( nth @ A @ Xs @ I5 )
= ( nth @ A @ Ys @ I5 ) ) )
=> ( ( take @ A @ K3 @ Xs )
= ( take @ A @ K3 @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1589_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_1590_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_1591_take__Cons_H,axiom,
! [A: $tType,N: nat,X3: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X3 @ Xs ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_1592_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_1593_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ? [N5: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N5 ) @ X3 ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1594_length__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N @ Xs ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_1595_subset__mset_Oelem__le__sum__list,axiom,
! [A: $tType,K3: nat,Ns: list @ ( multiset @ A )] :
( ( ord_less @ nat @ K3 @ ( size_size @ ( list @ ( multiset @ A ) ) @ Ns ) )
=> ( subseteq_mset @ A @ ( nth @ ( multiset @ A ) @ Ns @ K3 ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Ns ) ) ) ).
% subset_mset.elem_le_sum_list
thf(fact_1596_distinct__n__lists,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) ) ).
% distinct_n_lists
thf(fact_1597_lexord__take__index__conv,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X3 ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X3 ) @ Y )
= X3 ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X3 ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I4 @ X3 )
= ( take @ A @ I4 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X3 @ I4 ) @ ( nth @ A @ Y @ I4 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_1598_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_1599_refl__on__domain,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( refl_on @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ A @ A4 @ A3 )
& ( member @ A @ B3 @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_1600_take__update__last,axiom,
! [A: $tType,N: nat,List: list @ A,X3: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ List ) )
=> ( ( list_update @ A @ ( take @ A @ ( suc @ N ) @ List ) @ N @ X3 )
= ( append @ A @ ( take @ A @ N @ List ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% take_update_last
thf(fact_1601_append_Oassoc,axiom,
! [A: $tType,A4: list @ A,B3: list @ A,C2: list @ A] :
( ( append @ A @ ( append @ A @ A4 @ B3 ) @ C2 )
= ( append @ A @ A4 @ ( append @ A @ B3 @ C2 ) ) ) ).
% append.assoc
thf(fact_1602_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_1603_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_1604_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_1605_empty__append__eq__id,axiom,
! [A: $tType] :
( ( append @ A @ ( nil @ A ) )
= ( ^ [X4: list @ A] : X4 ) ) ).
% empty_append_eq_id
thf(fact_1606_append_Oright__neutral,axiom,
! [A: $tType,A4: list @ A] :
( ( append @ A @ A4 @ ( nil @ A ) )
= A4 ) ).
% append.right_neutral
thf(fact_1607_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_1608_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_1609_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_1610_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_1611_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_1612_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_1613_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_1614_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_1615_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_1616_foldl__append,axiom,
! [A: $tType,B: $tType,F3: A > B > A,A4: A,Xs: list @ B,Ys: list @ B] :
( ( foldl @ A @ B @ F3 @ A4 @ ( append @ B @ Xs @ Ys ) )
= ( foldl @ A @ B @ F3 @ ( foldl @ A @ B @ F3 @ A4 @ Xs ) @ Ys ) ) ).
% foldl_append
thf(fact_1617_removeAll__append,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] :
( ( removeAll @ A @ X3 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( removeAll @ A @ X3 @ Xs ) @ ( removeAll @ A @ X3 @ Ys ) ) ) ).
% removeAll_append
thf(fact_1618_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_1619_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_1620_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E22: A,L12: list @ A,E12: A,E23: A,L23: list @ A] :
( ( ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) )
= ( append @ A @ L12 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L23 ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_1621_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L12: list @ A,E12: A,E23: A,L23: list @ A,E1: A,E22: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L23 ) ) )
= ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_1622_list__se__match_I1_J,axiom,
! [A: $tType,L12: list @ A,L23: list @ A,A4: A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L23 )
= ( cons @ A @ A4 @ ( nil @ A ) ) )
= ( ( L12
= ( cons @ A @ A4 @ ( nil @ A ) ) )
& ( L23
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_1623_list__se__match_I2_J,axiom,
! [A: $tType,L23: list @ A,L12: list @ A,A4: A] :
( ( L23
!= ( nil @ A ) )
=> ( ( ( append @ A @ L12 @ L23 )
= ( cons @ A @ A4 @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( L23
= ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_1624_list__se__match_I3_J,axiom,
! [A: $tType,L12: list @ A,A4: A,L23: list @ A] :
( ( L12
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A4 @ ( nil @ A ) )
= ( append @ A @ L12 @ L23 ) )
= ( ( L12
= ( cons @ A @ A4 @ ( nil @ A ) ) )
& ( L23
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_1625_list__se__match_I4_J,axiom,
! [A: $tType,L23: list @ A,A4: A,L12: list @ A] :
( ( L23
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A4 @ ( nil @ A ) )
= ( append @ A @ L12 @ L23 ) )
= ( ( L12
= ( nil @ A ) )
& ( L23
= ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_1626_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E3: A,L12: list @ A,E4: A,L23: list @ A] :
( ( ( cons @ A @ E3 @ ( nil @ A ) )
= ( append @ A @ L12 @ ( cons @ A @ E4 @ L23 ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E4 = E3 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_1627_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L12: list @ A,E4: A,L23: list @ A,E3: A] :
( ( ( append @ A @ L12 @ ( cons @ A @ E4 @ L23 ) )
= ( cons @ A @ E3 @ ( nil @ A ) ) )
= ( ( L12
= ( nil @ A ) )
& ( E4 = E3 )
& ( L23
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_1628_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_1629_zip__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Us: list @ B,Ys: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Us ) )
=> ( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ ( append @ B @ Us @ Vs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Us ) @ ( zip @ A @ B @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_1630_List_Olast__in__set,axiom,
! [A: $tType,As2: list @ A] :
( ( As2
!= ( nil @ A ) )
=> ( member @ A @ ( last @ A @ As2 ) @ ( set2 @ A @ As2 ) ) ) ).
% List.last_in_set
thf(fact_1631_Misc_Olast__in__set,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( member @ A @ ( last @ A @ L ) @ ( set2 @ A @ L ) ) ) ).
% Misc.last_in_set
thf(fact_1632_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_1633_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_1634_nth__append__length,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_1635_nth__append__first,axiom,
! [A: $tType,I: nat,L: list @ A,L2: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( append @ A @ L @ L2 ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_append_first
thf(fact_1636_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_1637_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_1638_list__update__length,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A,Y: A] :
( ( list_update @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) @ Y )
= ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1639_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_1640_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= X3 ) ).
% last_snoc
thf(fact_1641_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_1642_lexord__cons__cons,axiom,
! [A: $tType,A4: A,X3: list @ A,B3: A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A4 @ X3 ) @ ( cons @ A @ B3 @ Y ) ) @ ( lexord @ A @ R2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
| ( ( A4 = B3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_1643_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R2 ) )
= ( ? [A5: A,X4: list @ A] :
( Y
= ( cons @ A @ A5 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_1644_list__assn__aux__append2,axiom,
! [A: $tType,B: $tType,L23: list @ A,L24: list @ B,P: A > B > assn,L12: list @ A,L13: list @ B] :
( ( ( size_size @ ( list @ A ) @ L23 )
= ( size_size @ ( list @ B ) @ L24 ) )
=> ( ( vEBT_List_list_assn @ A @ B @ P @ ( append @ A @ L12 @ L23 ) @ ( append @ B @ L13 @ L24 ) )
= ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ P @ L12 @ L13 ) @ ( vEBT_List_list_assn @ A @ B @ P @ L23 @ L24 ) ) ) ) ).
% list_assn_aux_append2
thf(fact_1645_list__assn__aux__append,axiom,
! [A: $tType,B: $tType,L12: list @ A,L13: list @ B,P: A > B > assn,L23: list @ A,L24: list @ B] :
( ( ( size_size @ ( list @ A ) @ L12 )
= ( size_size @ ( list @ B ) @ L13 ) )
=> ( ( vEBT_List_list_assn @ A @ B @ P @ ( append @ A @ L12 @ L23 ) @ ( append @ B @ L13 @ L24 ) )
= ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ P @ L12 @ L13 ) @ ( vEBT_List_list_assn @ A @ B @ P @ L23 @ L24 ) ) ) ) ).
% list_assn_aux_append
thf(fact_1646_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_1647_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_1648_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V2: list @ A,R2: set @ ( product_prod @ A @ A ),X3: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X3 @ U ) @ ( append @ A @ X3 @ V2 ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_leftI
thf(fact_1649_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_1650_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs5: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Xs5 @ Ss ) )
& ( Ys
= ( append @ A @ Ys4 @ Ss ) )
& ( ( Xs5
= ( nil @ A ) )
| ( Ys4
= ( nil @ A ) )
| ( ( last @ A @ Xs5 )
!= ( last @ A @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_1651_imp__le__cong,axiom,
! [X3: int,X7: int,P: $o,P3: $o] :
( ( X3 = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_1652_conj__le__cong,axiom,
! [X3: int,X7: int,P: $o,P3: $o] :
( ( X3 = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_1653_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs12: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs12 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs12 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_1654_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 ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_1655_incr__mult__lemma,axiom,
! [D3: int,P: int > $o,K3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X: int] :
( ( P @ X )
=> ( P @ ( plus_plus @ int @ X @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K3 )
=> ! [X8: int] :
( ( P @ X8 )
=> ( P @ ( plus_plus @ int @ X8 @ ( times_times @ int @ K3 @ D3 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1656_decr__mult__lemma,axiom,
! [D3: int,P: int > $o,K3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X: int] :
( ( P @ X )
=> ( P @ ( minus_minus @ int @ X @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K3 )
=> ! [X8: int] :
( ( P @ X8 )
=> ( P @ ( minus_minus @ int @ X8 @ ( times_times @ int @ K3 @ D3 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1657_minusinfinity,axiom,
! [D3: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X: int,K: int] :
( ( P1 @ X )
= ( P1 @ ( minus_minus @ int @ X @ ( times_times @ int @ K @ D3 ) ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less @ int @ X @ Z5 )
=> ( ( P @ X )
= ( P1 @ X ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1658_plusinfinity,axiom,
! [D3: int,P3: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X: int,K: int] :
( ( P3 @ X )
= ( P3 @ ( minus_minus @ int @ X @ ( times_times @ int @ K @ D3 ) ) ) )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less @ int @ Z5 @ X )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [X_12: int] : ( P3 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1659_lexord__append__leftD,axiom,
! [A: $tType,X3: list @ A,U: list @ A,V2: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X3 @ U ) @ ( append @ A @ X3 @ V2 ) ) @ ( lexord @ A @ R2 ) )
=> ( ! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_append_leftD
thf(fact_1660_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X3: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ? [B10: A,Z5: list @ A] :
( Y
= ( cons @ A @ B10 @ Z5 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ ( append @ A @ X3 @ Y ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_rightI
thf(fact_1661_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs: list @ A,Ys: list @ A,Qs: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R2 ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R2 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_1662_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_1663_append__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X3 @ Xs ) @ Ys )
= ( cons @ A @ X3 @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_1664_Cons__eq__appendI,axiom,
! [A: $tType,X3: A,Xs12: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X3 @ Xs12 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs12 @ Zs ) )
=> ( ( cons @ A @ X3 @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_1665_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_1666_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1667_append_Oleft__neutral,axiom,
! [A: $tType,A4: list @ A] :
( ( append @ A @ ( nil @ A ) @ A4 )
= A4 ) ).
% append.left_neutral
thf(fact_1668_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_1669_mset__le__addE,axiom,
! [A: $tType,Xs: multiset @ A,Ys: multiset @ A] :
( ( subseteq_mset @ A @ Xs @ Ys )
=> ~ ! [Zs2: multiset @ A] :
( Ys
!= ( plus_plus @ ( multiset @ A ) @ Xs @ Zs2 ) ) ) ).
% mset_le_addE
thf(fact_1670_mset__le__distrib,axiom,
! [A: $tType,X5: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ X5 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) )
=> ~ ! [Xa2: multiset @ A,Xb2: multiset @ A] :
( ( X5
= ( plus_plus @ ( multiset @ A ) @ Xa2 @ Xb2 ) )
=> ( ( subseteq_mset @ A @ Xa2 @ A3 )
=> ~ ( subseteq_mset @ A @ Xb2 @ B2 ) ) ) ) ).
% mset_le_distrib
thf(fact_1671_mset__union__subset,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C3: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ C3 )
=> ( ( subseteq_mset @ A @ A3 @ C3 )
& ( subseteq_mset @ A @ B2 @ C3 ) ) ) ).
% mset_union_subset
thf(fact_1672_mset__le__decr__left1,axiom,
! [A: $tType,A4: multiset @ A,C2: multiset @ A,B3: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C2 ) @ B3 )
=> ( subseteq_mset @ A @ A4 @ B3 ) ) ).
% mset_le_decr_left1
thf(fact_1673_mset__le__decr__left2,axiom,
! [A: $tType,C2: multiset @ A,A4: multiset @ A,B3: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A4 ) @ B3 )
=> ( subseteq_mset @ A @ A4 @ B3 ) ) ).
% mset_le_decr_left2
thf(fact_1674_mset__le__incr__right1,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A4 @ B3 )
=> ( subseteq_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ B3 @ C2 ) ) ) ).
% mset_le_incr_right1
thf(fact_1675_mset__le__incr__right2,axiom,
! [A: $tType,A4: multiset @ A,B3: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A4 @ B3 )
=> ( subseteq_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ C2 @ B3 ) ) ) ).
% mset_le_incr_right2
thf(fact_1676_mset__le__subtract,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C3: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A3 @ C3 ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ C3 ) ) ) ).
% mset_le_subtract
thf(fact_1677_lexord__append__left__rightI,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),U: list @ A,X3: list @ A,Y: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A4 @ X3 ) ) @ ( append @ A @ U @ ( cons @ A @ B3 @ Y ) ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_left_rightI
thf(fact_1678_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= Ys )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X3 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_1679_snoc__eq__iff__butlast_H,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,X3: A] :
( ( Ys
= ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X3 ) ) ) ).
% snoc_eq_iff_butlast'
thf(fact_1680_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R2 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R2 ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_1681_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W2: list @ A,R2: set @ ( product_prod @ A @ A ),V2: list @ A,Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W2 ) @ ( lexord @ A @ R2 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V2 ) @ ( append @ A @ W2 @ Z2 ) ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_sufI
thf(fact_1682_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_1683_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1684_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X3: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X3 @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X3 @ Xs ) ) )
| ? [Ys6: list @ A] :
( ( Ys
= ( cons @ A @ X3 @ Ys6 ) )
& ( ( append @ A @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1685_Cons__eq__append__conv,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X3 @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X3 @ Xs )
= Zs ) )
| ? [Ys6: list @ A] :
( ( ( cons @ A @ X3 @ Ys6 )
= Ys )
& ( Xs
= ( append @ A @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1686_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys5: list @ A,Y2: A] :
( Xs
!= ( append @ A @ Ys5 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_1687_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_1688_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X3: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X3 @ Xs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X3 @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X3 @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_1689_list__Cons__eq__append__cases,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X3 @ Xs )
= ( append @ A @ Ys @ Zs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X3 @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X3 @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_1690_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 ) )
=> ( ! [X: A,Y2: B] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) @ ( cons @ B @ Y2 @ ( nil @ B ) ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B] :
( ( Xs2
!= ( nil @ A ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) )
=> ( ! [X: A,Y2: B,Ys5: list @ B] :
( ( Ys5
!= ( nil @ B ) )
=> ( P @ ( cons @ A @ X @ ( nil @ A ) ) @ ( append @ B @ Ys5 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys5: list @ B] :
( ( P @ Xs2 @ Ys5 )
=> ( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys5
!= ( nil @ B ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ B @ Ys5 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_1691_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
= ( ? [Xs3: list @ A,X4: A] :
( L
= ( append @ A @ Xs3 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_1692_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 ) )
=> ( ! [X: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys5: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys5 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys5: list @ B] :
( ( P @ Xs2 @ Ys5 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ B @ Ys5 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_1693_neq__Nil__revE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [Ll2: list @ A,E2: A] :
( L
!= ( append @ A @ Ll2 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) ) ).
% neq_Nil_revE
thf(fact_1694_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_1695_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list @ A,X4: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y5: A] :
( ( member @ A @ Y5 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1696_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list @ A,X4: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y5: A] :
( ( member @ A @ Y5 @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1697_in__set__conv__decomp__first,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1698_in__set__conv__decomp__last,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1699_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
& ( P @ X8 ) )
=> ~ ! [Ys5: list @ A,X: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys5 @ ( cons @ A @ X @ Zs2 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys5 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1700_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
& ( P @ X8 ) )
=> ~ ! [Ys5: list @ A,X: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys5 @ ( cons @ A @ X @ Zs2 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1701_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
& ( P @ X8 ) )
=> ? [Ys5: list @ A,X: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys5 @ ( cons @ A @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys5 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_1702_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
& ( P @ X8 ) )
=> ? [Ys5: list @ A,X: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys5 @ ( cons @ A @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_1703_in__set__conv__decomp,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1704_append__Cons__eq__iff,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A,Xs4: list @ A,Ys7: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ~ ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ Ys ) )
= ( append @ A @ Xs4 @ ( cons @ A @ X3 @ Ys7 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1705_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
& ( P @ X8 ) )
=> ~ ! [Ys5: list @ A,X: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys5 @ ( cons @ A @ X @ Zs2 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_1706_split__list__first,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ? [Ys5: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys5 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Ys5 ) ) ) ) ).
% split_list_first
thf(fact_1707_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
& ( P @ X8 ) )
=> ? [Ys5: list @ A,X: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys5 @ ( cons @ A @ X @ Zs2 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_1708_split__list__last,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ? [Ys5: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys5 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ~ ( member @ A @ X3 @ ( set2 @ A @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1709_split__list,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ? [Ys5: list @ A,Zs2: list @ A] :
( Xs
= ( append @ A @ Ys5 @ ( cons @ A @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1710_in__set__list__format,axiom,
! [A: $tType,E3: A,L: list @ A] :
( ( member @ 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_1711_xy__in__set__cases,axiom,
! [A: $tType,X3: A,L: list @ A,Y: A] :
( ( member @ A @ X3 @ ( set2 @ A @ L ) )
=> ( ( member @ A @ Y @ ( set2 @ A @ L ) )
=> ( ( ( X3 = Y )
=> ! [L1: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ Y @ L22 ) ) ) )
=> ( ( ( X3 != Y )
=> ! [L1: list @ A,L22: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ X3 @ ( append @ A @ L22 @ ( cons @ A @ Y @ L32 ) ) ) ) ) )
=> ~ ( ( X3 != Y )
=> ! [L1: list @ A,L22: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L1 @ ( cons @ A @ Y @ ( append @ A @ L22 @ ( cons @ A @ X3 @ L32 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_1712_list__rest__coinc,axiom,
! [A: $tType,S22: list @ A,S1: list @ A,R12: list @ A,R23: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ S22 ) @ ( size_size @ ( list @ A ) @ S1 ) )
=> ( ( ( append @ A @ S1 @ R12 )
= ( append @ A @ S22 @ R23 ) )
=> ? [R1p: list @ A] :
( R23
= ( append @ A @ R1p @ R12 ) ) ) ) ).
% list_rest_coinc
thf(fact_1713_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_1714_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_1715_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 )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ ( set2 @ A @ Ys ) )
=> ( P @ X4 @ Y5 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1716_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_1717_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= X3 ) ) ).
% last_ConsL
thf(fact_1718_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= X3 ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_1719_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_1720_mset__le__subtract__right,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,X5: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ X5 )
=> ( ( subseteq_mset @ A @ A3 @ ( minus_minus @ ( multiset @ A ) @ X5 @ B2 ) )
& ( subseteq_mset @ A @ B2 @ X5 ) ) ) ).
% mset_le_subtract_right
thf(fact_1721_mset__le__subtract__left,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,X5: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ X5 )
=> ( ( subseteq_mset @ A @ B2 @ ( minus_minus @ ( multiset @ A ) @ X5 @ A3 ) )
& ( subseteq_mset @ A @ A3 @ X5 ) ) ) ).
% mset_le_subtract_left
thf(fact_1722_in__set__butlast__appendI,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X3 @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
| ( member @ A @ X3 @ ( set2 @ A @ ( butlast @ A @ Ys ) ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ ( butlast @ A @ ( append @ A @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1723_remove1__append,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X3 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remove1 @ A @ X3 @ Xs ) @ Ys ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X3 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( remove1 @ A @ X3 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1724_append__listrel1I,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R2 ) ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% append_listrel1I
thf(fact_1725_lex__append__leftI,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R2 ) ) ) ).
% lex_append_leftI
thf(fact_1726_lexord__linear,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: list @ A,Y: list @ A] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R2 )
| ( A6 = B5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) )
| ( X3 = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X3 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_linear
thf(fact_1727_lexord__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_irreflexive
thf(fact_1728_lexord__Nil__right,axiom,
! [A: $tType,X3: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) @ ( lexord @ A @ R2 ) ) ).
% lexord_Nil_right
thf(fact_1729_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,X: A,Xs5: list @ A,Y2: A,Ys4: list @ A] :
( ( X != Y2 )
& ( Xs
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ Xs5 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1730_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 )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X4 @ Y5 ) ) ) ) ) ) ).
% sorted_append
thf(fact_1731_not__distinct__decomp,axiom,
! [A: $tType,Ws: list @ A] :
( ~ ( distinct @ A @ Ws )
=> ? [Xs2: list @ A,Ys5: list @ A,Zs2: list @ A,Y2: A] :
( Ws
= ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ ( append @ A @ Ys5 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_1732_not__distinct__conv__prefix,axiom,
! [A: $tType,As2: list @ A] :
( ( ~ ( distinct @ A @ As2 ) )
= ( ? [Xs3: list @ A,Y5: A,Ys3: list @ A] :
( ( member @ A @ Y5 @ ( set2 @ A @ Xs3 ) )
& ( distinct @ A @ Xs3 )
& ( As2
= ( append @ A @ Xs3 @ ( cons @ A @ Y5 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1733_rev_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( rev @ A @ ( cons @ A @ X3 @ Xs ) )
= ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rev.simps(2)
thf(fact_1734_list__update__append1,axiom,
! [A: $tType,I: nat,Xs: list @ A,Ys: list @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ I @ X3 )
= ( append @ A @ ( list_update @ A @ Xs @ I @ X3 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1735_butlast__eq__cons__conv,axiom,
! [A: $tType,L: list @ A,X3: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X3 @ Xs ) )
= ( ? [Xl: A] :
( L
= ( cons @ A @ X3 @ ( append @ A @ Xs @ ( cons @ A @ Xl @ ( nil @ A ) ) ) ) ) ) ) ).
% butlast_eq_cons_conv
thf(fact_1736_butlast__eq__consE,axiom,
! [A: $tType,L: list @ A,X3: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X3 @ Xs ) )
=> ~ ! [Xl2: A] :
( L
!= ( cons @ A @ X3 @ ( append @ A @ Xs @ ( cons @ A @ Xl2 @ ( nil @ A ) ) ) ) ) ) ).
% butlast_eq_consE
thf(fact_1737_foldl__rule,axiom,
! [Sigma2: $tType,A: $tType,I3: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2] :
( ( I3 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X @ L22 ) ) )
=> ( ( I3 @ Sigma @ L1 @ ( cons @ A @ X @ L22 ) )
=> ( I3 @ ( F3 @ Sigma @ X ) @ ( append @ A @ L1 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ L22 ) ) )
=> ( I3 @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) @ L0 @ ( nil @ A ) ) ) ) ).
% foldl_rule
thf(fact_1738_foldl__rule__P,axiom,
! [Sigma2: $tType,A: $tType,I3: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I3 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X @ L22 ) ) )
=> ( ( I3 @ Sigma @ L1 @ ( cons @ A @ X @ L22 ) )
=> ( I3 @ ( F3 @ Sigma @ X ) @ ( append @ A @ L1 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ L22 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I3 @ Sigma @ L0 @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_P
thf(fact_1739_foldl__rule__aux,axiom,
! [Sigma2: $tType,A: $tType,I3: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2] :
( ( I3 @ Sigma_0 @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X @ L22 ) ) )
=> ( ( I3 @ Sigma @ ( cons @ A @ X @ L22 ) )
=> ( I3 @ ( F3 @ Sigma @ X ) @ L22 ) ) )
=> ( I3 @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) @ ( nil @ A ) ) ) ) ).
% foldl_rule_aux
thf(fact_1740_foldl__rule__aux__P,axiom,
! [Sigma2: $tType,A: $tType,I3: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I3 @ Sigma_0 @ L0 )
=> ( ! [L1: list @ A,L22: list @ A,X: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L1 @ ( cons @ A @ X @ L22 ) ) )
=> ( ( I3 @ Sigma @ ( cons @ A @ X @ L22 ) )
=> ( I3 @ ( F3 @ Sigma @ X ) @ L22 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I3 @ Sigma @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_aux_P
thf(fact_1741_remove1__split,axiom,
! [A: $tType,A4: A,Xs: list @ A,Ys: list @ A] :
( ( member @ A @ A4 @ ( set2 @ A @ Xs ) )
=> ( ( ( remove1 @ A @ A4 @ Xs )
= Ys )
= ( ? [Ls2: list @ A,Rs: list @ A] :
( ( Xs
= ( append @ A @ Ls2 @ ( cons @ A @ A4 @ Rs ) ) )
& ~ ( member @ A @ A4 @ ( set2 @ A @ Ls2 ) )
& ( Ys
= ( append @ A @ Ls2 @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1742_lex__append__leftD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_1743_lex__append__left__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_1744_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X3 @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_1745_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_1746_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs @ Ys ) ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append1
thf(fact_1747_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs: list @ A] :
( ! [X: A,Y2: A,Z3: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R2 ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_1748_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_1749_subset__mset_Osum__list__nonneg,axiom,
! [A: $tType,Xs: list @ ( multiset @ A )] :
( ! [X: multiset @ A] :
( ( member @ ( multiset @ A ) @ X @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) ) ) ).
% subset_mset.sum_list_nonneg
thf(fact_1750_subset__mset_Osum__list__nonpos,axiom,
! [A: $tType,Xs: list @ ( multiset @ A )] :
( ! [X: multiset @ A] :
( ( member @ ( multiset @ A ) @ X @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.sum_list_nonpos
thf(fact_1751_subset__mset_Omember__le__sum__list,axiom,
! [A: $tType,X3: multiset @ A,Xs: list @ ( multiset @ A )] :
( ( member @ ( multiset @ A ) @ X3 @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ X3 @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) ) ) ).
% subset_mset.member_le_sum_list
thf(fact_1752_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
! [A: $tType,Xs: list @ ( multiset @ A )] :
( ! [X: multiset @ A] :
( ( member @ ( multiset @ A ) @ X @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X ) )
=> ( ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( X4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.sum_list_nonneg_eq_0_iff
thf(fact_1753_lexord__lex,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lex @ A @ R2 ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) )
& ( ( size_size @ ( list @ A ) @ X3 )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_1754_length__append__singleton,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1755_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y5: A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y5 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1756_length__Suc__rev__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Ys3: list @ A,Y5: A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y5 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_rev_conv
thf(fact_1757_length__compl__rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [L3: list @ A,E2: A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L3 ) )
=> ( P @ Ll ) )
=> ( P @ ( append @ A @ L3 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) )
=> ( P @ L ) ) ) ).
% length_compl_rev_induct
thf(fact_1758_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ~ ( distinct @ A @ Xs )
=> ~ ! [Y2: A,Ys5: list @ A] :
( ( distinct @ A @ Ys5 )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Ys5 ) )
=> ! [Zs2: list @ A] :
( Xs
!= ( append @ A @ Ys5 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_1759_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_1760_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A,X3: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X3 )
= ( append @ A @ ( list_update @ A @ Xs @ N @ X3 ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X3 )
= ( append @ A @ Xs @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X3 ) ) ) ) ) ).
% list_update_append
thf(fact_1761_listrel1I,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( ( Xs
= ( append @ A @ Us @ ( cons @ A @ X3 @ Vs ) ) )
=> ( ( Ys
= ( append @ A @ Us @ ( cons @ A @ Y @ Vs ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_1762_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ~ ! [X: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 )
=> ! [Us3: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs2 ) ) )
=> ( Ys
!= ( append @ A @ Us3 @ ( cons @ A @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1763_sorted__append__bigger,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ Y ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_append_bigger
thf(fact_1764_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys: list @ A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
& ( X3 = Y ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1765_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_1766_last__list__update,axiom,
! [A: $tType,Xs: list @ A,K3: nat,X3: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( K3
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K3 @ X3 ) )
= X3 ) )
& ( ( K3
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K3 @ X3 ) )
= ( last @ A @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1767_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [N5: nat] : ( ord_less @ A @ X3 @ ( semiring_1_of_nat @ A @ N5 ) ) ) ).
% reals_Archimedean2
thf(fact_1768_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_1769_pos__int__cases,axiom,
! [K3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K3 )
=> ~ ! [N5: nat] :
( ( K3
= ( semiring_1_of_nat @ int @ N5 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ).
% pos_int_cases
thf(fact_1770_zero__less__imp__eq__int,axiom,
! [K3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K3 )
=> ? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( K3
= ( semiring_1_of_nat @ int @ N5 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1771_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K3: nat] :
( ( ord_less @ int @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K3 ) @ I ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K3 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1772_distinct__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ A @ ( concat @ A @ Xs ) )
= ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs ) )
& ! [Ys3: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys3 ) )
& ! [Ys3: list @ A,Zs3: list @ A] :
( ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( member @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( Ys3 != Zs3 ) )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% distinct_concat_iff
thf(fact_1773_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F3: B > A,A4: A,Xs: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A4 @ ( append @ B @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A4 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A4 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A4 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_1774_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_1775_distinct__product__lists,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ! [X: list @ A] :
( ( member @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( distinct @ A @ X ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_1776_distinct__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A ) @ Xs )
=> ( ! [Ys5: list @ A] :
( ( member @ ( list @ A ) @ Ys5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys5 ) )
=> ( ! [Ys5: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys5 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys5 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1777_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X4: list @ A] : X4 ) ) ).
% drop0
thf(fact_1778_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_1779_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_1780_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_1781_drop__Suc__Cons,axiom,
! [A: $tType,N: nat,X3: A,Xs: list @ A] :
( ( drop @ A @ ( suc @ N ) @ ( cons @ A @ X3 @ Xs ) )
= ( drop @ A @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1782_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_1783_drop__upd__irrelevant,axiom,
! [A: $tType,M2: nat,N: nat,L: list @ A,X3: A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( drop @ A @ N @ ( list_update @ A @ L @ M2 @ X3 ) )
= ( drop @ A @ N @ L ) ) ) ).
% drop_upd_irrelevant
thf(fact_1784_drop__update__cancel,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A,X3: A] :
( ( ord_less @ nat @ N @ M2 )
=> ( ( drop @ A @ M2 @ ( list_update @ A @ Xs @ N @ X3 ) )
= ( drop @ A @ M2 @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_1785_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_1786_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( ( concat @ A @ Xss2 )
= ( nil @ A ) )
= ( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( X4
= ( nil @ A ) ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1787_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( ( nil @ A )
= ( concat @ A @ Xss2 ) )
= ( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( X4
= ( nil @ A ) ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1788_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_1789_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_1790_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_1791_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_1792_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F3: B > A,A4: A,X3: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A4 @ ( cons @ B @ X3 @ Xs ) )
= ( plus_plus @ A @ ( F3 @ X3 ) @ ( times_times @ A @ A4 @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A4 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_1793_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_1794_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_1795_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_1796_drop__Nil,axiom,
! [A: $tType,N: nat] :
( ( drop @ A @ N @ ( nil @ A ) )
= ( nil @ A ) ) ).
% drop_Nil
thf(fact_1797_drop__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% drop_0
thf(fact_1798_in__set__dropD,axiom,
! [A: $tType,X3: A,N: nat,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1799_distinct__drop,axiom,
! [A: $tType,Xs: list @ A,I: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( drop @ A @ I @ Xs ) ) ) ).
% distinct_drop
thf(fact_1800_sorted__wrt__drop,axiom,
! [A: $tType,F3: A > A > $o,Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ F3 @ Xs )
=> ( sorted_wrt @ A @ F3 @ ( drop @ A @ N @ Xs ) ) ) ).
% sorted_wrt_drop
thf(fact_1801_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_1802_foldl__foldl__conv__concat,axiom,
! [A: $tType,B: $tType,F3: A > B > A,A4: A,Xs: list @ ( list @ B )] :
( ( foldl @ A @ ( list @ B ) @ ( foldl @ A @ B @ F3 ) @ A4 @ Xs )
= ( foldl @ A @ B @ F3 @ A4 @ ( concat @ B @ Xs ) ) ) ).
% foldl_foldl_conv_concat
thf(fact_1803_drop__eq__ConsD,axiom,
! [A: $tType,N: nat,Xs: list @ A,X3: A,Xs4: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( cons @ A @ X3 @ Xs4 ) )
=> ( ( drop @ A @ ( suc @ N ) @ Xs )
= Xs4 ) ) ).
% drop_eq_ConsD
thf(fact_1804_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_1805_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_1806_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_1807_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_1808_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_1809_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_1810_concat_Osimps_I2_J,axiom,
! [A: $tType,X3: list @ A,Xs: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X3 @ Xs ) )
= ( append @ A @ X3 @ ( concat @ A @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_1811_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_1812_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_1813_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_1814_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_1815_drop__update__swap,axiom,
! [A: $tType,M2: nat,N: nat,Xs: list @ A,X3: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( drop @ A @ M2 @ ( list_update @ A @ Xs @ N @ X3 ) )
= ( list_update @ A @ ( drop @ A @ M2 @ Xs ) @ ( minus_minus @ nat @ N @ M2 ) @ X3 ) ) ) ).
% drop_update_swap
thf(fact_1816_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_1817_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 ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs2: list @ A,Xs5: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append @ A @ Xs5 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_1818_drop__Cons_H,axiom,
! [A: $tType,N: nat,X3: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ Xs ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X3 @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_1819_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_1820_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_1821_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_1822_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_1823_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_1824_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_1825_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_1826_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 ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs3: list @ A,Xs6: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append @ A @ Xs6 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_1827_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_1828_in__set__drop__conv__nth,axiom,
! [A: $tType,X3: A,N: nat,L: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( drop @ A @ N @ L ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq @ nat @ N @ I4 )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) )
& ( X3
= ( nth @ A @ L @ I4 ) ) ) ) ) ).
% in_set_drop_conv_nth
thf(fact_1829_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_1830_foldl__list__update,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F3: B > A > B,A4: B,X3: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( foldl @ B @ A @ F3 @ A4 @ ( list_update @ A @ Xs @ N @ X3 ) )
= ( foldl @ B @ A @ F3 @ ( F3 @ ( foldl @ B @ A @ F3 @ A4 @ ( take @ A @ N @ Xs ) ) @ X3 ) @ ( drop @ A @ ( suc @ N ) @ Xs ) ) ) ) ).
% foldl_list_update
thf(fact_1831_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_1832_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_1833_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A,A4: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I @ A4 )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A4 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1834_in__set__product__lists__length,axiom,
! [A: $tType,Xs: list @ A,Xss2: list @ ( list @ A )] :
( ( member @ ( 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_1835_merge__list__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Ls3: list @ ( list @ A ),As2: list @ ( list @ A )] :
( ! [L3: list @ A] :
( ( member @ ( list @ A ) @ L3 @ ( set2 @ ( list @ A ) @ Ls3 ) )
=> ( ( distinct @ A @ L3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L3 ) ) )
=> ( ! [L3: list @ A] :
( ( member @ ( list @ A ) @ L3 @ ( set2 @ ( list @ A ) @ As2 ) )
=> ( ( distinct @ A @ L3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L3 ) ) )
=> ( ( distinct @ A @ ( merge_list @ A @ As2 @ Ls3 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge_list @ A @ As2 @ Ls3 ) )
& ( ( set2 @ A @ ( merge_list @ A @ As2 @ Ls3 ) )
= ( set2 @ A @ ( concat @ A @ ( append @ ( list @ A ) @ As2 @ Ls3 ) ) ) ) ) ) ) ) ).
% merge_list_correct
thf(fact_1836_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_1837_eucl__rel__int__iff,axiom,
! [K3: int,L: int,Q4: int,R2: int] :
( ( eucl_rel_int @ K3 @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
= ( ( K3
= ( plus_plus @ int @ ( times_times @ int @ L @ Q4 ) @ R2 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
& ( ord_less @ int @ R2 @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R2 )
& ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q4
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_1838_linear__order__on__singleton,axiom,
! [A: $tType,X3: A] : ( order_679001287576687338der_on @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_1839_Total__subset__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) )
=> ( ( R2
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
| ? [A6: A] :
( R2
= ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_1840_nat0__intermed__int__val,axiom,
! [N: nat,F3: nat > int,K3: int] :
( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( plus_plus @ nat @ I5 @ ( one_one @ nat ) ) ) @ ( F3 @ I5 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K3 )
=> ( ( ord_less_eq @ int @ K3 @ ( F3 @ N ) )
=> ? [I5: nat] :
( ( ord_less_eq @ nat @ I5 @ N )
& ( ( F3 @ I5 )
= K3 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1841_revg__fun,axiom,
! [A: $tType] :
( ( revg @ A )
= ( ^ [A5: list @ A] : ( append @ A @ ( rev @ A @ A5 ) ) ) ) ).
% revg_fun
thf(fact_1842_abs__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A4 ) )
= ( abs_abs @ A @ A4 ) ) ) ).
% abs_abs
thf(fact_1843_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A4 ) )
= ( abs_abs @ A @ A4 ) ) ) ).
% abs_idempotent
thf(fact_1844_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_1845_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A4 ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_1846_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ( abs_abs @ A @ A4 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_1847_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_1848_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_add_abs
thf(fact_1849_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ A4 ) )
= ( times_times @ A @ A4 @ A4 ) ) ) ).
% abs_mult_self_eq
thf(fact_1850_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_1851_pair__in__Id__conv,axiom,
! [A: $tType,A4: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( id2 @ A ) )
= ( A4 = B3 ) ) ).
% pair_in_Id_conv
thf(fact_1852_IdI,axiom,
! [A: $tType,A4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_1853_total__on__diff__Id,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A3 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
= ( total_on @ A @ A3 @ R2 ) ) ).
% total_on_diff_Id
thf(fact_1854_bijective__Id,axiom,
! [A: $tType] : ( bijective @ A @ A @ ( id2 @ A ) ) ).
% bijective_Id
thf(fact_1855_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_1856_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ A4 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% abs_le_self_iff
thf(fact_1857_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( abs_abs @ A @ A4 )
= A4 ) ) ) ).
% abs_of_nonneg
thf(fact_1858_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A4 ) )
= ( A4
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_1859_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_1860_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_1861_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A4 ) @ N ) )
= ( ( A4
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_1862_unique__remainder,axiom,
! [A4: int,B3: int,Q4: int,R2: int,Q7: int,R7: int] :
( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q7 @ R7 ) )
=> ( R2 = R7 ) ) ) ).
% unique_remainder
thf(fact_1863_unique__quotient,axiom,
! [A4: int,B3: int,Q4: int,R2: int,Q7: int,R7: int] :
( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q7 @ R7 ) )
=> ( Q4 = Q7 ) ) ) ).
% unique_quotient
thf(fact_1864_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ B3 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% abs_le_D1
thf(fact_1865_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( abs_abs @ A @ A4 ) ) ) ).
% abs_ge_self
thf(fact_1866_abs__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( ( abs_abs @ A @ A4 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0_iff
thf(fact_1867_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A,B3: A] :
( ( abs_abs @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_mult
thf(fact_1868_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_1869_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A4 @ B3 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A4 ) ) ) ) ).
% abs_minus_commute
thf(fact_1870_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_1871_Linear__order__in__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_1872_BNF__Greatest__Fixpoint_OIdD,axiom,
! [A: $tType,A4: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( id2 @ A ) )
=> ( A4 = B3 ) ) ).
% BNF_Greatest_Fixpoint.IdD
thf(fact_1873_IdE,axiom,
! [A: $tType,P4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P4 @ ( id2 @ A ) )
=> ~ ! [X: A] :
( P4
!= ( product_Pair @ A @ A @ X @ X ) ) ) ).
% IdE
thf(fact_1874_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_1875_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_1876_eucl__rel__int__by0,axiom,
! [K3: int] : ( eucl_rel_int @ K3 @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) ).
% eucl_rel_int_by0
thf(fact_1877_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A4 ) ) ) ).
% abs_ge_zero
thf(fact_1878_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( abs_abs @ A @ A4 )
= A4 ) ) ) ).
% abs_of_pos
thf(fact_1879_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A4 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_1880_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A4 @ B3 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_1881_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ A4 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1882_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_1883_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_1884_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,C2: A,B3: A,D3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A4 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B3 ) @ D3 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) @ ( times_times @ A @ C2 @ D3 ) ) ) ) ) ).
% abs_mult_less
thf(fact_1885_hd__in__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ).
% hd_in_set
thf(fact_1886_list_Oset__sel_I1_J,axiom,
! [A: $tType,A4: list @ A] :
( ( A4
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ A4 ) @ ( set2 @ A @ A4 ) ) ) ).
% list.set_sel(1)
thf(fact_1887_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_1888_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ps2: list @ A,Xs5: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Ps2 @ Xs5 ) )
& ( Ys
= ( append @ A @ Ps2 @ Ys4 ) )
& ( ( Xs5
= ( nil @ A ) )
| ( Ys4
= ( nil @ A ) )
| ( ( hd @ A @ Xs5 )
!= ( hd @ A @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_1889_hd__Nil__eq__last,axiom,
! [A: $tType] :
( ( hd @ A @ ( nil @ A ) )
= ( last @ A @ ( nil @ A ) ) ) ).
% hd_Nil_eq_last
thf(fact_1890_last__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( last @ A @ ( rev @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% last_rev
thf(fact_1891_hd__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( rev @ A @ Xs ) )
= ( last @ A @ Xs ) ) ).
% hd_rev
thf(fact_1892_revg_Osimps_I2_J,axiom,
! [A: $tType,A4: A,As2: list @ A,B3: list @ A] :
( ( revg @ A @ ( cons @ A @ A4 @ As2 ) @ B3 )
= ( revg @ A @ As2 @ ( cons @ A @ A4 @ B3 ) ) ) ).
% revg.simps(2)
thf(fact_1893_revg_Osimps_I1_J,axiom,
! [A: $tType,B3: list @ A] :
( ( revg @ A @ ( nil @ A ) @ B3 )
= B3 ) ).
% revg.simps(1)
thf(fact_1894_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ E2 ) )
=> ( X3
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_1895_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_1896_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A4: A,B3: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
| ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
| ( ord_less_eq @ A @ B3 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_1897_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X3 )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X3 ) ) ) ) ) ).
% abs_mult_pos
thf(fact_1898_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,A4: A,R2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A4 ) ) @ R2 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ R2 ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( plus_plus @ A @ A4 @ R2 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1899_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A4 @ B3 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_1900_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( plus_plus @ A @ C2 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A4 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1901_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,A4: A,R2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ A4 ) ) @ R2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A4 @ R2 ) @ X3 )
& ( ord_less @ A @ X3 @ ( plus_plus @ A @ A4 @ R2 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1902_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_1903_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_1904_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_1905_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_1906_eucl__rel__int__dividesI,axiom,
! [L: int,K3: int,Q4: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K3
= ( times_times @ int @ Q4 @ L ) )
=> ( eucl_rel_int @ K3 @ L @ ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_1907_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_1908_merge__list_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc22: list @ ( list @ A ),L12: list @ A,L23: list @ A,Ls3: list @ ( list @ A )] :
( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls3 ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ Acc22 ) @ Ls3 ) ) ) ).
% merge_list.simps(5)
thf(fact_1909_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X3 ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_1910_sorted__hd__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ! [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( hd @ A @ Xs ) @ X8 ) ) ) ) ) ).
% sorted_hd_min
thf(fact_1911_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_1912_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_1913_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_1914_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_1915_revg_Oelims,axiom,
! [A: $tType,X3: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X3 @ Xa )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [A6: A,As: list @ A] :
( ( X3
= ( cons @ A @ A6 @ As ) )
=> ( Y
!= ( revg @ A @ As @ ( cons @ A @ A6 @ Xa ) ) ) ) ) ) ).
% revg.elims
thf(fact_1916_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F3: nat > int,K3: int] :
( ! [I5: nat] :
( ( ( ord_less_eq @ nat @ M2 @ I5 )
& ( ord_less @ nat @ I5 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I5 ) ) @ ( F3 @ I5 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ int @ ( F3 @ M2 ) @ K3 )
=> ( ( ord_less_eq @ int @ K3 @ ( F3 @ N ) )
=> ? [I5: nat] :
( ( ord_less_eq @ nat @ M2 @ I5 )
& ( ord_less_eq @ nat @ I5 @ N )
& ( ( F3 @ I5 )
= K3 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1917_incr__lemma,axiom,
! [D3: int,Z2: int,X3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X3 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X3 @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) ) ) ).
% incr_lemma
thf(fact_1918_decr__lemma,axiom,
! [D3: int,X3: int,Z2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ ( minus_minus @ int @ X3 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X3 @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_1919_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_1920_merge__list_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X3 @ Xa )
= Y )
=> ( ( ( X3
= ( nil @ ( list @ A ) ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ A ) ) ) )
=> ( ( ( X3
= ( nil @ ( list @ A ) ) )
=> ! [L3: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
=> ( Y != L3 ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( X3
= ( 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 )] :
( ( X3
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ! [L3: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
=> ( Y
!= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) ) ) ) )
=> ~ ! [L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) )
=> ( Y
!= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ X3 ) @ Ls ) ) ) ) ) ) ) ) ) ).
% merge_list.elims
thf(fact_1921_nat__ivt__aux,axiom,
! [N: nat,F3: nat > int,K3: int] :
( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I5 ) ) @ ( F3 @ I5 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K3 )
=> ( ( ord_less_eq @ int @ K3 @ ( F3 @ N ) )
=> ? [I5: nat] :
( ( ord_less_eq @ nat @ I5 @ N )
& ( ( F3 @ I5 )
= K3 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1922_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J2: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J2 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) @ ( cons @ int @ J2 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1923_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A13: int,A23: int,A32: product_prod @ int @ int] :
( ? [K4: int] :
( ( A13 = K4 )
& ( A23
= ( zero_zero @ int ) )
& ( A32
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K4 ) ) )
| ? [L4: int,K4: int,Q8: int] :
( ( A13 = K4 )
& ( A23 = L4 )
& ( A32
= ( product_Pair @ int @ int @ Q8 @ ( zero_zero @ int ) ) )
& ( L4
!= ( zero_zero @ int ) )
& ( K4
= ( times_times @ int @ Q8 @ L4 ) ) )
| ? [R5: int,L4: int,K4: int,Q8: int] :
( ( A13 = K4 )
& ( A23 = L4 )
& ( A32
= ( product_Pair @ int @ int @ Q8 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L4 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L4 ) )
& ( K4
= ( plus_plus @ int @ ( times_times @ int @ Q8 @ L4 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_1924_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A33: product_prod @ int @ int] :
( ( eucl_rel_int @ A1 @ A22 @ A33 )
=> ( ( ( A22
= ( zero_zero @ int ) )
=> ( A33
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q6: int] :
( ( A33
= ( product_Pair @ int @ int @ Q6 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q6 @ A22 ) ) ) )
=> ~ ! [R3: int,Q6: int] :
( ( A33
= ( product_Pair @ int @ int @ Q6 @ R3 ) )
=> ( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q6 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_1925_listrel1__subset__listrel,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R7 )
=> ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R7 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel @ A @ A @ R7 ) ) ) ) ).
% listrel1_subset_listrel
thf(fact_1926_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X3: B,Xs: list @ B,F3: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X3 @ Xs ) @ F3 )
= ( append @ A @ ( F3 @ X3 ) @ ( bind @ B @ A @ Xs @ F3 ) ) ) ).
% bind_simps(2)
thf(fact_1927_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K3: nat,Xs: list @ A,X3: A] :
( ( ord_less @ nat @ K3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K3 @ X3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X3 ) @ ( nth @ A @ Xs @ K3 ) ) ) ) ) ).
% sum_list_update
thf(fact_1928_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_1929_UNIV__I,axiom,
! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_1930_sgn__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A4 ) )
= ( sgn_sgn @ A @ A4 ) ) ) ).
% sgn_sgn
thf(fact_1931_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A4: A] :
( ( inf_inf @ A @ A4 @ ( top_top @ A ) )
= A4 ) ) ).
% inf_top.right_neutral
thf(fact_1932_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A4: A,B3: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ A4 @ B3 ) )
= ( ( A4
= ( top_top @ A ) )
& ( B3
= ( top_top @ A ) ) ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1933_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A4: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ A4 )
= A4 ) ) ).
% inf_top.left_neutral
thf(fact_1934_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A4: A,B3: A] :
( ( ( inf_inf @ A @ A4 @ B3 )
= ( top_top @ A ) )
= ( ( A4
= ( top_top @ A ) )
& ( B3
= ( top_top @ A ) ) ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1935_top__eq__inf__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X3: A,Y: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ X3 @ Y ) )
= ( ( X3
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% top_eq_inf_iff
thf(fact_1936_inf__eq__top__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X3: A,Y: A] :
( ( ( inf_inf @ A @ X3 @ Y )
= ( top_top @ A ) )
= ( ( X3
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% inf_eq_top_iff
thf(fact_1937_inf__top__right,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( top_top @ A ) )
= X3 ) ) ).
% inf_top_right
thf(fact_1938_inf__top__left,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ X3 )
= X3 ) ) ).
% inf_top_left
thf(fact_1939_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_1940_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X3 )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_1941_Int__UNIV,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( top_top @ ( set @ A ) ) )
= ( ( A3
= ( top_top @ ( set @ A ) ) )
& ( B2
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_1942_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_1943_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_min @ A @ X3 @ ( top_top @ A ) )
= X3 ) ) ).
% min_top2
thf(fact_1944_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X3 )
= X3 ) ) ).
% min_top
thf(fact_1945_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_1946_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F3: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F3 )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_1947_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_1948_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A4 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% sgn_greater
thf(fact_1949_Diff__UNIV,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_1950_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ns ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_1951_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( sgn_sgn @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_1952_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_1953_UNIV__witness,axiom,
! [A: $tType] :
? [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_1954_UNIV__eq__I,axiom,
! [A: $tType,A3: set @ A] :
( ! [X: A] : ( member @ A @ X @ A3 )
=> ( ( top_top @ ( set @ A ) )
= A3 ) ) ).
% UNIV_eq_I
thf(fact_1955_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
=> ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_1956_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
= ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_1957_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_1958_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A4 ) ) ).
% top.extremum_strict
thf(fact_1959_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( A4
!= ( top_top @ A ) )
= ( ord_less @ A @ A4 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_1960_sgn__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( ( sgn_sgn @ A @ A4 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% sgn_eq_0_iff
thf(fact_1961_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ( sgn_sgn @ A @ A4 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_1962_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: A,A4: A] :
( ( ( sgn_sgn @ A @ B3 )
= ( sgn_sgn @ A @ A4 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A4 @ B3 ) )
= ( sgn_sgn @ A @ A4 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_1963_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A,B3: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A4 ) @ ( sgn_sgn @ A @ B3 ) ) ) ) ).
% sgn_mult
thf(fact_1964_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_1965_subset__UNIV,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_1966_total__lexord,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lexord @ A @ R2 ) ) ) ).
% total_lexord
thf(fact_1967_insert__UNIV,axiom,
! [A: $tType,X3: A] :
( ( insert3 @ A @ X3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_1968_Int__UNIV__right,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= A3 ) ).
% Int_UNIV_right
thf(fact_1969_Int__UNIV__left,axiom,
! [A: $tType,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Int_UNIV_left
thf(fact_1970_Un__UNIV__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_1971_Un__UNIV__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_1972_total__lenlex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lenlex @ A @ R2 ) ) ) ).
% total_lenlex
thf(fact_1973_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% member_le_sum_list
thf(fact_1974_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: A,A4: A] :
( ( ( sgn_sgn @ A @ B3 )
= ( sgn_sgn @ A @ A4 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A4 @ B3 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A4 ) @ ( abs_abs @ A @ B3 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_1975_mult__sgn__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ X3 ) @ ( abs_abs @ A @ X3 ) )
= X3 ) ) ).
% mult_sgn_abs
thf(fact_1976_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A4 ) @ ( abs_abs @ A @ A4 ) )
= A4 ) ) ).
% sgn_mult_abs
thf(fact_1977_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( abs_abs @ A @ A4 ) @ ( sgn_sgn @ A @ A4 ) )
= A4 ) ) ).
% abs_mult_sgn
thf(fact_1978_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K4: A] : ( times_times @ A @ K4 @ ( sgn_sgn @ A @ K4 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_1979_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F3: A > ( list @ B ),G: A > ( list @ B )] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( F3 @ X )
= ( G @ X ) ) )
=> ( ( bind @ A @ B @ Xs @ F3 )
= ( bind @ A @ B @ Ys @ G ) ) ) ) ).
% list_bind_cong
thf(fact_1980_refl__Id,axiom,
! [A: $tType] : ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ ( id2 @ A ) ) ).
% refl_Id
thf(fact_1981_UNIV__coset,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( coset @ A @ ( nil @ A ) ) ) ).
% UNIV_coset
thf(fact_1982_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A4: A,X3: A,Y: A] :
( ( ( inf_inf @ A @ A4 @ X3 )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A4 @ X3 )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A4 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A4 @ Y )
= ( top_top @ A ) )
=> ( X3 = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1983_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_1984_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_1985_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_1986_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ( sgn_sgn @ A @ A4 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% sgn_1_pos
thf(fact_1987_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ( A4
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A4 ) )
= ( zero_zero @ A ) ) )
& ( ( A4
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_1988_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [K3: nat,Ns: list @ A] :
( ( ord_less @ nat @ K3 @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K3 ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_1989_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K3: int,Q4: int] :
( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L ) )
=> ( ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q4 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K3 @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_1990_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
= ( ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R2 ) )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ! [Y5: A] :
( ( member @ A @ Y5 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_1991_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_1992_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_1993_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M: nat] :
( if @ A
@ ( M
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_1994_list__assn_Opelims,axiom,
! [A: $tType,C: $tType,X3: A > C > assn,Xa: list @ A,Xb: list @ C,Y: assn] :
( ( ( vEBT_List_list_assn @ A @ C @ X3 @ 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 ) ) @ X3 @ ( 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( nil @ C ) ) ) ) ) ) )
=> ( ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [C5: C,Cs2: list @ C] :
( ( Xb
= ( cons @ C @ C5 @ Cs2 ) )
=> ( ( Y
= ( times_times @ assn @ ( X3 @ A6 @ C5 ) @ ( vEBT_List_list_assn @ A @ C @ X3 @ As @ Cs2 ) ) )
=> ~ ( 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ A6 @ As ) @ ( cons @ C @ C5 @ Cs2 ) ) ) ) ) ) )
=> ( ! [V3: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ C ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: C,Va: list @ C] :
( ( Xb
= ( cons @ C @ V3 @ Va ) )
=> ( ( 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ C ) @ ( nil @ A ) @ ( cons @ C @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% list_assn.pelims
thf(fact_1995_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_1996_int__power__div__base,axiom,
! [M2: nat,K3: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K3 )
=> ( ( divide_divide @ int @ ( power_power @ int @ K3 @ M2 ) @ K3 )
= ( power_power @ int @ K3 @ ( minus_minus @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1997_merge__true__star,axiom,
( ( times_times @ assn @ ( top_top @ assn ) @ ( top_top @ assn ) )
= ( top_top @ assn ) ) ).
% merge_true_star
thf(fact_1998_assn__basic__inequalities_I1_J,axiom,
( ( top_top @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(1)
thf(fact_1999_entt__true,axiom,
! [A3: assn] : ( entailst @ A3 @ ( top_top @ assn ) ) ).
% entt_true
thf(fact_2000_assn__basic__inequalities_I5_J,axiom,
( ( top_top @ assn )
!= ( bot_bot @ assn ) ) ).
% assn_basic_inequalities(5)
thf(fact_2001_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_2002_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ A4 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_2003_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% times_divide_eq_right
thf(fact_2004_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( divide_divide @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ).
% divide_divide_eq_right
thf(fact_2005_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 )
= ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_2006_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( times_times @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( divide_divide @ A @ ( times_times @ A @ B3 @ A4 ) @ C2 ) ) ) ).
% times_divide_eq_left
thf(fact_2007_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ A4 @ ( one_one @ A ) )
= A4 ) ) ).
% div_by_1
thf(fact_2008_entt__star__true__simp_I1_J,axiom,
! [A3: assn,B2: assn] :
( ( entailst @ A3 @ ( times_times @ assn @ B2 @ ( top_top @ assn ) ) )
= ( entailst @ A3 @ B2 ) ) ).
% entt_star_true_simp(1)
thf(fact_2009_entt__star__true__simp_I2_J,axiom,
! [A3: assn,B2: assn] :
( ( entailst @ ( times_times @ assn @ A3 @ ( top_top @ assn ) ) @ B2 )
= ( entailst @ A3 @ B2 ) ) ).
% entt_star_true_simp(2)
thf(fact_2010_wf__listrel1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ ( list @ A ) @ ( listrel1 @ A @ R2 ) )
= ( wf @ A @ R2 ) ) ).
% wf_listrel1_iff
thf(fact_2011_wf__lex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ).
% wf_lex
thf(fact_2012_wf__lenlex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lenlex @ A @ R2 ) ) ) ).
% wf_lenlex
thf(fact_2013_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ A4 )
= B3 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2014_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ B3 )
= A4 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2015_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2016_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2017_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2018_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2019_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2020_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A4 @ A4 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_2021_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( divide_divide @ A @ A4 @ ( sgn_sgn @ A @ B3 ) )
= ( times_times @ A @ A4 @ ( sgn_sgn @ A @ B3 ) ) ) ) ).
% divide_sgn
thf(fact_2022_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_2023_butlast__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( tl @ A @ Xs ) ) ) ).
% butlast_rev
thf(fact_2024_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_2025_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_2026_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_2027_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_2028_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_2029_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_2030_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B3 @ ( times_times @ A @ A4 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A4 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_2031_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ A4 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_2032_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_2033_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_2034_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_2035_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_2036_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list @ A,X3: A] :
( ( L
!= ( nil @ A ) )
=> ( ( ( X3
= ( hd @ A @ L ) )
| ( member @ A @ X3 @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
= ( member @ A @ X3 @ ( set2 @ A @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_2037_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_2038_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_2039_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_2040_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_2041_wf__lexn,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lexn @ A @ R2 @ N ) ) ) ).
% wf_lexn
thf(fact_2042_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semila1105856199041335345_order @ A @ F3 @ Z2 @ Less_eq @ Less )
=> ( ( ( F3 @ A4 @ B3 )
= Z2 )
= ( ( A4 = Z2 )
& ( B3 = Z2 ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_2043_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semila1105856199041335345_order @ A @ F3 @ Z2 @ Less_eq @ Less )
=> ( ( Z2
= ( F3 @ A4 @ B3 ) )
= ( ( A4 = Z2 )
& ( B3 = Z2 ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_2044_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_2045_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_2046_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_2047_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_2048_in__set__tlD,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( tl @ A @ Xs ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_tlD
thf(fact_2049_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,Y: A,Z2: A,W2: A] :
( ( times_times @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ Y @ W2 ) ) ) ) ).
% times_divide_times_eq
thf(fact_2050_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,Y: A,Z2: A,W2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X3 @ W2 ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_2051_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 )
= ( divide_divide @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_2052_diff__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A4 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ).
% diff_divide_distrib
thf(fact_2053_distinct__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( tl @ A @ Xs ) ) ) ).
% distinct_tl
thf(fact_2054_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_2055_ent__true,axiom,
! [P: assn] : ( entails @ P @ ( top_top @ assn ) ) ).
% ent_true
thf(fact_2056_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_2057_butlast__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% butlast_tl
thf(fact_2058_norm__assertion__simps_I12_J,axiom,
! [X3: assn] :
( ( sup_sup @ assn @ X3 @ ( top_top @ assn ) )
= ( top_top @ assn ) ) ).
% norm_assertion_simps(12)
thf(fact_2059_norm__assertion__simps_I11_J,axiom,
! [X3: assn] :
( ( sup_sup @ assn @ ( top_top @ assn ) @ X3 )
= ( top_top @ assn ) ) ).
% norm_assertion_simps(11)
thf(fact_2060_norm__assertion__simps_I4_J,axiom,
! [X3: assn] :
( ( inf_inf @ assn @ X3 @ ( top_top @ assn ) )
= X3 ) ).
% norm_assertion_simps(4)
thf(fact_2061_norm__assertion__simps_I3_J,axiom,
! [X3: assn] :
( ( inf_inf @ assn @ ( top_top @ assn ) @ X3 )
= X3 ) ).
% norm_assertion_simps(3)
thf(fact_2062_wf__measures,axiom,
! [A: $tType,Fs: list @ ( A > nat )] : ( wf @ A @ ( measures @ A @ Fs ) ) ).
% wf_measures
thf(fact_2063_fact__less__mono__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_2064_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X4: A] :
( Xs
= ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_2065_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X4: A] :
( Xs
= ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_2066_tl__obtain__elem,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( tl @ A @ Xs )
= ( nil @ A ) )
=> ~ ! [E2: A] :
( Xs
!= ( cons @ A @ E2 @ ( nil @ A ) ) ) ) ) ).
% tl_obtain_elem
thf(fact_2067_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_neg_neg
thf(fact_2068_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_2069_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_2070_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_pos_pos
thf(fact_2071_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A4 @ B3 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_2072_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A4 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A4 @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A4 ) )
& ( C2
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_2073_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A4 @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_2074_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( divide_divide @ A @ A4 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_2075_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A4 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_2076_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X3: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X3 @ Y )
= ( divide_divide @ A @ W2 @ Z2 ) )
= ( ( times_times @ A @ X3 @ Z2 )
= ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2077_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ( divide_divide @ A @ B3 @ C2 )
= A4 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A4 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_2078_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ C2 )
= B3 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_2079_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B3
= ( times_times @ A @ A4 @ C2 ) )
=> ( ( divide_divide @ A @ B3 @ C2 )
= A4 ) ) ) ) ).
% divide_eq_imp
thf(fact_2080_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A4 @ C2 )
= B3 )
=> ( A4
= ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_2081_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B3 @ C2 )
= A4 )
= ( B3
= ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2082_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( A4
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( times_times @ A @ A4 @ C2 )
= B3 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2083_list_Oset__sel_I2_J,axiom,
! [A: $tType,A4: list @ A,X3: A] :
( ( A4
!= ( nil @ A ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ ( tl @ A @ A4 ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ A4 ) ) ) ) ).
% list.set_sel(2)
thf(fact_2084_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_2085_fact__gt__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_gt_zero
thf(fact_2086_fact__not__neg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] :
~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% fact_not_neg
thf(fact_2087_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_2088_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_2089_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_2090_not__hd__in__tl,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( X3
!= ( hd @ A @ Xs ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X3 @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% not_hd_in_tl
thf(fact_2091_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_2092_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_2093_mod__star__trueI,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H2 )
=> ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 ) ) ).
% mod_star_trueI
thf(fact_2094_mod__star__trueE,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 )
=> ~ ! [H5: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H5 ) ) ).
% mod_star_trueE
thf(fact_2095_ent__star__mono__true,axiom,
! [A3: assn,A8: assn,B2: assn,B11: assn] :
( ( entails @ A3 @ ( times_times @ assn @ A8 @ ( top_top @ assn ) ) )
=> ( ( entails @ B2 @ ( times_times @ assn @ B11 @ ( top_top @ assn ) ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ A3 @ B2 ) @ ( top_top @ assn ) ) @ ( times_times @ assn @ ( times_times @ assn @ A8 @ B11 ) @ ( top_top @ assn ) ) ) ) ) ).
% ent_star_mono_true
thf(fact_2096_ent__refl__true,axiom,
! [A3: assn] : ( entails @ A3 @ ( times_times @ assn @ A3 @ ( top_top @ assn ) ) ) ).
% ent_refl_true
thf(fact_2097_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_2098_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_2099_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_2100_div__int__unique,axiom,
! [K3: int,L: int,Q4: int,R2: int] :
( ( eucl_rel_int @ K3 @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( ( divide_divide @ int @ K3 @ L )
= Q4 ) ) ).
% div_int_unique
thf(fact_2101_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ B3 )
=> ( ( divide_divide @ A @ A4 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2102_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_positive
thf(fact_2103_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X3: A,W2: A,Z2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_le
thf(fact_2104_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A,W2: A,Z2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less
thf(fact_2105_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A,W2: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less @ A @ W2 @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_2106_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A4 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A4 @ B3 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_2107_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_2108_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_2109_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_2110_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_2111_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2112_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_2113_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_2114_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_2115_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_2116_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_2117_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_2118_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X3: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_2119_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z2 @ Y ) @ X3 )
=> ( ord_less @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_2120_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A4 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_2121_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A4 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_2122_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ B3 @ A4 ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A4 @ B3 ) )
| ( A4
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_2123_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ A4 @ B3 ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_2124_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A4 @ Z2 ) @ B3 )
= B3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A4 @ Z2 ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ B3 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2125_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A4 @ ( divide_divide @ A @ B3 @ Z2 ) )
= A4 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A4 @ ( divide_divide @ A @ B3 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A4 @ Z2 ) @ B3 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2126_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X3: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_2127_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X3: A,Z2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_2128_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X3: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_2129_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X3 @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_2130_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X3 @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_2131_tl__subset,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( tl @ A @ Xs ) ) @ A3 ) ) ) ).
% tl_subset
thf(fact_2132_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A4 @ ( divide_divide @ A @ B3 @ Z2 ) )
= A4 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A4 @ ( divide_divide @ A @ B3 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A4 @ Z2 ) @ B3 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2133_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X3: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2134_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X3 @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_2135_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X3 @ Z2 ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X3 @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_2136_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B3 ) ) ) ).
% gt_half_sum
thf(fact_2137_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ A4 @ ( divide_divide @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_2138_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_2139_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X3 ) @ Y )
= ( abs_abs @ A @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% abs_div_pos
thf(fact_2140_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 ) )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ B @ Ys @ N5 ) @ ( nth @ A @ Xs @ N5 ) ) )
=> ! [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_2141_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_2142_fact__less__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ A @ ( semiring_char_0_fact @ A @ M2 ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ) ).
% fact_less_mono
thf(fact_2143_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_2144_distinct__hd__tl,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( X3
= ( hd @ A @ Xs ) )
=> ~ ( member @ A @ X3 @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% distinct_hd_tl
thf(fact_2145_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_2146_mod__h__bot__iff_I2_J,axiom,
! [H2: heap_ext @ product_unit] : ( rep_assn @ ( top_top @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_h_bot_iff(2)
thf(fact_2147_remove1__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( remove1 @ A @ ( hd @ A @ Xs ) @ Xs )
= ( tl @ A @ Xs ) ) ) ).
% remove1_tl
thf(fact_2148_entt__def__true,axiom,
( entailst
= ( ^ [P2: assn,Q2: assn] : ( entails @ ( times_times @ assn @ P2 @ ( top_top @ assn ) ) @ ( times_times @ assn @ Q2 @ ( top_top @ assn ) ) ) ) ) ).
% entt_def_true
thf(fact_2149_entailst__def,axiom,
( entailst
= ( ^ [A7: assn,B7: assn] : ( entails @ A7 @ ( times_times @ assn @ B7 @ ( top_top @ assn ) ) ) ) ) ).
% entailst_def
thf(fact_2150_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_2151_enttI,axiom,
! [A3: assn,B2: assn] :
( ( entails @ A3 @ ( times_times @ assn @ B2 @ ( top_top @ assn ) ) )
=> ( entailst @ A3 @ B2 ) ) ).
% enttI
thf(fact_2152_enttD,axiom,
! [A3: assn,B2: assn] :
( ( entailst @ A3 @ B2 )
=> ( entails @ A3 @ ( times_times @ assn @ B2 @ ( top_top @ assn ) ) ) ) ).
% enttD
thf(fact_2153_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A7: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R5 ) )
& ( A7
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ! [Y5: A] :
( ( member @ A @ Y5 @ A7 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_2154_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_2155_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A4 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_2156_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ Y ) @ X3 )
=> ( ord_less_eq @ A @ Z2 @ ( divide_divide @ A @ X3 @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_2157_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X3: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X3 @ ( times_times @ A @ Z2 @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_2158_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_2159_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_2160_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_2161_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_2162_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A4 ) @ ( divide_divide @ A @ C2 @ B3 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_2163_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_2164_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_2165_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ B3 @ A4 ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A4 @ B3 ) )
| ( A4
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_2166_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B3 @ A4 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less_eq @ A @ A4 @ B3 ) )
| ( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_2167_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X3: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_2168_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X3: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X3 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X3 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_2169_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_2170_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V2: A,R2: A,S3: A] :
( ( ord_less_eq @ A @ U @ V2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ( ord_less_eq @ A @ R2 @ S3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V2 @ U ) ) @ S3 ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_2171_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
=> ~ ! [N5: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N5 ) ) ) @ E3 ) ) ) ).
% nat_approx_posE
thf(fact_2172_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_2173_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_2174_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self1
thf(fact_2175_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self2
thf(fact_2176_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B3 ) @ A4 ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self3
thf(fact_2177_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ C2 ) @ A4 ) @ B3 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_self4
thf(fact_2178_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% div_mult_mult1
thf(fact_2179_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% div_mult_mult2
thf(fact_2180_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2181_top1I,axiom,
! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% top1I
thf(fact_2182_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( divide_divide @ nat @ M2 @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_2183_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_2184_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_2185_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_2186_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( times_times @ nat @ I @ N ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_2187_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ord_less_eq @ nat @ N @ M2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_2188_div__le__mono2,axiom,
! [M2: nat,N: nat,K3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K3 @ N ) @ ( divide_divide @ nat @ K3 @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_2189_div__less__iff__less__mult,axiom,
! [Q4: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ Q4 ) @ N )
= ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_2190_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ( divide_divide @ nat @ M2 @ N )
= M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_2191_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_2192_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M: nat,N2: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M @ N2 )
| ( N2
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_2193_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
=> ( ( ord_less_eq @ nat @ M2 @ ( divide_divide @ nat @ N @ Q4 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M2 @ Q4 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_2194_div__nat__eqI,axiom,
! [N: nat,Q4: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q4 ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ ( suc @ Q4 ) ) )
=> ( ( divide_divide @ nat @ M2 @ N )
= Q4 ) ) ) ).
% div_nat_eqI
thf(fact_2195_split__div,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J2: nat] :
( ( ord_less @ nat @ J2 @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J2 ) )
=> ( P @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_2196_dividend__less__div__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_2197_dividend__less__times__div,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M2 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_2198_nat__mult__div__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K3 @ M2 ) @ ( times_times @ nat @ K3 @ N ) )
= ( divide_divide @ nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_2199_le__div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ nat @ M2 @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_2200_split__div_H,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q8: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q8 ) @ M2 )
& ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ ( suc @ Q8 ) ) )
& ( P @ Q8 ) ) ) ) ).
% split_div'
thf(fact_2201_div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ~ ( ord_less @ nat @ M2 @ N )
=> ( ( divide_divide @ nat @ M2 @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_2202_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( divide_divide @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_2203_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_2204_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_2205_wf__bounded__measure,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > nat,F3: A > nat] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R2 )
=> ( ( ord_less_eq @ nat @ ( Ub @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less_eq @ nat @ ( F3 @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less @ nat @ ( F3 @ A6 ) @ ( F3 @ B5 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_measure
thf(fact_2206_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
~ ? [F7: nat > A] :
! [I4: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F7 @ ( suc @ I4 ) ) @ ( F7 @ I4 ) ) @ R5 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_2207_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),F3: nat > A] :
( ( wf @ A @ R2 )
=> ~ ! [K: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ ( suc @ K ) ) @ ( F3 @ K ) ) @ R2 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_2208_wfE__min_H,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Q: set @ A] :
( ( wf @ A @ R )
=> ( ( Q
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
=> ~ ( member @ A @ Y3 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_2209_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_2210_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A4: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_2211_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A4: A] :
( ( modulo_modulo @ A @ A4 @ ( zero_zero @ A ) )
= A4 ) ) ).
% mod_by_0
thf(fact_2212_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A4: A] :
( ( modulo_modulo @ A @ A4 @ A4 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_2213_minus__mod__self2,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,B3: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A4 @ B3 ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% minus_mod_self2
thf(fact_2214_mod__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= M2 ) ) ).
% mod_less
thf(fact_2215_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,A4: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B3 @ A4 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_2216_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ B3 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_2217_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A4: A] :
( ( modulo_modulo @ A @ A4 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_2218_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self1
thf(fact_2219_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self2
thf(fact_2220_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B3 ) @ A4 ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self3
thf(fact_2221_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ C2 ) @ A4 ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% mod_mult_self4
thf(fact_2222_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% mod_mult_right_eq
thf(fact_2223_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A4 @ C2 ) @ B3 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% mod_mult_left_eq
thf(fact_2224_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( times_times @ A @ C2 @ ( modulo_modulo @ A @ A4 @ B3 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ).
% mult_mod_right
thf(fact_2225_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% mod_mult_mult2
thf(fact_2226_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C2: A,A9: A,B3: A,B8: A] :
( ( ( modulo_modulo @ A @ A4 @ C2 )
= ( modulo_modulo @ A @ A9 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B3 @ C2 )
= ( modulo_modulo @ A @ B8 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A9 @ B8 ) @ C2 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_2227_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A4 @ C2 ) @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% mod_mult_eq
thf(fact_2228_mod__diff__right__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A4 @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% mod_diff_right_eq
thf(fact_2229_mod__diff__left__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A4 @ C2 ) @ B3 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% mod_diff_left_eq
thf(fact_2230_mod__diff__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,C2: A,A9: A,B3: A,B8: A] :
( ( ( modulo_modulo @ A @ A4 @ C2 )
= ( modulo_modulo @ A @ A9 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B3 @ C2 )
= ( modulo_modulo @ A @ B8 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A9 @ B8 ) @ C2 ) ) ) ) ) ).
% mod_diff_cong
thf(fact_2231_mod__diff__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A4 @ C2 ) @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 ) ) ) ).
% mod_diff_eq
thf(fact_2232_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A4 @ B3 ) @ B3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_2233_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ( modulo_modulo @ A @ A4 @ C2 )
= ( modulo_modulo @ A @ B3 @ C2 ) )
=> ~ ! [D4: A] :
( B3
!= ( plus_plus @ A @ A4 @ ( times_times @ A @ C2 @ D4 ) ) ) ) ) ).
% mod_eqE
thf(fact_2234_mod__less__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( modulo_modulo @ nat @ M2 @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_2235_div__less__mono,axiom,
! [A3: nat,B2: nat,N: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A3 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B2 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ N ) @ ( divide_divide @ nat @ B2 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_2236_mod__induct,axiom,
! [P: nat > $o,N: nat,P4: nat,M2: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ N @ P4 )
=> ( ( ord_less @ nat @ M2 @ P4 )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ P4 )
=> ( ( P @ N5 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N5 ) @ P4 ) ) ) )
=> ( P @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_2237_mod__geq,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less @ nat @ M2 @ N )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ).
% mod_geq
thf(fact_2238_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M: nat,N2: nat] : ( if @ nat @ ( ord_less @ nat @ M @ N2 ) @ M @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_2239_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ B3 )
=> ( ( modulo_modulo @ A @ A4 @ B3 )
= A4 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_2240_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_2241_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A4 @ B3 ) ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) @ C2 )
= ( plus_plus @ A @ A4 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_2242_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) @ C2 )
= ( plus_plus @ A @ A4 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_2243_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( A4
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) ) ) ).
% mod_div_decomp
thf(fact_2244_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) @ ( modulo_modulo @ A @ A4 @ B3 ) )
= A4 ) ) ).
% div_mult_mod_eq
thf(fact_2245_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ B3 ) @ ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) )
= A4 ) ) ).
% mod_div_mult_eq
thf(fact_2246_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ B3 ) @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A4 @ B3 ) ) )
= A4 ) ) ).
% mod_mult_div_eq
thf(fact_2247_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B3: A,A4: A] :
( ( plus_plus @ A @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A4 @ B3 ) ) @ ( modulo_modulo @ A @ A4 @ B3 ) )
= A4 ) ) ).
% mult_div_mod_eq
thf(fact_2248_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A4 @ ( modulo_modulo @ A @ B3 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_2249_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_2250_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( modulo_modulo @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_2251_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( modulo_modulo @ A @ A4 @ B3 ) )
= ( times_times @ A @ B3 @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_2252_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( times_times @ A @ B3 @ ( divide_divide @ A @ A4 @ B3 ) ) )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_2253_field__char__0__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M2 @ N ) ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_2254_mod__le__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M2 @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_2255_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A4 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A4 @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2256_split__mod,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( modulo_modulo @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ M2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J2: nat] :
( ( ord_less @ nat @ J2 @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J2 ) )
=> ( P @ J2 ) ) ) ) ) ) ).
% split_mod
thf(fact_2257_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B3 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) @ ( modulo_modulo @ A @ A4 @ B3 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2258_verit__le__mono__div,axiom,
! [A3: nat,B2: nat,N: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A3 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B2 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B2 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_2259_Suc__times__mod__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M2 @ N ) ) @ M2 )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_2260_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_2261_wf__induct__rule,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( wf @ A @ R2 )
=> ( ! [X: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R2 )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A4 ) ) ) ).
% wf_induct_rule
thf(fact_2262_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [Q2: set @ A] :
( ? [X4: A] : ( member @ A @ X4 @ Q2 )
=> ? [X4: A] :
( ( member @ A @ X4 @ Q2 )
& ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 )
=> ~ ( member @ A @ Y5 @ Q2 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_2263_wf__not__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( wf @ A @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R2 ) ) ).
% wf_not_refl
thf(fact_2264_wf__not__sym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,X3: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A4 ) @ R2 ) ) ) ).
% wf_not_sym
thf(fact_2265_wf__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( wf @ A @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R2 ) ) ).
% wf_irrefl
thf(fact_2266_wf__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( wf @ A @ R2 )
=> ( ! [X: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R2 )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A4 ) ) ) ).
% wf_induct
thf(fact_2267_wf__asym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,X3: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A4 ) @ R2 ) ) ) ).
% wf_asym
thf(fact_2268_wfUNIVI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [P6: A > $o,X: A] :
( ! [Xa3: A] :
( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa3 ) @ R2 )
=> ( P6 @ Y2 ) )
=> ( P6 @ Xa3 ) )
=> ( P6 @ X ) )
=> ( wf @ A @ R2 ) ) ).
% wfUNIVI
thf(fact_2269_wfI__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X: A,Q5: set @ A] :
( ( member @ A @ X @ Q5 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ Q5 )
& ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa3 ) @ R )
=> ~ ( member @ A @ Y2 @ Q5 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_min
thf(fact_2270_wfE__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X3: A,Q: set @ A] :
( ( wf @ A @ R )
=> ( ( member @ A @ X3 @ Q )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
=> ~ ( member @ A @ Y3 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_2271_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [P2: A > $o] :
( ! [X4: A] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 )
=> ( P2 @ Y5 ) )
=> ( P2 @ X4 ) )
=> ! [X9: A] : ( P2 @ X9 ) ) ) ) ).
% wf_def
thf(fact_2272_in__measure,axiom,
! [A: $tType,X3: A,Y: A,F3: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( measure @ A @ F3 ) )
= ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ).
% in_measure
thf(fact_2273_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [M6: nat] : ( P @ M6 @ ( zero_zero @ nat ) )
=> ( ! [M6: nat,N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( P @ N5 @ ( modulo_modulo @ nat @ M6 @ N5 ) )
=> ( P @ M6 @ N5 ) ) )
=> ( P @ M2 @ N ) ) ) ).
% gcd_nat_induct
thf(fact_2274_in__lex__prod,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,A9: A,B8: B,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Pair @ A @ B @ A9 @ B8 ) ) @ ( lex_prod @ A @ B @ R2 @ S3 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A9 ) @ R2 )
| ( ( A4 = A9 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B8 ) @ S3 ) ) ) ) ).
% in_lex_prod
thf(fact_2275_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
( ( wf @ A @ R )
=> ( ! [F5: A > B,G3: A > B,X: A,R3: B] :
( ! [Z5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ X ) @ R )
=> ( ( F5 @ Z5 )
= ( G3 @ Z5 ) ) )
=> ( ( P @ F5 @ X @ R3 )
= ( P @ G3 @ X @ R3 ) ) )
=> ( ! [X: A,F5: A > B] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R )
=> ( P @ F5 @ Y3 @ ( F5 @ Y3 ) ) )
=> ? [X_12: B] : ( P @ F5 @ X @ X_12 ) )
=> ? [F5: A > B] :
! [X8: A] : ( P @ F5 @ X8 @ ( F5 @ X8 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_2276_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K3: nat,A4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ( gbinomial @ A @ A4 @ K3 )
= ( times_times @ A @ ( divide_divide @ A @ A4 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_2277_rotate__drop__take,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N2: nat,Xs3: list @ A] : ( append @ A @ ( drop @ A @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) @ ( take @ A @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ) ).
% rotate_drop_take
thf(fact_2278_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_2279_set__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( set2 @ A @ ( rotate @ A @ N @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate
thf(fact_2280_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_2281_distinct__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( distinct @ A @ ( rotate @ A @ N @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rotate
thf(fact_2282_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_2283_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_2284_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_2285_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_2286_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_2287_mod__int__unique,axiom,
! [K3: int,L: int,Q4: int,R2: int] :
( ( eucl_rel_int @ K3 @ L @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( ( modulo_modulo @ int @ K3 @ L )
= R2 ) ) ).
% mod_int_unique
thf(fact_2288_rotate__append,axiom,
! [A: $tType,L: list @ A,Q4: list @ A] :
( ( rotate @ A @ ( size_size @ ( list @ A ) @ L ) @ ( append @ A @ L @ Q4 ) )
= ( append @ A @ Q4 @ L ) ) ).
% rotate_append
thf(fact_2289_rotate__conv__mod,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N2: nat,Xs3: list @ A] : ( rotate @ A @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_2290_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K3: nat] :
( ( gbinomial @ A @ A4 @ ( suc @ K3 ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ ( suc @ K3 ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ K3 ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_2291_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K3: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( gbinomial @ A @ A4 @ K3 ) )
= ( times_times @ A @ A4 @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ K3 ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_2292_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K3: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A4 @ K3 ) @ A4 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( gbinomial @ A @ A4 @ K3 ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K3 ) ) @ ( gbinomial @ A @ A4 @ ( suc @ K3 ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_2293_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K3: nat] :
( ( times_times @ A @ A4 @ ( gbinomial @ A @ A4 @ K3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( gbinomial @ A @ A4 @ K3 ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K3 ) ) @ ( gbinomial @ A @ A4 @ ( suc @ K3 ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_2294_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R3: B,F5: A > B,G3: A > B,X: A] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X )
=> ( ( F5 @ Y3 )
= ( G3 @ Y3 ) ) )
=> ( ( P @ F5 @ X @ R3 )
= ( P @ G3 @ X @ R3 ) ) )
=> ( ! [X: A,F5: A > B] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X )
=> ( P @ F5 @ Y3 @ ( F5 @ Y3 ) ) )
=> ? [X_12: B] : ( P @ F5 @ X @ X_12 ) )
=> ? [F5: A > B] :
! [X8: A] : ( P @ F5 @ X8 @ ( F5 @ X8 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_2295_eucl__rel__int,axiom,
! [K3: int,L: int] : ( eucl_rel_int @ K3 @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K3 @ L ) @ ( modulo_modulo @ int @ K3 @ L ) ) ) ).
% eucl_rel_int
thf(fact_2296_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K3: nat,A4: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K3 ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( suc @ K3 ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A4 @ K3 ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_2297_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K3: nat,A4: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K3 ) ) @ ( gbinomial @ A @ A4 @ ( suc @ K3 ) ) )
= ( times_times @ A @ A4 @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ K3 ) ) ) ) ).
% gbinomial_absorption
thf(fact_2298_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K3: nat,M2: nat,A4: A] :
( ( ord_less_eq @ nat @ K3 @ M2 )
=> ( ( times_times @ A @ ( gbinomial @ A @ A4 @ M2 ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M2 ) @ K3 ) )
= ( times_times @ A @ ( gbinomial @ A @ A4 @ K3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( minus_minus @ nat @ M2 @ K3 ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_2299_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K3: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( suc @ K3 ) )
= ( times_times @ A @ ( gbinomial @ A @ A4 @ K3 ) @ ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K3 ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_2300_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K3: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( suc @ K3 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K3 ) ) ) @ ( gbinomial @ A @ A4 @ K3 ) ) ) ) ).
% gbinomial_factors
thf(fact_2301_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K3: nat,A4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ( gbinomial @ A @ A4 @ K3 )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_2302_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_2303_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_2304_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_2305_bezw__0,axiom,
! [X3: nat] :
( ( bezw @ X3 @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_2306_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ K4 ) ) @ ( one_one @ A ) ) @ K4 ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_2307_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K3: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A4 ) @ K3 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ).
% gbinomial_minus
thf(fact_2308_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F23: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F23 @ List ) )
= ( ( ( List
= ( nil @ A ) )
=> ( P @ F1 ) )
& ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
=> ( P @ ( F23 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% list.split_sel
thf(fact_2309_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F23: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F23 @ List ) )
= ( ~ ( ( ( List
= ( nil @ A ) )
& ~ ( P @ F1 ) )
| ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
& ~ ( P @ ( F23 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% list.split_sel_asm
thf(fact_2310_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( ( gcd_Gcd @ A @ A3 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert3 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_2311_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A4 ) )
= A4 ) ) ).
% add.inverse_inverse
thf(fact_2312_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( uminus_uminus @ A @ A4 )
= ( uminus_uminus @ A @ B3 ) )
= ( A4 = B3 ) ) ) ).
% neg_equal_iff_equal
thf(fact_2313_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X4: A] : ( uminus_uminus @ B @ ( A7 @ X4 ) ) ) ) ) ).
% uminus_apply
thf(fact_2314_Compl__anti__mono,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_2315_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_2316_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% neg_le_iff_le
thf(fact_2317_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_2318_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A4 ) )
= ( ( zero_zero @ A )
= A4 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_2319_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( ( uminus_uminus @ A @ A4 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_2320_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( A4
= ( uminus_uminus @ A @ A4 ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_2321_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ( uminus_uminus @ A @ A4 )
= A4 )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_2322_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% neg_less_iff_less
thf(fact_2323_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X3 ) ) ) ).
% compl_less_compl_iff
thf(fact_2324_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ A4 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ B3 ) )
= B3 ) ) ).
% add_minus_cancel
thf(fact_2325_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ ( plus_plus @ A @ A4 @ B3 ) )
= B3 ) ) ).
% minus_add_cancel
thf(fact_2326_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A4 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ).
% minus_add_distrib
thf(fact_2327_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% mult_minus_right
thf(fact_2328_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ ( uminus_uminus @ A @ B3 ) )
= ( times_times @ A @ A4 @ B3 ) ) ) ).
% minus_mult_minus
thf(fact_2329_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( uminus_uminus @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% mult_minus_left
thf(fact_2330_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A4 @ B3 ) )
= ( minus_minus @ A @ B3 @ A4 ) ) ) ).
% minus_diff_eq
thf(fact_2331_Compl__disjoint2,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_2332_Compl__disjoint,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_2333_abs__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A4 ) )
= ( abs_abs @ A @ A4 ) ) ) ).
% abs_minus
thf(fact_2334_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A4 ) )
= ( abs_abs @ A @ A4 ) ) ) ).
% abs_minus_cancel
thf(fact_2335_Diff__Compl,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ).
% Diff_Compl
thf(fact_2336_inter__compl__diff__conv,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ).
% inter_compl_diff_conv
thf(fact_2337_Compl__Diff__eq,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ B2 ) ) ).
% Compl_Diff_eq
thf(fact_2338_sgn__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A4 ) )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ A4 ) ) ) ) ).
% sgn_minus
thf(fact_2339_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ A4 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_2340_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_2341_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% neg_le_0_iff_le
thf(fact_2342_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_2343_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% neg_less_0_iff_less
thf(fact_2344_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_2345_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A4 ) @ A4 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% neg_less_pos
thf(fact_2346_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A4: A] :
( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ A4 ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_2347_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ A4 @ ( uminus_uminus @ A @ A4 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_2348_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_2349_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A4 )
= ( uminus_uminus @ A @ A4 ) ) ) ).
% diff_0
thf(fact_2350_verit__minus__simplify_I3_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B3: B] :
( ( minus_minus @ B @ ( zero_zero @ B ) @ B3 )
= ( uminus_uminus @ B @ B3 ) ) ) ).
% verit_minus_simplify(3)
thf(fact_2351_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_2352_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_2353_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( minus_minus @ A @ B3 @ A4 ) ) ) ).
% uminus_add_conv_diff
thf(fact_2354_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
= ( plus_plus @ A @ A4 @ B3 ) ) ) ).
% diff_minus_eq_add
thf(fact_2355_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_2356_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_2357_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( uminus_uminus @ A @ X3 ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_right
thf(fact_2358_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_left
thf(fact_2359_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X3 ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_2360_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_2361_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ ( inf_inf @ A @ X3 @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_2362_subset__Compl__singleton,axiom,
! [A: $tType,A3: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B3 @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_2363_minus__mod__self1,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [B3: A,A4: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ B3 @ A4 ) @ B3 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A4 ) @ B3 ) ) ) ).
% minus_mod_self1
thf(fact_2364_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_2365_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_2366_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A4 )
= ( uminus_uminus @ A @ A4 ) ) ) ) ).
% abs_of_nonpos
thf(fact_2367_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A4: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A4 ) )
= A4 ) ) ).
% left_minus_one_mult_self
thf(fact_2368_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_2369_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A4 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_2370_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( uminus_uminus @ A @ B3 ) )
= ( B3
= ( uminus_uminus @ A @ A4 ) ) ) ) ).
% equation_minus_iff
thf(fact_2371_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( uminus_uminus @ A @ A4 )
= B3 )
= ( ( uminus_uminus @ A @ B3 )
= A4 ) ) ) ).
% minus_equation_iff
thf(fact_2372_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X4: A] : ( uminus_uminus @ B @ ( A7 @ X4 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_2373_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% le_minus_iff
thf(fact_2374_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A4 ) ) ) ).
% minus_le_iff
thf(fact_2375_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% le_imp_neg_le
thf(fact_2376_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ A4 ) ) ) ).
% minus_less_iff
thf(fact_2377_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less @ A @ B3 @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% less_minus_iff
thf(fact_2378_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X3 ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_2379_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X3 ) )
=> ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_2380_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_2381_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K3: A,A4: A] :
( ( A3
= ( plus_plus @ A @ K3 @ A4 ) )
=> ( ( uminus_uminus @ A @ A3 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K3 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_2382_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A4 @ B3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A4 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_2383_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( times_times @ A @ A4 @ ( uminus_uminus @ A @ B3 ) ) ) ) ).
% minus_mult_commute
thf(fact_2384_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A4: A,B3: A] :
( ( ( times_times @ A @ A4 @ A4 )
= ( times_times @ A @ B3 @ B3 ) )
= ( ( A4 = B3 )
| ( A4
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% square_eq_iff
thf(fact_2385_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B3: A,A4: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B3 ) @ A4 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A4 ) @ B3 ) ) ) ).
% minus_diff_commute
thf(fact_2386_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X3: A,Y: A] :
( ( ( abs_abs @ A @ X3 )
= ( abs_abs @ A @ Y ) )
= ( ( X3 = Y )
| ( X3
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% abs_eq_iff
thf(fact_2387_minus__assn__def,axiom,
( ( minus_minus @ assn )
= ( ^ [A5: assn,B4: assn] : ( inf_inf @ assn @ A5 @ ( uminus_uminus @ assn @ B4 ) ) ) ) ).
% minus_assn_def
thf(fact_2388_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,K3: nat] :
( ( ord_less @ nat @ N @ K3 )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K3 )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_2389_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N: nat,K3: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K3 )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N @ K3 ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_2390_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A4 @ N )
= ( zero_zero @ A ) )
= ( ? [K4: nat] :
( ( ord_less @ nat @ K4 @ N )
& ( A4
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K4 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_2391_list_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F23: A > ( list @ A ) > B,X21: A,X222: list @ A] :
( ( case_list @ B @ A @ F1 @ F23 @ ( cons @ A @ X21 @ X222 ) )
= ( F23 @ X21 @ X222 ) ) ).
% list.simps(5)
thf(fact_2392_list_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F23: A > ( list @ A ) > B] :
( ( case_list @ B @ A @ F1 @ F23 @ ( nil @ A ) )
= F1 ) ).
% list.simps(4)
thf(fact_2393_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( uminus_uminus @ A @ A4 )
= B3 )
= ( ( plus_plus @ A @ A4 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_2394_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( uminus_uminus @ A @ B3 ) )
= ( ( plus_plus @ A @ A4 @ B3 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_2395_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A4 )
= B3 ) ) ) ).
% add.inverse_unique
thf(fact_2396_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ A4 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_2397_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A4: A,B3: A] :
( ( ( plus_plus @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
= ( B3
= ( uminus_uminus @ A @ A4 ) ) ) ) ).
% add_eq_0_iff
thf(fact_2398_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_2399_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_2400_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X3: A] :
( ( ( times_times @ A @ X3 @ X3 )
= ( one_one @ A ) )
= ( ( X3
= ( one_one @ A ) )
| ( X3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_2401_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,K3: A,B3: A,A4: A] :
( ( B2
= ( plus_plus @ A @ K3 @ B3 ) )
=> ( ( minus_minus @ A @ A4 @ B2 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K3 ) @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_2402_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_2403_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2404_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,A4: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ A4 ) @ ( inf_inf @ A @ X3 @ B3 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_2405_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,A4: A,B3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ A4 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X3 ) @ B3 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_2406_diff__eq,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( minus_minus @ A )
= ( ^ [X4: A,Y5: A] : ( inf_inf @ A @ X4 @ ( uminus_uminus @ A @ Y5 ) ) ) ) ) ).
% diff_eq
thf(fact_2407_subset__Compl__self__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_2408_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ ( abs_abs @ A @ A4 ) ) ) ).
% abs_ge_minus_self
thf(fact_2409_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ B3 )
= ( ( ord_less_eq @ A @ A4 @ B3 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B3 ) ) ) ) ).
% abs_le_iff
thf(fact_2410_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B3 ) ) ) ).
% abs_le_D2
thf(fact_2411_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A4 ) @ B3 ) ) ) ) ).
% abs_leI
thf(fact_2412_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_2413_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_2414_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A4 ) @ B3 )
= ( ( ord_less @ A @ A4 @ B3 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A4 ) @ B3 ) ) ) ) ).
% abs_less_iff
thf(fact_2415_Compl__partition2,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ A3 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_2416_Compl__partition,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_2417_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_2418_sgn__not__eq__imp,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: A,A4: A] :
( ( ( sgn_sgn @ A @ B3 )
!= ( sgn_sgn @ A @ A4 ) )
=> ( ( ( sgn_sgn @ A @ A4 )
!= ( zero_zero @ A ) )
=> ( ( ( sgn_sgn @ A @ B3 )
!= ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A4 )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ B3 ) ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_2419_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X3 @ N ) ) ) ) ).
% pochhammer_pos
thf(fact_2420_Compl__Un,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ).
% Compl_Un
thf(fact_2421_Compl__Int,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ).
% Compl_Int
thf(fact_2422_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_2423_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R2: A,K3: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R2 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R2 ) @ K3 ) )
= ( times_times @ A @ R2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R2 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_2424_coset__def,axiom,
! [A: $tType] :
( ( coset @ A )
= ( ^ [Xs3: list @ A] : ( uminus_uminus @ ( set @ A ) @ ( set2 @ A @ Xs3 ) ) ) ) ).
% coset_def
thf(fact_2425_compl__coset,axiom,
! [A: $tType,Xs: list @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( coset @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% compl_coset
thf(fact_2426_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_2427_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B3: A,K3: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B3 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B3 ) @ K3 ) ) ) ) ).
% pochhammer_minus'
thf(fact_2428_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B3: A,K3: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B3 ) @ K3 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B3 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ).
% pochhammer_minus
thf(fact_2429_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_2430_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_2431_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4
= ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ C2 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_2432_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) )
= A4 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B3 )
= ( times_times @ A @ A4 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_2433_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B3 ) )
= C2 )
= ( ( uminus_uminus @ A @ A4 )
= ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_2434_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B3 ) ) )
= ( ( times_times @ A @ C2 @ B3 )
= ( uminus_uminus @ A @ A4 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_2435_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ( inf_inf @ A @ X3 @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% inf_shunt
thf(fact_2436_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A4 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_2437_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( abs_abs @ A @ B3 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ( B3 = A4 )
| ( B3
= ( uminus_uminus @ A @ A4 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_2438_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A4: A,B3: A] :
( ( ( abs_abs @ A @ A4 )
= B3 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
& ( ( A4 = B3 )
| ( A4
= ( uminus_uminus @ A @ B3 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_2439_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if_raw
thf(fact_2440_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if
thf(fact_2441_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A4: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A4 )
= ( uminus_uminus @ A @ A4 ) ) ) ) ).
% abs_of_neg
thf(fact_2442_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A4: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A4 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A4 @ N ) ) ) ) ).
% power_minus
thf(fact_2443_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A5 ) @ K4 ) ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_2444_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_2445_Compl__insert,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_2446_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X3 @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_2447_int__cases4,axiom,
! [M2: int] :
( ! [N5: nat] :
( M2
!= ( semiring_1_of_nat @ int @ N5 ) )
=> ~ ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( M2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ) ).
% int_cases4
thf(fact_2448_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_2449_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_2450_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_2451_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_2452_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_2453_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_2454_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X3 @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X3 ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2455_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ Z2 ) ) @ B3 )
= B3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ Z2 ) ) @ B3 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A4 ) @ ( times_times @ A @ B3 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2456_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X3 @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X3 ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2457_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A4 @ Z2 ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A4 @ Z2 ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ A4 @ ( times_times @ A @ B3 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2458_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A4: A,B3: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ Z2 ) ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ Z2 ) ) @ B3 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A4 ) @ ( times_times @ A @ B3 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2459_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X3: A,Y: A] :
( ( ( inf_inf @ A @ X3 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ X3 @ Y )
= ( top_top @ A ) )
=> ( ( uminus_uminus @ A @ X3 )
= Y ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_2460_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X4: A] :
( if @ A
@ ( X4
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X4 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_2461_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ( sgn_sgn @ A @ A4 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_2462_int__cases3,axiom,
! [K3: int] :
( ( K3
!= ( zero_zero @ int ) )
=> ( ! [N5: nat] :
( ( K3
= ( semiring_1_of_nat @ int @ N5 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 ) )
=> ~ ! [N5: nat] :
( ( K3
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N5 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ).
% int_cases3
thf(fact_2463_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A4 @ ( suc @ N ) )
= ( times_times @ A @ A4 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_2464_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A4 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A4 @ N ) @ ( plus_plus @ A @ A4 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_2465_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_2466_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_2467_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_2468_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_2469_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_2470_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_2471_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) @ A4 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_2472_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B3 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A4 @ C2 ) @ ( uminus_uminus @ A @ B3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_2473_list_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_list @ B @ A )
= ( ^ [F13: B,F24: A > ( list @ A ) > B,List2: list @ A] :
( if @ B
@ ( List2
= ( nil @ A ) )
@ F13
@ ( F24 @ ( hd @ A @ List2 ) @ ( tl @ A @ List2 ) ) ) ) ) ).
% list.case_eq_if
thf(fact_2474_neg__int__cases,axiom,
! [K3: int] :
( ( ord_less @ int @ K3 @ ( zero_zero @ int ) )
=> ~ ! [N5: nat] :
( ( K3
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N5 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ).
% neg_int_cases
thf(fact_2475_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_2476_zminus1__lemma,axiom,
! [A4: int,B3: int,Q4: int,R2: int] :
( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( ( B3
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A4 ) @ B3
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q4 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q4 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B3 @ R2 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_2477_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K3: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K3 ) )
= ( 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 @ K3 ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% gbinomial_index_swap
thf(fact_2478_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A5 ) @ ( one_one @ A ) ) @ K4 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_2479_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X3 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X3 @ N ) ) ) ) ) ).
% power_minus'
thf(fact_2480_lenlex__append2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs ) @ ( append @ A @ Us @ Ys ) ) @ ( lenlex @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append2
thf(fact_2481_entails__solve__finalize_I1_J,axiom,
! [M5: list @ ( product_prod @ assn @ assn ),P: assn] : ( fI_RESULT @ M5 @ P @ ( one_one @ assn ) @ ( top_top @ assn ) ) ).
% entails_solve_finalize(1)
thf(fact_2482_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_2483_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X3 ) ) ) ).
% zero_le_ceiling
thf(fact_2484_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_2485_ComplI,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% ComplI
thf(fact_2486_Compl__iff,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( ~ ( member @ A @ C2 @ A3 ) ) ) ).
% Compl_iff
thf(fact_2487_Compl__eq__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A3 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) )
= ( A3 = B2 ) ) ).
% Compl_eq_Compl_iff
thf(fact_2488_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: num,N: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ M2 @ N ) ) ) ).
% numeral_less_iff
thf(fact_2489_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_2490_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V2: num,W2: num,Z2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Z2 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W2 ) ) @ Z2 ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_2491_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,M2: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ M2 ) ) @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_2492_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,M2: num,N: num,B3: A] :
( ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ M2 ) ) @ ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ N ) ) @ B3 ) )
= ( times_times @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) @ B3 ) ) ) ).
% power_add_numeral2
thf(fact_2493_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V2: num,B3: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ).
% distrib_left_numeral
thf(fact_2494_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A4: A,B3: A,V2: num] :
( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( numeral_numeral @ A @ V2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_2495_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V2: num,B3: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_2496_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A4: A,B3: A,V2: num] :
( ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( numeral_numeral @ A @ V2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_2497_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less @ num @ N @ M2 ) ) ) ).
% neg_numeral_less_iff
thf(fact_2498_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_2499_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_2500_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_2501_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(170)
thf(fact_2502_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(171)
thf(fact_2503_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W2 ) ) @ Y ) ) ) ).
% semiring_norm(172)
thf(fact_2504_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_2505_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_2506_cSup__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A] :
( ( complete_Sup_Sup @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% cSup_singleton
thf(fact_2507_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) ) @ B3 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_2508_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) @ A4 )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_2509_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) )
= A4 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2510_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( A4
= ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) )
= B3 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2511_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) ) @ B3 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_2512_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ W2 ) ) @ A4 )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_2513_nth__Cons__numeral,axiom,
! [A: $tType,X3: A,Xs: list @ A,V2: num] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ ( numeral_numeral @ nat @ V2 ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_2514_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X3 ) ) ) ).
% numeral_less_ceiling
thf(fact_2515_take__Cons__numeral,axiom,
! [A: $tType,V2: num,X3: A,Xs: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_2516_drop__Cons__numeral,axiom,
! [A: $tType,V2: num,X3: A,Xs: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X3 @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_2517_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_2518_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A] :
( ( ord_less @ A @ X3 @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X3 ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_2519_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( irrefl @ A @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_2520_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( ord_less_eq @ A @ A4 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B3 @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_2521_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A4 )
= ( ord_less_eq @ A @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B3 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_2522_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A4 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_2523_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( A4
= ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B3 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A4
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_2524_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,W2: num] :
( ( ord_less @ A @ A4 @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B3 @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_2525_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,W2: num,A4: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A4 )
= ( ord_less @ A @ ( times_times @ A @ A4 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B3 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_2526_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) ) ) ).
% zero_less_ceiling
thf(fact_2527_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) ) ) ).
% one_le_ceiling
thf(fact_2528_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ).
% one_less_ceiling
thf(fact_2529_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_2530_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_2531_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% numeral_le_ceiling
thf(fact_2532_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X3: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X3 ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X3 ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_2533_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_2534_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X3 ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_2535_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_2536_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X3: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_2537_ComplD,axiom,
! [A: $tType,C2: A,A3: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
=> ~ ( member @ A @ C2 @ A3 ) ) ).
% ComplD
thf(fact_2538_double__complement,axiom,
! [A: $tType,A3: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= A3 ) ).
% double_complement
thf(fact_2539_lexord__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( irrefl @ ( list @ A ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irrefl
thf(fact_2540_irrefl__lex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( irrefl @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ).
% irrefl_lex
thf(fact_2541_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_2542_irreflI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R )
=> ( irrefl @ A @ R ) ) ).
% irreflI
thf(fact_2543_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R5 ) ) ) ).
% irrefl_def
thf(fact_2544_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,Z2: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ X5 )
=> ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X5 ) @ Z2 ) ) ) ) ).
% cSup_least
thf(fact_2545_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A4: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ X5 )
=> ( ord_less_eq @ A @ X @ A4 ) )
=> ( ! [Y2: A] :
( ! [X8: A] :
( ( member @ A @ X8 @ X5 )
=> ( ord_less_eq @ A @ X8 @ Y2 ) )
=> ( ord_less_eq @ A @ A4 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X5 )
= A4 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_2546_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 ) ) )
=> ~ ! [X: A] :
( ( member @ A @ X @ X5 )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ) ) ).
% less_cSupE
thf(fact_2547_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 ) )
=> ? [X: A] :
( ( member @ A @ X @ X5 )
& ( ord_less @ A @ Z2 @ X ) ) ) ) ) ).
% less_cSupD
thf(fact_2548_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B2: set @ A,A4: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B2 ) @ A4 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ( ( inf_inf @ A @ X4 @ A4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_2549_not__numeral__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_less_zero
thf(fact_2550_zero__less__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% zero_less_numeral
thf(fact_2551_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ ( archimedean_ceiling @ A @ Y ) )
=> ( ord_less @ A @ X3 @ Y ) ) ) ).
% ceiling_less_cancel
thf(fact_2552_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_2553_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_2554_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_less_numeral
thf(fact_2555_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W2: num,X3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ ( uminus_uminus @ A @ X3 ) )
= ( times_times @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_2556_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ A4 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S ) ) @ A4 ) ) ) ) ).
% cSup_abs_le
thf(fact_2557_neg__numeral__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_less_zero
thf(fact_2558_not__zero__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_2559_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B3 @ C2 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2560_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 )
= B3 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2561_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_2562_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_2563_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_2564_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_2565_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_2566_lexl__not__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: list @ A] :
( ( irrefl @ A @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ X3 ) @ ( lex @ A @ R2 ) ) ) ).
% lexl_not_refl
thf(fact_2567_irrefl__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( irrefl @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ).
% irrefl_diff_Id
thf(fact_2568_frame__inference__finalize,axiom,
! [M5: list @ ( product_prod @ assn @ assn ),F: assn] : ( fI_RESULT @ M5 @ F @ ( one_one @ assn ) @ F ) ).
% frame_inference_finalize
thf(fact_2569_entails__solve__finalize_I2_J,axiom,
! [M5: list @ ( product_prod @ assn @ assn )] : ( fI_RESULT @ M5 @ ( one_one @ assn ) @ ( one_one @ assn ) @ ( one_one @ assn ) ) ).
% entails_solve_finalize(2)
thf(fact_2570_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S ) @ L ) ) @ E3 ) ) ) ) ).
% cSup_asclose
thf(fact_2571_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2572_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_2573_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 )
= B3 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2574_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B3 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B3
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_2575_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A4 @ B3 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A4 ) @ ( archimedean_ceiling @ A @ B3 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_2576_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_2577_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_2578_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B3: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B3 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2579_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2580_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X3: B,Y: B,C2: A,A4: A,B3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_2581_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X3: B,Y: B,A4: A,B3: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ A4 )
=> ( ( times_times @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_2582_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X3: B,Y: B,A4: A,B3: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ A4 )
=> ( ( times_times @ A @ A4 @ ( minus_minus @ A @ B3 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_2583_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X3: B,Y: B,C2: A,A4: A,B3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X3 @ Y ) @ C2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_2584_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B3: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B3 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B3 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_2585_Sup__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: A,A3: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( sup_sup @ A @ A4 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_insert
thf(fact_2586_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X3: A] :
( ( complete_Sup_Sup @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% ccpo_Sup_singleton
thf(fact_2587_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_2588_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Sup_Sup @ A @ A3 )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ X4 @ ( top_top @ A ) )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A3 )
& ( ord_less @ A @ X4 @ Y5 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_2589_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( ( complete_Sup_Sup @ A @ A3 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( X4
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_2590_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( X4
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_2591_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_2592_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: A,S: set @ A] :
( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ S ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S )
& ( ord_less @ A @ A4 @ X4 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_2593_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_2594_Union__empty__conv,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A3 )
=> ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_2595_empty__Union__conv,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A3 )
=> ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_2596_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X3: A,A3: set @ A] :
( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ A3 ) )
= ( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less @ A @ Y5 @ X4 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_2597_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A3 )
=> ( ord_less_eq @ A @ U @ V3 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_2598_Union__disjoint,axiom,
! [A: $tType,C3: set @ ( set @ A ),A3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C3 )
=> ( ( inf_inf @ ( set @ A ) @ X4 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_2599_insert__partition,axiom,
! [A: $tType,X3: set @ A,F: set @ ( set @ A )] :
( ~ ( member @ ( set @ A ) @ X3 @ F )
=> ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ ( insert3 @ ( set @ A ) @ X3 @ F ) )
=> ! [Xa4: set @ A] :
( ( member @ ( set @ A ) @ Xa4 @ ( insert3 @ ( set @ A ) @ X3 @ F ) )
=> ( ( X != Xa4 )
=> ( ( inf_inf @ ( set @ A ) @ X @ Xa4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ ( complete_Sup_Sup @ ( set @ A ) @ F ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_partition
thf(fact_2600_Sup__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A4: A,A3: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( sup_sup @ A @ A4 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_finite_insert
thf(fact_2601_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: int,X3: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) )
= ( ord_less @ int @ A4 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_2602_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N: nat,A4: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X3 ) ) @ N ) @ ( ring_1_of_int @ A @ A4 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X3 ) ) @ N ) @ A4 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_2603_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_2604_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_2605_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z2: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ W2 @ Z2 ) ) ) ).
% of_int_less_iff
thf(fact_2606_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z2: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W2 @ Z2 ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_mult
thf(fact_2607_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z2: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W2 @ Z2 ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_diff
thf(fact_2608_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less @ int @ J @ I )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_2609_upto__Nil2,axiom,
! [I: int,J: int] :
( ( ( nil @ int )
= ( upto @ I @ J ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil2
thf(fact_2610_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil
thf(fact_2611_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons @ int @ I @ ( nil @ int ) ) ) ).
% upto_single
thf(fact_2612_floor__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 ) ) ) ).
% floor_diff_of_int
thf(fact_2613_ceiling__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 ) ) ) ).
% ceiling_diff_of_int
thf(fact_2614_nth__upto,axiom,
! [I: int,K3: nat,J: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K3 ) ) @ J )
=> ( ( nth @ int @ ( upto @ I @ J ) @ K3 )
= ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K3 ) ) ) ) ).
% nth_upto
thf(fact_2615_of__int__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) )
= ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ).
% of_int_less_0_iff
thf(fact_2616_of__int__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% of_int_0_less_iff
thf(fact_2617_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z2: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z2 ) ) ) ).
% of_int_numeral_less_iff
thf(fact_2618_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int,N: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ int @ Z2 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_2619_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_2620_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_2621_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X3 @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_2622_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_2623_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% floor_less_numeral
thf(fact_2624_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_2625_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X3 @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_diff_numeral
thf(fact_2626_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_2627_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: int,B3: int,W2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X3 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 ) )
= ( ord_less @ int @ X3 @ ( power_power @ int @ B3 @ W2 ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_2628_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B3: int,W2: nat,X3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B3 ) @ W2 ) @ ( ring_1_of_int @ A @ X3 ) )
= ( ord_less @ int @ ( power_power @ int @ B3 @ W2 ) @ X3 ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_2629_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_2630_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_2631_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: int,X3: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) )
= ( ord_less @ int @ A4 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_2632_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: num,N: nat,A4: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X3 ) @ N ) @ ( ring_1_of_int @ A @ A4 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) @ A4 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_2633_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_2634_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_2635_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_2636_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 )
= ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% floor_less_iff
thf(fact_2637_distinct__upto,axiom,
! [I: int,J: int] : ( distinct @ int @ ( upto @ I @ J ) ) ).
% distinct_upto
thf(fact_2638_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X3 ) ) ).
% ex_of_int_less
thf(fact_2639_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] : ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_less_of_int
thf(fact_2640_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: int,Y: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X3 ) @ Y )
= ( times_times @ A @ Y @ ( ring_1_of_int @ A @ X3 ) ) ) ) ).
% mult_of_int_commute
thf(fact_2641_sorted__upto,axiom,
! [M2: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M2 @ N ) ) ).
% sorted_upto
thf(fact_2642_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_2643_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 )
=> ( ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X3 )
= Z2 ) ) ) ) ).
% floor_unique
thf(fact_2644_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A4: int] :
( ( ( archim6421214686448440834_floor @ A @ X3 )
= A4 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A4 ) @ X3 )
& ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A4 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_2645_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I4 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% floor_split
thf(fact_2646_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ Z2 )
= ( ord_less @ A @ X3 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_2647_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_2648_upto__aux__def,axiom,
( upto_aux
= ( ^ [I4: int,J2: int] : ( append @ int @ ( upto @ I4 @ J2 ) ) ) ) ).
% upto_aux_def
thf(fact_2649_upto__code,axiom,
( upto
= ( ^ [I4: int,J2: int] : ( upto_aux @ I4 @ J2 @ ( nil @ int ) ) ) ) ).
% upto_code
thf(fact_2650_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) )
=> ( ord_less @ A @ X3 @ Y ) ) ) ).
% floor_less_cancel
thf(fact_2651_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ) ).
% floor_divide_lower
thf(fact_2652_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) ) ) ) ).
% floor_divide_upper
thf(fact_2653_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less @ int @ Z2 @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X3 ) ) ) ).
% less_ceiling_iff
thf(fact_2654_upto__split2,axiom,
! [I: int,J: int,K3: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K3 )
=> ( ( upto @ I @ K3 )
= ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K3 ) ) ) ) ) ).
% upto_split2
thf(fact_2655_upto__split1,axiom,
! [I: int,J: int,K3: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K3 )
=> ( ( upto @ I @ K3 )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K3 ) ) ) ) ) ).
% upto_split1
thf(fact_2656_of__int__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_pos
thf(fact_2657_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X3 )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% of_int_lessD
thf(fact_2658_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [X: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X @ ( one_one @ int ) ) ) )
& ! [Y3: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y3 ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y3 @ ( one_one @ int ) ) ) ) )
=> ( Y3 = X ) ) ) ) ).
% floor_exists1
thf(fact_2659_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X3 )
& ( ord_less @ A @ X3 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_2660_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,X3: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X3 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X3 ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_2661_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A4 ) @ ( archim6421214686448440834_floor @ A @ B3 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_2662_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) @ ( one_one @ A ) ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X3 ) ) ) ) ) ).
% ceiling_correct
thf(fact_2663_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ Z2 ) )
=> ( ( archimedean_ceiling @ A @ X3 )
= Z2 ) ) ) ) ).
% ceiling_unique
thf(fact_2664_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A4: int] :
( ( ( archimedean_ceiling @ A @ X3 )
= A4 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A4 ) @ ( one_one @ A ) ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( ring_1_of_int @ A @ A4 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_2665_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I4: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I4 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% ceiling_split
thf(fact_2666_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_2667_upto_Osimps,axiom,
( upto
= ( ^ [I4: int,J2: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I4 @ J2 ) @ ( cons @ int @ I4 @ ( upto @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J2 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_2668_upto_Oelims,axiom,
! [X3: int,Xa: int,Y: list @ int] :
( ( ( upto @ X3 @ Xa )
= Y )
=> ( ( ( ord_less_eq @ int @ X3 @ Xa )
=> ( Y
= ( cons @ int @ X3 @ ( upto @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X3 @ Xa )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_2669_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_2670_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Z2: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X3 ) @ Z2 )
= ( ord_less_eq @ A @ X3 @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_2671_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X3: A] :
( ( ord_less_eq @ int @ Z2 @ ( archimedean_ceiling @ A @ X3 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X3 ) ) ) ).
% le_ceiling_iff
thf(fact_2672_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2673_upto__split3,axiom,
! [I: int,J: int,K3: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K3 )
=> ( ( upto @ I @ K3 )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K3 ) ) ) ) ) ) ).
% upto_split3
thf(fact_2674_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) @ P4 ) ) ) ).
% ceiling_divide_lower
thf(fact_2675_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_2676_numeral__power__less__nat__cancel__iff,axiom,
! [X3: num,N: nat,A4: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) @ ( nat2 @ A4 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) @ A4 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_2677_nat__less__numeral__power__cancel__iff,axiom,
! [A4: int,X3: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A4 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X3 ) @ N ) )
= ( ord_less @ int @ A4 @ ( power_power @ int @ ( numeral_numeral @ int @ X3 ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_2678_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A4: B,B3: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A4 )
=> ( ( member @ B @ A4 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A4 @ B3 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A4 ) @ ( archimedean_ceiling @ B @ B3 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_2679_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A4: B,B3: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A4 )
=> ( ( member @ B @ A4 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A4 ) @ ( archim6421214686448440834_floor @ B @ B3 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A4 @ B3 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_2680_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_2681_zless__nat__conj,axiom,
! [W2: int,Z2: int] :
( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
& ( ord_less @ int @ W2 @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_2682_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X3 ) )
= ( ~ ( member @ A @ X3 @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_2683_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_2684_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_2685_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_2686_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_2687_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_2688_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_2689_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A4: A,B3: A] :
( ( member @ A @ A4 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A4 @ B3 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_2690_Ints__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A4: A,B3: A] :
( ( member @ A @ A4 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_diff
thf(fact_2691_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A4: A] :
( ( times_times @ A @ A4 @ ( numeral_numeral @ A @ one2 ) )
= A4 ) ) ).
% mult_numeral_1_right
thf(fact_2692_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A4 )
= A4 ) ) ).
% mult_numeral_1
thf(fact_2693_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X3 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X3 ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X3 ) ) ) ) ) ) ).
% frac_neg
thf(fact_2694_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X3 ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_2695_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,A4: A] :
( ( ( archimedean_frac @ A @ X3 )
= A4 )
= ( ( member @ A @ ( minus_minus @ A @ X3 @ A4 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
& ( ord_less @ A @ A4 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_2696_nat__mono__iff,axiom,
! [Z2: int,W2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ W2 @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_2697_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z2: int] :
( ( ord_less @ nat @ M2 @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ M2 ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_2698_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B3: A] :
( ( times_times @ A @ B3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
= ( uminus_uminus @ A @ B3 ) ) ) ).
% mult_1s_ring_1(2)
thf(fact_2699_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B3 )
= ( uminus_uminus @ A @ B3 ) ) ) ).
% mult_1s_ring_1(1)
thf(fact_2700_nat__less__eq__zless,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ W2 @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_2701_frac__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_frac @ A )
= ( ^ [X4: A] : ( minus_minus @ A @ X4 @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X4 ) ) ) ) ) ) ).
% frac_def
thf(fact_2702_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( member @ A @ A4 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_2703_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) )
=> ( X3
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_2704_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( member @ A @ X3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X3 = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_2705_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_2706_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ( archimedean_frac @ A @ X3 )
= X3 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
& ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_2707_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X3 @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X3 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_2708_butlast__upd__last__eq,axiom,
! [A: $tType,L: list @ A,X3: 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 ) ) ) @ X3 )
= ( append @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ L ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% butlast_upd_last_eq
thf(fact_2709_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_2710_neg__eucl__rel__int__mult__2,axiom,
! [B3: int,A4: int,Q4: int,R2: int] :
( ( ord_less_eq @ int @ B3 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A4 @ ( one_one @ int ) ) @ B3 @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A4 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( product_Pair @ int @ int @ Q4 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_2711_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_2712_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2713_pos__eucl__rel__int__mult__2,axiom,
! [B3: int,A4: int,Q4: int,R2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( eucl_rel_int @ A4 @ B3 @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A4 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B3 ) @ ( product_Pair @ int @ int @ Q4 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_2714_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_2715_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_2716_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A4
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_2717_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_2718_mod2__gr__0,axiom,
! [M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ nat ) ) ) ).
% mod2_gr_0
thf(fact_2719_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A4: A,B3: A] :
( ( plus_plus @ A @ A4 @ ( plus_plus @ A @ A4 @ B3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) @ B3 ) ) ) ).
% left_add_twice
thf(fact_2720_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_2721_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_2722_power4__eq__xxxx,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X3: A] :
( ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X3 @ X3 ) @ X3 ) @ X3 ) ) ) ).
% power4_eq_xxxx
thf(fact_2723_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A4 @ A4 ) ) ) ).
% power2_eq_square
thf(fact_2724_power2__commute,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ ( minus_minus @ A @ Y @ X3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_commute
thf(fact_2725_less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% less_exp
thf(fact_2726_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_2727_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% half_gt_zero_iff
thf(fact_2728_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
~ ( ord_less @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_2729_list__decomp__2,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ? [A6: A,B5: A] :
( L
= ( cons @ A @ A6 @ ( cons @ A @ B5 @ ( nil @ A ) ) ) ) ) ).
% list_decomp_2
thf(fact_2730_less__2__cases,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( N
= ( zero_zero @ nat ) )
| ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases
thf(fact_2731_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases_iff
thf(fact_2732_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
=> ( ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2733_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ X3 @ Y ) ) ) ) ).
% power2_less_imp_less
thf(fact_2734_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_2735_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X3
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2736_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X3: A,Y: A] :
( ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( 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 ) ) @ X3 ) @ Y ) ) ) ) ).
% power2_sum
thf(fact_2737_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X3 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2738_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A4: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A4 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A4 ) @ N ) )
= ( power_power @ A @ A4 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_2739_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A,N: nat] :
( ( power_power @ A @ A4 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A4 @ ( power_power @ A @ ( power_power @ A @ A4 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_2740_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_2741_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_2742_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2743_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X3 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X3 @ ( 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 ) ) @ X3 ) @ Y ) ) ) ) ).
% power2_diff
thf(fact_2744_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A4 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_2745_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_2746_ex__power__ivl1,axiom,
! [B3: nat,K3: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K3 )
=> ? [N5: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B3 @ N5 ) @ K3 )
& ( ord_less @ nat @ K3 @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_2747_ex__power__ivl2,axiom,
! [B3: nat,K3: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 )
=> ? [N5: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B3 @ N5 ) @ K3 )
& ( ord_less_eq @ nat @ K3 @ ( power_power @ nat @ B3 @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_2748_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: A,X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( modulo_modulo @ A @ X3 @ M2 ) )
| ( ( modulo_modulo @ A @ X3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X3 @ M2 ) @ M2 ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2749_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ B3 )
= ( modulo_modulo @ A @ A4 @ B3 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2750_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R2: A,Q4: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R2 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R2 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R2 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_2751_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat,A4: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2752_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N5: nat] :
( ( P @ N5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) )
=> ( ! [N5: nat] :
( ( P @ N5 )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_2753_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( 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 @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X3 )
= Y ) ) ) ) ).
% round_unique
thf(fact_2754_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X3 )
= N ) ) ) ).
% round_unique'
thf(fact_2755_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) @ X3 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_2756_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: A,M2: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z2 ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ M2 ) ) ) ) ) ).
% round_diff_minimal
thf(fact_2757_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ).
% of_int_round_ge
thf(fact_2758_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X3: A] : ( ord_less @ A @ ( minus_minus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X3 ) ) ) ) ).
% of_int_round_gt
thf(fact_2759_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A4 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2760_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A4 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_2761_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A4 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_2762_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A4 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_2763_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A4 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_2764_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A5: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2765_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A4 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_2766_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2767_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_2768_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_2769_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A5: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_2770_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M2 )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M2 @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2771_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_2772_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A] : ( dvd_dvd @ A @ A4 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_2773_dvd__0__left__iff,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A4 )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left_iff
thf(fact_2774_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( plus_plus @ A @ A4 @ B3 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_2775_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( plus_plus @ A @ B3 @ A4 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_2776_dvd__minus__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y: A] :
( ( dvd_dvd @ A @ X3 @ ( uminus_uminus @ A @ Y ) )
= ( dvd_dvd @ A @ X3 @ Y ) ) ) ).
% dvd_minus_iff
thf(fact_2777_minus__dvd__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y: A] :
( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X3 ) @ Y )
= ( dvd_dvd @ A @ X3 @ Y ) ) ) ).
% minus_dvd_iff
thf(fact_2778_div__dvd__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ A4 @ C2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ B3 @ A4 ) @ ( divide_divide @ A @ C2 @ A4 ) )
= ( dvd_dvd @ A @ B3 @ C2 ) ) ) ) ) ).
% div_dvd_div
thf(fact_2779_abs__dvd__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: A,K3: A] :
( ( dvd_dvd @ A @ ( abs_abs @ A @ M2 ) @ K3 )
= ( dvd_dvd @ A @ M2 @ K3 ) ) ) ).
% abs_dvd_iff
thf(fact_2780_dvd__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: A,K3: A] :
( ( dvd_dvd @ A @ M2 @ ( abs_abs @ A @ K3 ) )
= ( dvd_dvd @ A @ M2 @ K3 ) ) ) ).
% dvd_abs_iff
thf(fact_2781_of__bool__less__eq__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P: $o,Q: $o] :
( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( P
=> Q ) ) ) ).
% of_bool_less_eq_iff
thf(fact_2782_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A @ $false )
= ( zero_zero @ A ) ) ) ).
% of_bool_eq(1)
thf(fact_2783_of__bool__eq__0__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( zero_zero @ A ) )
= ~ P ) ) ).
% of_bool_eq_0_iff
thf(fact_2784_of__bool__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P: $o,Q: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( ~ P
& Q ) ) ) ).
% of_bool_less_iff
thf(fact_2785_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_2786_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_2787_abs__bool__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: $o] :
( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% abs_bool_eq
thf(fact_2788_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_2789_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_2790_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ ( times_times @ A @ A4 @ C2 ) )
= ( dvd_dvd @ A @ B3 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_2791_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B3 @ A4 ) @ ( times_times @ A @ C2 @ A4 ) )
= ( dvd_dvd @ A @ B3 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_2792_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A4 ) @ B3 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_2793_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( plus_plus @ A @ B3 @ ( times_times @ A @ C2 @ A4 ) ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_2794_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_2795_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ A4 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A4 @ B3 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A4 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ) ).
% div_add
thf(fact_2796_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ A4 ) @ A4 )
= B3 ) ) ) ).
% dvd_div_mult_self
thf(fact_2797_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ A4 ) )
= B3 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_2798_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A4 @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_2799_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_2800_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) )
= A4 ) ) ) ).
% unit_div_1_div_1
thf(fact_2801_div__diff,axiom,
! [A: $tType] :
( ( idom_modulo @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ A4 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( divide_divide @ A @ ( minus_minus @ A @ A4 @ B3 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A4 @ C2 ) @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ) ) ).
% div_diff
thf(fact_2802_dvd__imp__mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( modulo_modulo @ A @ B3 @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% dvd_imp_mod_0
thf(fact_2803_take__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% take_bit_of_Suc_0
thf(fact_2804_zero__less__of__bool__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) )
= P ) ) ).
% zero_less_of_bool_iff
thf(fact_2805_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_2806_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_2807_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ A4 ) @ A4 )
= B3 ) ) ) ).
% unit_div_mult_self
thf(fact_2808_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B3 @ ( divide_divide @ A @ ( one_one @ A ) @ A4 ) )
= ( divide_divide @ A @ B3 @ A4 ) ) ) ) ).
% unit_mult_div_div
thf(fact_2809_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A4: A,B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A4 @ N ) @ ( power_power @ A @ B3 @ N ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_2810_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A4 ) @ ( sgn_sgn @ A @ A4 ) )
= ( zero_neq_one_of_bool @ A
@ ( A4
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_2811_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( sgn_sgn @ A @ ( abs_abs @ A @ A4 ) )
= ( zero_neq_one_of_bool @ A
@ ( A4
!= ( zero_zero @ A ) ) ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_2812_sgn__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A4: A] :
( ( abs_abs @ A @ ( sgn_sgn @ A @ A4 ) )
= ( zero_neq_one_of_bool @ A
@ ( A4
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_abs
thf(fact_2813_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_2814_sgn__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat] :
( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sgn_of_nat
thf(fact_2815_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_2816_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_2817_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A4 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_2818_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A] : ( dvd_dvd @ A @ A4 @ A4 ) ) ).
% dvd_refl
thf(fact_2819_dvd__trans,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ C2 )
=> ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% dvd_trans
thf(fact_2820_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P4: $o,Q4: $o] :
( ( ( zero_neq_one_of_bool @ A @ P4 )
= ( zero_neq_one_of_bool @ A @ Q4 ) )
= ( P4 = Q4 ) ) ) ).
% of_bool_eq_iff
thf(fact_2821_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_2822_dvd__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A4 )
=> ( A4
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left
thf(fact_2823_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ A4 @ C2 )
=> ( dvd_dvd @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ) ).
% dvd_add
thf(fact_2824_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ C2 )
=> ( ( dvd_dvd @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_2825_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ A4 @ ( plus_plus @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_2826_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A] : ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ A4 ) ) ) ).
% dvd_triv_right
thf(fact_2827_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
=> ( dvd_dvd @ A @ B3 @ C2 ) ) ) ).
% dvd_mult_right
thf(fact_2828_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ C2 @ D3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_2829_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A] : ( dvd_dvd @ A @ A4 @ ( times_times @ A @ A4 @ B3 ) ) ) ).
% dvd_triv_left
thf(fact_2830_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
=> ( dvd_dvd @ A @ A4 @ C2 ) ) ) ).
% dvd_mult_left
thf(fact_2831_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% dvd_mult2
thf(fact_2832_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ C2 )
=> ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% dvd_mult
thf(fact_2833_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B4: A,A5: A] :
? [K4: A] :
( A5
= ( times_times @ A @ B4 @ K4 ) ) ) ) ) ).
% dvd_def
thf(fact_2834_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A4: A,B3: A,K3: A] :
( ( A4
= ( times_times @ A @ B3 @ K3 ) )
=> ( dvd_dvd @ A @ B3 @ A4 ) ) ) ).
% dvdI
thf(fact_2835_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ A4 )
=> ~ ! [K: A] :
( A4
!= ( times_times @ A @ B3 @ K ) ) ) ) ).
% dvdE
thf(fact_2836_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
=> ? [B12: A,C7: A] :
( ( A4
= ( times_times @ A @ B12 @ C7 ) )
& ( dvd_dvd @ A @ B12 @ B3 )
& ( dvd_dvd @ A @ C7 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_2837_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P4: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ P4 @ ( times_times @ A @ A4 @ B3 ) )
=> ~ ! [X: A,Y2: A] :
( ( P4
= ( times_times @ A @ X @ Y2 ) )
=> ( ( dvd_dvd @ A @ X @ A4 )
=> ~ ( dvd_dvd @ A @ Y2 @ B3 ) ) ) ) ) ).
% dvd_productE
thf(fact_2838_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A4 ) ) ).
% one_dvd
thf(fact_2839_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B3 @ A4 ) ) ) ).
% unit_imp_dvd
thf(fact_2840_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A4 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_2841_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( dvd_dvd @ A @ X3 @ Y )
=> ( ( dvd_dvd @ A @ X3 @ Z2 )
=> ( dvd_dvd @ A @ X3 @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ) ).
% dvd_diff
thf(fact_2842_dvd__diff__commute,axiom,
! [A: $tType] :
( ( euclid5891614535332579305n_ring @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( minus_minus @ A @ C2 @ B3 ) )
= ( dvd_dvd @ A @ A4 @ ( minus_minus @ A @ B3 @ C2 ) ) ) ) ).
% dvd_diff_commute
thf(fact_2843_dvd__div__eq__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ A4 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( ( divide_divide @ A @ A4 @ C2 )
= ( divide_divide @ A @ B3 @ C2 ) )
= ( A4 = B3 ) ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_2844_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ( divide_divide @ A @ A4 @ C2 )
= ( divide_divide @ A @ B3 @ C2 ) )
=> ( ( dvd_dvd @ A @ C2 @ A4 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( A4 = B3 ) ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_2845_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D3: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ D3 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A4 @ D3 ) @ ( divide_divide @ A @ B3 @ D3 ) )
= ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% div_div_div_same
thf(fact_2846_dvd__mod__iff,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A4 @ B3 ) )
= ( dvd_dvd @ A @ C2 @ A4 ) ) ) ) ).
% dvd_mod_iff
thf(fact_2847_dvd__mod__imp__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A4 @ B3 ) )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( dvd_dvd @ A @ C2 @ A4 ) ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_2848_dvd__if__abs__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [L: A,K3: A] :
( ( ( abs_abs @ A @ L )
= ( abs_abs @ A @ K3 ) )
=> ( dvd_dvd @ A @ L @ K3 ) ) ) ).
% dvd_if_abs_eq
thf(fact_2849_uminus__dvd__conv_I1_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D6: int] : ( dvd_dvd @ int @ ( uminus_uminus @ int @ D6 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_2850_uminus__dvd__conv_I2_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D6: int,T3: int] : ( dvd_dvd @ int @ D6 @ ( uminus_uminus @ int @ T3 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_2851_zero__less__eq__of__bool,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% zero_less_eq_of_bool
thf(fact_2852_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_2853_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_2854_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_2855_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P7: $o] : ( if @ A @ P7 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_2856_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_2857_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S3: B] :
? [Z3: B] :
! [X8: B] :
( ( ord_less @ B @ Z3 @ X8 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X8 @ S3 ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X8 @ S3 ) ) ) ) ) ).
% pinf(9)
thf(fact_2858_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S3: B] :
? [Z3: B] :
! [X8: B] :
( ( ord_less @ B @ Z3 @ X8 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X8 @ S3 ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X8 @ S3 ) ) ) ) ) ) ).
% pinf(10)
thf(fact_2859_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S3: B] :
? [Z3: B] :
! [X8: B] :
( ( ord_less @ B @ X8 @ Z3 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X8 @ S3 ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X8 @ S3 ) ) ) ) ) ).
% minf(9)
thf(fact_2860_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S3: B] :
? [Z3: B] :
! [X8: B] :
( ( ord_less @ B @ X8 @ Z3 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X8 @ S3 ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X8 @ S3 ) ) ) ) ) ) ).
% minf(10)
thf(fact_2861_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_2862_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) )
= ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_2863_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_2864_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_2865_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ B3 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_2866_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A4 @ B3 )
= ( times_times @ A @ A4 @ C2 ) )
= ( B3 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_2867_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B3 @ A4 )
= ( times_times @ A @ C2 @ A4 ) )
= ( B3 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_2868_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( ( divide_divide @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_2869_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B3 @ C2 ) @ A4 )
= ( divide_divide @ A @ ( times_times @ A @ B3 @ A4 ) @ C2 ) ) ) ) ).
% dvd_div_mult
thf(fact_2870_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ) ).
% div_mult_swap
thf(fact_2871_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( divide_divide @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_2872_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B3 @ C2 ) @ A4 )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_2873_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 )
=> ( dvd_dvd @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_2874_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,D3: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( dvd_dvd @ A @ D3 @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ ( divide_divide @ A @ C2 @ D3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ D3 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_2875_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B3 @ A4 )
= ( divide_divide @ A @ C2 @ A4 ) )
= ( B3 = C2 ) ) ) ) ).
% unit_div_cancel
thf(fact_2876_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_2877_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ ( divide_divide @ A @ C2 @ B3 ) )
= ( dvd_dvd @ A @ A4 @ C2 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_2878_mod__0__imp__dvd,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A4: A,B3: A] :
( ( ( modulo_modulo @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ B3 @ A4 ) ) ) ).
% mod_0_imp_dvd
thf(fact_2879_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B4: A] :
( ( modulo_modulo @ A @ B4 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_2880_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A4: A,B3: A] :
( ( ( modulo_modulo @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ B3 @ A4 ) ) ) ).
% mod_eq_0_iff_dvd
thf(fact_2881_dvd__neg__div,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A4 ) @ B3 )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% dvd_neg_div
thf(fact_2882_dvd__div__neg,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( divide_divide @ A @ A4 @ ( uminus_uminus @ A @ B3 ) )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A4 @ B3 ) ) ) ) ) ).
% dvd_div_neg
thf(fact_2883_mod__eq__dvd__iff,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( ( modulo_modulo @ A @ A4 @ C2 )
= ( modulo_modulo @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ C2 @ ( minus_minus @ A @ A4 @ B3 ) ) ) ) ).
% mod_eq_dvd_iff
thf(fact_2884_dvd__minus__mod,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B3: A,A4: A] : ( dvd_dvd @ A @ B3 @ ( minus_minus @ A @ A4 @ ( modulo_modulo @ A @ A4 @ B3 ) ) ) ) ).
% dvd_minus_mod
thf(fact_2885_nat__dvd__not__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) ).
% nat_dvd_not_less
thf(fact_2886_dvd__pos__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 ) ) ) ).
% dvd_pos_nat
thf(fact_2887_dvd__minus__self,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ M2 ) )
= ( ( ord_less @ nat @ N @ M2 )
| ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% dvd_minus_self
thf(fact_2888_zdvd__mono,axiom,
! [K3: int,M2: int,T2: int] :
( ( K3
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ M2 @ T2 )
= ( dvd_dvd @ int @ ( times_times @ int @ K3 @ M2 ) @ ( times_times @ int @ K3 @ T2 ) ) ) ) ).
% zdvd_mono
thf(fact_2889_subseqs__refl,axiom,
! [A: $tType,Xs: list @ A] : ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs_refl
thf(fact_2890_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X4: A] : ( P @ ( times_times @ A @ L @ X4 ) ) )
= ( ? [X4: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X4 @ ( zero_zero @ A ) ) )
& ( P @ X4 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_2891_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ~ ( ( A4
!= ( zero_zero @ A ) )
=> ! [C5: A] :
( B3
!= ( times_times @ A @ A4 @ C5 ) ) ) ) ) ).
% unit_dvdE
thf(fact_2892_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D2: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D2 )
=> ! [X8: A,K2: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X8 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K2 @ D2 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_2893_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D2: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D2 )
=> ! [X8: A,K2: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X8 @ T2 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X8 @ ( times_times @ A @ K2 @ D2 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_2894_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( ( divide_divide @ A @ B3 @ A4 )
= C2 )
= ( B3
= ( times_times @ A @ C2 @ A4 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_2895_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 )
= ( dvd_dvd @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_2896_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ( dvd_dvd @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_2897_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C2: A,B3: A,D3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ C2 @ D3 )
=> ( ( ( divide_divide @ A @ B3 @ A4 )
= ( divide_divide @ A @ D3 @ C2 ) )
= ( ( times_times @ A @ B3 @ C2 )
= ( times_times @ A @ A4 @ D3 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_2898_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_2899_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_2900_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A4 @ ( divide_divide @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_2901_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ) ).
% unit_div_commute
thf(fact_2902_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A4 @ B3 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_2903_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( A4
= ( divide_divide @ A @ C2 @ B3 ) )
= ( ( times_times @ A @ A4 @ B3 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_2904_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A4 @ B3 )
= C2 )
= ( A4
= ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_2905_dvd__imp__le,axiom,
! [K3: nat,N: nat] :
( ( dvd_dvd @ nat @ K3 @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ K3 @ N ) ) ) ).
% dvd_imp_le
thf(fact_2906_nat__mult__dvd__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K3 @ M2 ) @ ( times_times @ nat @ K3 @ N ) )
= ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_2907_dvd__mult__cancel,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K3 @ M2 ) @ ( times_times @ nat @ K3 @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_2908_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K3: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K3 ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K3 @ N ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_2909_mod__greater__zero__iff__not__dvd,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M2 @ N ) )
= ( ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_2910_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A,N: nat] :
( ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A4 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A4 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_2911_Cons__in__subseqsD,axiom,
! [A: $tType,Y: A,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Y @ Ys ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_2912_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ B3 @ A4 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_2913_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A4 @ ( times_times @ A @ A4 @ B3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B3 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_2914_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ~ ( ( A4
!= ( zero_zero @ A ) )
=> ! [B5: A] :
( ( B5
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B5 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A4 )
= B5 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B5 )
= A4 )
=> ( ( ( times_times @ A @ A4 @ B5 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A4 )
!= ( times_times @ A @ C2 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_2915_subseqs__distinctD,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( ( distinct @ A @ Xs )
=> ( distinct @ A @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_2916_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X3: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X3
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X3 @ ( power_power @ A @ X3 @ N ) ) ) ) ).
% dvd_power
thf(fact_2917_dvd__mult__cancel2,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M2 ) @ M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_2918_dvd__mult__cancel1,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M2 @ N ) @ M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_2919_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K3: nat,N: nat] :
( ( ord_less_eq @ nat @ K3 @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K3 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K3 ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_2920_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_2921_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ R2 @ N )
=> ( ( ord_less_eq @ nat @ R2 @ M2 )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M2 @ R2 ) )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_2922_power__dvd__imp__le,axiom,
! [I: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M2 ) @ ( power_power @ nat @ I @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_2923_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K3: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K3 ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_2924_take__bit__nat__eq__self,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 )
= M2 ) ) ).
% take_bit_nat_eq_self
thf(fact_2925_take__bit__nat__less__exp,axiom,
! [N: nat,M2: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_nat_less_exp
thf(fact_2926_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M2: nat] :
( ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 )
= M2 )
= ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_2927_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_2928_take__bit__nat__less__self__iff,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 ) @ M2 )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M2 ) ) ).
% take_bit_nat_less_self_iff
thf(fact_2929_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A4: A] :
( ! [A6: A] :
( ( ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: A,B5: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% bits_induct
thf(fact_2930_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_2931_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_2932_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A4 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_2933_take__bit__numeral__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K3: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K3 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K3 ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_2934_take__bit__int__less__eq,axiom,
! [N: nat,K3: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K3 ) @ ( minus_minus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_2935_even__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suc @ ( zero_zero @ nat ) ) )
=> ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_2936_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_2937_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_2938_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,W2: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A4
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_2939_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_2940_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A4 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_2941_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A4 @ N ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% even_power
thf(fact_2942_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,W2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( ( numeral_numeral @ nat @ W2 )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A4
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_2943_even__diff__nat,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ).
% even_diff_nat
thf(fact_2944_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A4 @ B3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% even_mult_iff
thf(fact_2945_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A4 @ B3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A4 @ B3 ) ) ) ) ).
% even_diff
thf(fact_2946_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,W2: num] :
( ( ord_less @ A @ ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_2947_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( power_power @ A @ A4 @ N ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_2948_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ~ ! [B5: A] :
( A4
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) ) ) ) ).
% evenE
thf(fact_2949_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% odd_pos
thf(fact_2950_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A4 ) ) ) ).
% even_two_times_div_two
thf(fact_2951_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A4: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ~ ! [B5: A] :
( A4
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_2952_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A4 @ N ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A4
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_2953_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A4: A,N: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A4 @ N ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ A4 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A4
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_2954_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_2955_Divides_Oadjust__div__eq,axiom,
! [Q4: int,R2: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q4 @ R2 ) )
= ( plus_plus @ int @ Q4
@ ( zero_neq_one_of_bool @ int
@ ( R2
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_2956_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_2957_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M: num,N2: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M @ N2 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M ) ) @ ( unique1321980374590559556d_step @ A @ N2 @ ( unique8689654367752047608divmod @ A @ M @ ( bit0 @ N2 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2958_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_2959_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A4 )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) ) ).
% one_xor_eq
thf(fact_2960_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_2961_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num] :
( ( unique8689654367752047608divmod @ A @ M2 @ one2 )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M2 ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_2962_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_2963_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_2964_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( ( ord_less_eq @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_2965_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_2966_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_2967_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( ( ord_less @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_2968_xor__num_Ocases,axiom,
! [X3: product_prod @ num @ num] :
( ( X3
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N5: num] :
( X3
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N5 ) ) )
=> ( ! [N5: num] :
( X3
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N5 ) ) )
=> ( ! [M6: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M6 ) @ one2 ) )
=> ( ! [M6: num,N5: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M6 ) @ ( bit0 @ N5 ) ) )
=> ( ! [M6: num,N5: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit0 @ M6 ) @ ( bit1 @ N5 ) ) )
=> ( ! [M6: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M6 ) @ one2 ) )
=> ( ! [M6: num,N5: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M6 ) @ ( bit0 @ N5 ) ) )
=> ~ ! [M6: num,N5: num] :
( X3
!= ( product_Pair @ num @ num @ ( bit1 @ M6 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_2969_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A4: A] :
( ( power_power @ A @ A4 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A4 @ A4 ) @ A4 ) ) ) ).
% power3_eq_cube
thf(fact_2970_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M: num,N2: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_2971_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M: num,N2: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).
% divmod_int_def
thf(fact_2972_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M: num,N2: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ) ) ).
% divmod_def
thf(fact_2973_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K3: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K3 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_2974_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A4 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A4 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) ) ).
% xor_one_eq
thf(fact_2975_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K3: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K3 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_2976_divmod__BitM__2__eq,axiom,
! [M2: num] :
( ( unique8689654367752047608divmod @ int @ ( bitM @ M2 ) @ ( bit0 @ one2 ) )
= ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% divmod_BitM_2_eq
thf(fact_2977_mset__size2elem,axiom,
! [A: $tType,P: multiset @ A,Q4: A,Q7: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q7 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ P )
=> ( P
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q7 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% mset_size2elem
thf(fact_2978_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_2979_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_2980_less__numeral__Suc,axiom,
! [K3: num,N: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( suc @ N ) )
= ( ord_less @ nat @ ( pred_numeral @ K3 ) @ N ) ) ).
% less_numeral_Suc
thf(fact_2981_less__Suc__numeral,axiom,
! [N: nat,K3: num] :
( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K3 ) )
= ( ord_less @ nat @ N @ ( pred_numeral @ K3 ) ) ) ).
% less_Suc_numeral
thf(fact_2982_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% or_numerals(3)
thf(fact_2983_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_2984_mset__le__add__mset__decr__left1,axiom,
! [A: $tType,C2: A,A4: multiset @ A,B3: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A4 ) @ B3 )
=> ( subseteq_mset @ A @ A4 @ B3 ) ) ).
% mset_le_add_mset_decr_left1
thf(fact_2985_mset__union__2__elem,axiom,
! [A: $tType,A4: A,B3: A,C2: A,M5: multiset @ A] :
( ( ( add_mset @ A @ A4 @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( add_mset @ A @ C2 @ M5 ) )
=> ( ( ( ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) )
= M5 )
& ( B3 = C2 ) )
| ( ( A4 = C2 )
& ( ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) )
= M5 ) ) ) ) ).
% mset_union_2_elem
thf(fact_2986_mset__le__add__mset,axiom,
! [A: $tType,X3: A,B2: multiset @ A,C3: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X3 @ B2 ) @ C3 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X3 @ ( zero_zero @ ( multiset @ A ) ) ) @ C3 )
& ( subseteq_mset @ A @ B2 @ C3 ) ) ) ).
% mset_le_add_mset
thf(fact_2987_mset__le__single__cases,axiom,
! [A: $tType,M5: multiset @ A,A4: A] :
( ( subseteq_mset @ A @ M5 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( ( M5
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( M5
= ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_le_single_cases
thf(fact_2988_mset__le__add__mset__decr__left2,axiom,
! [A: $tType,C2: A,A4: multiset @ A,B3: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A4 ) @ B3 )
=> ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) ) @ B3 ) ) ).
% mset_le_add_mset_decr_left2
thf(fact_2989_mset__le__subtract__add__mset__right,axiom,
! [A: $tType,X3: A,B2: multiset @ A,X5: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X3 @ B2 ) @ X5 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ X5 @ B2 ) )
& ( subseteq_mset @ A @ B2 @ X5 ) ) ) ).
% mset_le_subtract_add_mset_right
thf(fact_2990_mset__le__subtract__add__mset__left,axiom,
! [A: $tType,X3: A,B2: multiset @ A,X5: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X3 @ B2 ) @ X5 )
=> ( ( subseteq_mset @ A @ B2 @ ( minus_minus @ ( multiset @ A ) @ X5 @ ( add_mset @ A @ X3 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
& ( subseteq_mset @ A @ ( add_mset @ A @ X3 @ ( zero_zero @ ( multiset @ A ) ) ) @ X5 ) ) ) ).
% mset_le_subtract_add_mset_left
thf(fact_2991_mset__unplusm__dist__cases2,axiom,
! [A: $tType,B2: multiset @ A,C3: multiset @ A,S3: A,A3: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ B2 @ C3 )
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 ) )
=> ( ( ( B2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C3 ) ) )
=> ~ ( ( C3
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ B2 @ ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases2
thf(fact_2992_mset__unplusm__dist__cases,axiom,
! [A: $tType,S3: A,A3: multiset @ A,B2: multiset @ A,C3: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 )
= ( plus_plus @ ( multiset @ A ) @ B2 @ C3 ) )
=> ( ( ( B2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C3 ) ) )
=> ~ ( ( C3
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ B2 @ ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases
thf(fact_2993_mset__single__cases2_H,axiom,
! [A: $tType,S3: A,C2: multiset @ A,R7: A,C8: multiset @ A] :
( ( ( add_mset @ A @ S3 @ C2 )
= ( add_mset @ A @ R7 @ C8 ) )
=> ( ( ( S3 = R7 )
=> ( C2 != C8 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C8
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases2'
thf(fact_2994_mset__single__cases2,axiom,
! [A: $tType,S3: A,C2: multiset @ A,R7: A,C8: multiset @ A] :
( ( ( add_mset @ A @ S3 @ C2 )
= ( add_mset @ A @ R7 @ C8 ) )
=> ( ( ( S3 = R7 )
=> ( C2 != C8 ) )
=> ~ ( ( C8
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases2
thf(fact_2995_mset__single__cases_H,axiom,
! [A: $tType,S3: A,C2: multiset @ A,R7: A,C8: multiset @ A] :
( ( ( add_mset @ A @ S3 @ C2 )
= ( add_mset @ A @ R7 @ C8 ) )
=> ( ( ( S3 = R7 )
=> ( C2 != C8 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C8
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases'
thf(fact_2996_mset__single__cases,axiom,
! [A: $tType,S3: A,C2: multiset @ A,R7: A,C8: multiset @ A] :
( ( ( add_mset @ A @ S3 @ C2 )
= ( add_mset @ A @ R7 @ C8 ) )
=> ( ( ( S3 = R7 )
=> ( C2 != C8 ) )
=> ~ ( ( C8
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C8 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases
thf(fact_2997_fact__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K3: num] :
( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ K3 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K3 ) @ ( semiring_char_0_fact @ A @ ( pred_numeral @ K3 ) ) ) ) ) ).
% fact_numeral
thf(fact_2998_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_2999_mset__size__le1__cases,axiom,
! [A: $tType,M5: multiset @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ M5 ) @ ( suc @ ( zero_zero @ nat ) ) )
=> ( ( M5
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [M6: A] :
( M5
!= ( add_mset @ A @ M6 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size_le1_cases
thf(fact_3000_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K3: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K3 ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_3001_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_3002_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_3003_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M: nat,N2: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M @ N2 ) @ ( modulo_modulo @ nat @ M @ N2 ) ) ) ) ).
% divmod_nat_def
thf(fact_3004_mset__zip__take__Cons__drop__twice,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,J: nat,X3: 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 @ X3 @ ( 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 @ X3 @ Y ) @ ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ) ).
% mset_zip_take_Cons_drop_twice
thf(fact_3005_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( 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 @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_3006_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% mask_nat_positive_iff
thf(fact_3007_size__mset,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( multiset @ A ) @ ( mset @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% size_mset
thf(fact_3008_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(3)
thf(fact_3009_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(6)
thf(fact_3010_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X3: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X3 ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X3 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(4)
thf(fact_3011_ex__mset__zip__left,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xs4: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( mset @ A @ Xs4 )
= ( mset @ A @ Xs ) )
=> ? [Ys4: list @ B] :
( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ A ) @ Xs4 ) )
& ( ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs4 @ Ys4 ) )
= ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ) ).
% ex_mset_zip_left
thf(fact_3012_mset__eq__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% mset_eq_length
thf(fact_3013_mset__eq__setD,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) )
=> ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys ) ) ) ).
% mset_eq_setD
thf(fact_3014_mset__eq__imp__distinct__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) )
=> ( ( distinct @ A @ Xs )
= ( distinct @ A @ Ys ) ) ) ).
% mset_eq_imp_distinct_iff
thf(fact_3015_set__eq__iff__mset__eq__distinct,axiom,
! [A: $tType,X3: list @ A,Y: list @ A] :
( ( distinct @ A @ X3 )
=> ( ( distinct @ A @ Y )
=> ( ( ( set2 @ A @ X3 )
= ( set2 @ A @ Y ) )
= ( ( mset @ A @ X3 )
= ( mset @ A @ Y ) ) ) ) ) ).
% set_eq_iff_mset_eq_distinct
thf(fact_3016_less__mask,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ) ).
% less_mask
thf(fact_3017_mset__swap,axiom,
! [A: $tType,I: nat,Ls3: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls3 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Ls3 ) )
=> ( ( mset @ A @ ( list_update @ A @ ( list_update @ A @ Ls3 @ J @ ( nth @ A @ Ls3 @ I ) ) @ I @ ( nth @ A @ Ls3 @ J ) ) )
= ( mset @ A @ Ls3 ) ) ) ) ).
% mset_swap
thf(fact_3018_mask__nat__less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% mask_nat_less_exp
thf(fact_3019_mset__update,axiom,
! [A: $tType,I: nat,Ls3: list @ A,V2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls3 ) )
=> ( ( mset @ A @ ( list_update @ A @ Ls3 @ I @ V2 ) )
= ( add_mset @ A @ V2 @ ( minus_minus @ ( multiset @ A ) @ ( mset @ A @ Ls3 ) @ ( add_mset @ A @ ( nth @ A @ Ls3 @ I ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% mset_update
thf(fact_3020_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N2: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_3021_and__int_Opelims,axiom,
! [X3: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X3 @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X3 @ ( insert3 @ int @ ( zero_zero @ int ) @ ( insert3 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert3 @ int @ ( zero_zero @ int ) @ ( insert3 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X3 @ ( insert3 @ int @ ( zero_zero @ int ) @ ( insert3 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert3 @ int @ ( zero_zero @ int ) @ ( insert3 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X3 @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_3022_and__int_Opsimps,axiom,
! [K3: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K3 @ L ) )
=> ( ( ( ( member @ int @ K3 @ ( insert3 @ int @ ( zero_zero @ int ) @ ( insert3 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert3 @ int @ ( zero_zero @ int ) @ ( insert3 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K3 @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K3 @ ( insert3 @ int @ ( zero_zero @ int ) @ ( insert3 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert3 @ int @ ( zero_zero @ int ) @ ( insert3 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K3 @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_3023_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A4: A,B3: A,N: nat] :
( ! [J3: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A4 @ ( suc @ J3 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_3024_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_3025_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_3026_diff__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ M2 @ N ) ) ) ).
% diff_numeral_simps(1)
thf(fact_3027_diff__numeral__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ M2 ) ) ) ).
% diff_numeral_simps(4)
thf(fact_3028_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_3029_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_3030_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_3031_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A4: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M2 @ A4 ) @ N )
= ( ( ord_less @ nat @ N @ M2 )
& ( bit_se5641148757651400278ts_bit @ A @ A4 @ N ) ) ) ) ).
% bit_take_bit_iff
thf(fact_3032_neg__numeral__class_Osub__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_sub @ A )
= ( ^ [K4: num,L4: num] : ( minus_minus @ A @ ( numeral_numeral @ A @ K4 ) @ ( numeral_numeral @ A @ L4 ) ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_3033_sub__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M2: num] :
( ( ord_less @ A @ ( neg_numeral_sub @ A @ N @ M2 ) @ ( zero_zero @ A ) )
= ( ord_less @ num @ N @ M2 ) ) ) ).
% sub_negative
thf(fact_3034_sub__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M2 ) )
= ( ord_less @ num @ M2 @ N ) ) ) ).
% sub_positive
thf(fact_3035_bit__imp__take__bit__positive,axiom,
! [N: nat,M2: nat,K3: int] :
( ( ord_less @ nat @ N @ M2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K3 @ N )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M2 @ K3 ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_3036_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Bs: list @ $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( list @ $o ) @ Bs ) )
& ( nth @ $o @ Bs @ N ) ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_3037_int__bit__bound,axiom,
! [K3: int] :
~ ! [N5: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ N5 @ M3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K3 @ M3 )
= ( bit_se5641148757651400278ts_bit @ int @ K3 @ N5 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K3 @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N5 ) ) ) ) ) ).
% int_bit_bound
thf(fact_3038_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [K: int,L3: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L3 ) )
=> ( ( ~ ( ( member @ int @ K @ ( insert3 @ int @ ( zero_zero @ int ) @ ( insert3 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L3 @ ( insert3 @ int @ ( zero_zero @ int ) @ ( insert3 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K @ L3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_3039_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A4 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_3040_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_3041_upto_Opelims,axiom,
! [X3: int,Xa: int,Y: list @ int] :
( ( ( upto @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X3 @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X3 @ Xa )
=> ( Y
= ( cons @ int @ X3 @ ( upto @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X3 @ Xa )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X3 @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_3042_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I5: int,J3: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I5 @ J3 ) )
=> ( ( ( ord_less_eq @ int @ I5 @ J3 )
=> ( P @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ J3 ) )
=> ( P @ I5 @ J3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_3043_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A5 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ N2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_3044_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_3045_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_3046_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A4: A,B3: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A4 @ B3 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A4 ) @ B3 ) ) ) ).
% not_diff_distrib
thf(fact_3047_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A4: A,B3: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A4 @ B3 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A4 ) @ B3 ) ) ) ).
% not_add_distrib
thf(fact_3048_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_3049_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A5: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_3050_disjunctive__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [B3: A,A4: A] :
( ! [N5: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ B3 @ N5 )
=> ( bit_se5641148757651400278ts_bit @ A @ A4 @ N5 ) )
=> ( ( minus_minus @ A @ A4 @ B3 )
= ( bit_se5824344872417868541ns_and @ A @ A4 @ ( bit_ri4277139882892585799ns_not @ A @ B3 ) ) ) ) ) ).
% disjunctive_diff
thf(fact_3051_take__bit__not__eq__mask__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A4 ) )
= ( minus_minus @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A4 ) ) ) ) ).
% take_bit_not_eq_mask_diff
thf(fact_3052_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one2 ) ) ) ).
% num_of_nat.simps(2)
thf(fact_3053_binomial__code,axiom,
( binomial
= ( ^ [N2: nat,K4: nat] : ( if @ nat @ ( ord_less @ nat @ N2 @ K4 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) ) @ ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K4 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ K4 ) @ ( one_one @ nat ) ) @ N2 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K4 ) ) ) ) ) ) ).
% binomial_code
thf(fact_3054_set__decode__plus__power__2,axiom,
! [N: nat,Z2: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z2 ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z2 ) )
= ( insert3 @ nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_3055_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A4 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_3056_size__diff__se,axiom,
! [A: $tType,T2: A,S: multiset @ A] :
( ( member @ A @ T2 @ ( set_mset @ A @ S ) )
=> ( ( size_size @ ( multiset @ A ) @ S )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% size_diff_se
thf(fact_3057_set__mset__empty,axiom,
! [A: $tType] :
( ( set_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% set_mset_empty
thf(fact_3058_set__mset__eq__empty__iff,axiom,
! [A: $tType,M5: multiset @ A] :
( ( ( set_mset @ A @ M5 )
= ( bot_bot @ ( set @ A ) ) )
= ( M5
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% set_mset_eq_empty_iff
thf(fact_3059_set__mset__add__mset__insert,axiom,
! [A: $tType,A4: A,A3: multiset @ A] :
( ( set_mset @ A @ ( add_mset @ A @ A4 @ A3 ) )
= ( insert3 @ A @ A4 @ ( set_mset @ A @ A3 ) ) ) ).
% set_mset_add_mset_insert
thf(fact_3060_binomial__eq__0__iff,axiom,
! [N: nat,K3: nat] :
( ( ( binomial @ N @ K3 )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N @ K3 ) ) ).
% binomial_eq_0_iff
thf(fact_3061_set__mset__mset,axiom,
! [A: $tType,Xs: list @ A] :
( ( set_mset @ A @ ( mset @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_mset_mset
thf(fact_3062_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_3063_zero__less__binomial__iff,axiom,
! [N: nat,K3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K3 ) )
= ( ord_less_eq @ nat @ K3 @ N ) ) ).
% zero_less_binomial_iff
thf(fact_3064_mset__diff__cancel1elem,axiom,
! [A: $tType,A4: A,B2: multiset @ A] :
( ~ ( member @ A @ A4 @ ( set_mset @ A @ B2 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ B2 )
= ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% mset_diff_cancel1elem
thf(fact_3065_mset__un__cases,axiom,
! [A: $tType,A4: A,A3: multiset @ A,B2: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) )
=> ( ~ ( member @ A @ A4 @ ( set_mset @ A @ A3 ) )
=> ( member @ A @ A4 @ ( set_mset @ A @ B2 ) ) ) ) ).
% mset_un_cases
thf(fact_3066_mset__left__cancel__union,axiom,
! [A: $tType,A4: A,A3: multiset @ A,B2: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) )
=> ( ~ ( member @ A @ A4 @ ( set_mset @ A @ A3 ) )
=> ( member @ A @ A4 @ ( set_mset @ A @ B2 ) ) ) ) ).
% mset_left_cancel_union
thf(fact_3067_mset__right__cancel__union,axiom,
! [A: $tType,A4: A,A3: multiset @ A,B2: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) )
=> ( ~ ( member @ A @ A4 @ ( set_mset @ A @ B2 ) )
=> ( member @ A @ A4 @ ( set_mset @ A @ A3 ) ) ) ) ).
% mset_right_cancel_union
thf(fact_3068_ex__Melem__conv,axiom,
! [A: $tType,A3: multiset @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ ( set_mset @ A @ A3 ) ) )
= ( A3
!= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% ex_Melem_conv
thf(fact_3069_binomial__eq__0,axiom,
! [N: nat,K3: nat] :
( ( ord_less @ nat @ N @ K3 )
=> ( ( binomial @ N @ K3 )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_3070_in__multiset__in__set,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set_mset @ A @ ( mset @ A @ Xs ) ) )
= ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_3071_zero__less__binomial,axiom,
! [K3: nat,N: nat] :
( ( ord_less_eq @ nat @ K3 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K3 ) ) ) ).
% zero_less_binomial
thf(fact_3072_mset__right__cancel__elem,axiom,
! [A: $tType,A4: A,A3: multiset @ A,B3: A] :
( ( member @ A @ A4 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( A4 != B3 )
=> ( member @ A @ A4 @ ( set_mset @ A @ A3 ) ) ) ) ).
% mset_right_cancel_elem
thf(fact_3073_mset__left__cancel__elem,axiom,
! [A: $tType,A4: A,B3: A,A3: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 ) ) )
=> ( ( A4 != B3 )
=> ( member @ A @ A4 @ ( set_mset @ A @ A3 ) ) ) ) ).
% mset_left_cancel_elem
thf(fact_3074_prod__decode__aux_Ocases,axiom,
! [X3: product_prod @ nat @ nat] :
~ ! [K: nat,M6: nat] :
( X3
!= ( product_Pair @ nat @ nat @ K @ M6 ) ) ).
% prod_decode_aux.cases
thf(fact_3075_set__mset__single,axiom,
! [A: $tType,B3: A] :
( ( set_mset @ A @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_mset_single
thf(fact_3076_mset__2dist2__cases,axiom,
! [A: $tType,A4: A,B3: A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ A3 )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ B2 )
=> ( ( ( member @ A @ A4 @ ( set_mset @ A @ A3 ) )
=> ~ ( member @ A @ B3 @ ( set_mset @ A @ B2 ) ) )
=> ~ ( ( member @ A @ A4 @ ( set_mset @ A @ B2 ) )
=> ~ ( member @ A @ B3 @ ( set_mset @ A @ A3 ) ) ) ) ) ) ) ).
% mset_2dist2_cases
thf(fact_3077_mset__union__subset__s,axiom,
! [A: $tType,A4: A,B2: multiset @ A,C3: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ B2 ) @ C3 )
=> ( ( member @ A @ A4 @ ( set_mset @ A @ C3 ) )
& ( subseteq_mset @ A @ B2 @ C3 ) ) ) ).
% mset_union_subset_s
thf(fact_3078_mset__le__mono__add__single,axiom,
! [A: $tType,A4: A,Ys: multiset @ A,B3: A,Ws: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ Ys ) )
=> ( ( member @ A @ B3 @ ( set_mset @ A @ Ws ) )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ Ys @ Ws ) ) ) ) ).
% mset_le_mono_add_single
thf(fact_3079_nth__mem__mset,axiom,
! [A: $tType,I: nat,Ls3: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls3 ) )
=> ( member @ A @ ( nth @ A @ Ls3 @ I ) @ ( set_mset @ A @ ( mset @ A @ Ls3 ) ) ) ) ).
% nth_mem_mset
thf(fact_3080_mset__contains__eq,axiom,
! [A: $tType,M2: A,M5: multiset @ A] :
( ( member @ A @ M2 @ ( set_mset @ A @ M5 ) )
= ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ M2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ M5 @ ( add_mset @ A @ M2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= M5 ) ) ).
% mset_contains_eq
thf(fact_3081_mset__union__diff__comm,axiom,
! [A: $tType,T2: A,S: multiset @ A,T4: multiset @ A] :
( ( member @ A @ T2 @ ( set_mset @ A @ S ) )
=> ( ( plus_plus @ ( multiset @ A ) @ T4 @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ T4 @ S ) @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_union_diff_comm
thf(fact_3082_diff__union__single__conv2,axiom,
! [A: $tType,A4: A,J4: multiset @ A,I3: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ J4 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ J4 @ I3 ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ J4 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ I3 ) ) ) ).
% diff_union_single_conv2
thf(fact_3083_mset__un__single__un__cases,axiom,
! [A: $tType,A4: A,A3: multiset @ A,B2: multiset @ A,C3: multiset @ A] :
( ( ( add_mset @ A @ A4 @ A3 )
= ( plus_plus @ ( multiset @ A ) @ B2 @ C3 ) )
=> ( ( ( member @ A @ A4 @ ( set_mset @ A @ B2 ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C3 ) ) )
=> ~ ( ( member @ A @ A4 @ ( set_mset @ A @ C3 ) )
=> ( A3
!= ( plus_plus @ ( multiset @ A ) @ B2 @ ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_un_single_un_cases
thf(fact_3084_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A4: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A4 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_3085_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( numeral_numeral @ nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_3086_choose__reduce__nat,axiom,
! [N: nat,K3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ( binomial @ N @ K3 )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K3 ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_3087_times__binomial__minus1__eq,axiom,
! [K3: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ( times_times @ nat @ K3 @ ( binomial @ N @ K3 ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_3088_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F7: nat > A > A,A5: nat,B4: nat,Acc3: A] : ( if @ A @ ( ord_less @ nat @ B4 @ A5 ) @ Acc3 @ ( set_fo6178422350223883121st_nat @ A @ F7 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B4 @ ( F7 @ A5 @ Acc3 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_3089_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X3: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X3 @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X3 @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X3 @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_3090_mset__size1elem,axiom,
! [A: $tType,P: multiset @ A,Q4: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( one_one @ nat ) )
=> ( ( member @ A @ Q4 @ ( set_mset @ A @ P ) )
=> ( P
= ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size1elem
thf(fact_3091_size__Diff1__less,axiom,
! [A: $tType,X3: A,M5: multiset @ A] :
( ( member @ A @ X3 @ ( set_mset @ A @ M5 ) )
=> ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M5 @ ( add_mset @ A @ X3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( size_size @ ( multiset @ A ) @ M5 ) ) ) ).
% size_Diff1_less
thf(fact_3092_size__Diff2__less,axiom,
! [A: $tType,X3: A,M5: multiset @ A,Y: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ M5 ) )
=> ( ( member @ A @ Y @ ( set_mset @ A @ M5 ) )
=> ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M5 @ ( add_mset @ A @ X3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( size_size @ ( multiset @ A ) @ M5 ) ) ) ) ).
% size_Diff2_less
thf(fact_3093_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N2: nat,A5: A] : ( times_times @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_3094_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_3095_num__of__nat__plus__distrib,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M2 @ N ) )
= ( plus_plus @ num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_3096_binomial__addition__formula,axiom,
! [N: nat,K3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( binomial @ N @ ( suc @ K3 ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K3 ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K3 ) ) ) ) ).
% binomial_addition_formula
thf(fact_3097_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K3: nat,N: nat] :
( ( ord_less_eq @ nat @ K3 @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K3 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K3 ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_3098_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K3: nat,N: nat] :
( ( ord_less_eq @ nat @ K3 @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K3 ) ) ) ) ) ) ).
% fact_binomial
thf(fact_3099_at__most__one__mset__mset__diff,axiom,
! [A: $tType,A4: A,M5: multiset @ A] :
( ~ ( member @ A @ A4 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M5 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M5 @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( set_mset @ A @ M5 ) @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_3100_prod__decode__aux_Oelims,axiom,
! [X3: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X3 @ Xa )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa @ X3 )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X3 @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X3 )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa @ ( suc @ X3 ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_3101_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K4: nat,M: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M @ K4 ) @ ( product_Pair @ nat @ nat @ M @ ( minus_minus @ nat @ K4 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus @ nat @ M @ ( suc @ K4 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_3102_set__mset__replicate__mset__subset,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N @ X3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N @ X3 ) )
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_3103_mult1E,axiom,
! [A: $tType,N7: multiset @ A,M5: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ M5 ) @ ( mult1 @ A @ R2 ) )
=> ~ ! [A6: A,M0: multiset @ A] :
( ( M5
= ( add_mset @ A @ A6 @ M0 ) )
=> ! [K5: multiset @ A] :
( ( N7
= ( plus_plus @ ( multiset @ A ) @ M0 @ K5 ) )
=> ~ ! [B10: A] :
( ( member @ A @ B10 @ ( set_mset @ A @ K5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B10 @ A6 ) @ R2 ) ) ) ) ) ).
% mult1E
thf(fact_3104_mult1I,axiom,
! [A: $tType,M5: multiset @ A,A4: A,M02: multiset @ A,N7: multiset @ A,K6: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
( ( M5
= ( add_mset @ A @ A4 @ M02 ) )
=> ( ( N7
= ( plus_plus @ ( multiset @ A ) @ M02 @ K6 ) )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ ( set_mset @ A @ K6 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A4 ) @ R2 ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ M5 ) @ ( mult1 @ A @ R2 ) ) ) ) ) ).
% mult1I
thf(fact_3105_in__replicate__mset,axiom,
! [A: $tType,X3: A,N: nat,Y: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ ( replicate_mset @ A @ N @ Y ) ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( X3 = Y ) ) ) ).
% in_replicate_mset
thf(fact_3106_not__less__empty,axiom,
! [A: $tType,M5: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M5 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R2 ) ) ).
% not_less_empty
thf(fact_3107_mult1__union,axiom,
! [A: $tType,B2: multiset @ A,D2: multiset @ A,R2: set @ ( product_prod @ A @ A ),C3: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ B2 @ D2 ) @ ( mult1 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C3 @ B2 ) @ ( plus_plus @ ( multiset @ A ) @ C3 @ D2 ) ) @ ( mult1 @ A @ R2 ) ) ) ).
% mult1_union
thf(fact_3108_less__add,axiom,
! [A: $tType,N7: multiset @ A,A4: A,M02: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ ( add_mset @ A @ A4 @ M02 ) ) @ ( mult1 @ A @ R2 ) )
=> ( ? [M7: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M7 @ M02 ) @ ( mult1 @ A @ R2 ) )
& ( N7
= ( add_mset @ A @ A4 @ M7 ) ) )
| ? [K5: multiset @ A] :
( ! [B10: A] :
( ( member @ A @ B10 @ ( set_mset @ A @ K5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B10 @ A4 ) @ R2 ) )
& ( N7
= ( plus_plus @ ( multiset @ A ) @ M02 @ K5 ) ) ) ) ) ).
% less_add
thf(fact_3109_one__step__implies__mult,axiom,
! [A: $tType,J4: multiset @ A,K6: multiset @ A,R2: set @ ( product_prod @ A @ A ),I3: multiset @ A] :
( ( J4
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set_mset @ A @ K6 ) )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ ( set_mset @ A @ J4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Xa3 ) @ R2 ) ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I3 @ K6 ) @ ( plus_plus @ ( multiset @ A ) @ I3 @ J4 ) ) @ ( mult @ A @ R2 ) ) ) ) ).
% one_step_implies_mult
thf(fact_3110_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F3: nat > A > A,A4: nat,B3: nat,Acc4: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B3 @ Acc4 ) ) ) )
=> ( ( ( ord_less @ nat @ B3 @ A4 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A4 @ B3 @ Acc4 )
= Acc4 ) )
& ( ~ ( ord_less @ nat @ B3 @ A4 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A4 @ B3 @ Acc4 )
= ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B3 @ ( F3 @ A4 @ Acc4 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_3111_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X3: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X3 @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X3 @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X3 @ Xa @ Xc ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_3112_set__sorted__list__of__multiset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M5: multiset @ A] :
( ( set2 @ A @ ( linord6283353356039996273ltiset @ A @ M5 ) )
= ( set_mset @ A @ M5 ) ) ) ).
% set_sorted_list_of_multiset
thf(fact_3113_cppi,axiom,
! [D2: int,P: int > $o,P3: int > $o,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less @ int @ Z5 @ X )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ! [X: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( plus_plus @ int @ X @ D2 ) ) ) )
=> ( ! [X: int,K: int] :
( ( P3 @ X )
= ( P3 @ ( minus_minus @ int @ X @ ( times_times @ int @ K @ D2 ) ) ) )
=> ( ( ? [X9: int] : ( P @ X9 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P3 @ X4 ) )
| ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ? [Y5: int] :
( ( member @ int @ Y5 @ A3 )
& ( P @ ( minus_minus @ int @ Y5 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_3114_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3115_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( ~ ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3116_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_3117_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( insert3 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A4 = B3 )
& ( B3 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_3118_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] :
( ( set_or1337092689740270186AtMost @ A @ A4 @ A4 )
= ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_3119_atLeastAtMost__upto,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I4: int,J2: int] : ( set2 @ int @ ( upto @ I4 @ J2 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_3120_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( A4 = B3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_3121_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X3: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X3 @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_3122_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ~ ( ord_less_eq @ A @ A4 @ B3 )
| ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less_eq @ A @ B3 @ D3 )
& ( ( ord_less @ A @ C2 @ A4 )
| ( ord_less @ A @ B3 @ D3 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3123_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_3124_bset_I1_J,axiom,
! [D2: int,B2: set @ int,P: int > $o,Q: int > $o] :
( ! [X: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( minus_minus @ int @ X @ D2 ) ) ) )
=> ( ! [X: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( Q @ X )
=> ( Q @ ( minus_minus @ int @ X @ D2 ) ) ) )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
=> ( ( P @ ( minus_minus @ int @ X8 @ D2 ) )
& ( Q @ ( minus_minus @ int @ X8 @ D2 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_3125_bset_I2_J,axiom,
! [D2: int,B2: set @ int,P: int > $o,Q: int > $o] :
( ! [X: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( minus_minus @ int @ X @ D2 ) ) ) )
=> ( ! [X: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( Q @ X )
=> ( Q @ ( minus_minus @ int @ X @ D2 ) ) ) )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
=> ( ( P @ ( minus_minus @ int @ X8 @ D2 ) )
| ( Q @ ( minus_minus @ int @ X8 @ D2 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_3126_aset_I1_J,axiom,
! [D2: int,A3: set @ int,P: int > $o,Q: int > $o] :
( ! [X: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( plus_plus @ int @ X @ D2 ) ) ) )
=> ( ! [X: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( Q @ X )
=> ( Q @ ( plus_plus @ int @ X @ D2 ) ) ) )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ A3 )
=> ( X8
!= ( minus_minus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ( P @ X8 )
& ( Q @ X8 ) )
=> ( ( P @ ( plus_plus @ int @ X8 @ D2 ) )
& ( Q @ ( plus_plus @ int @ X8 @ D2 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_3127_aset_I2_J,axiom,
! [D2: int,A3: set @ int,P: int > $o,Q: int > $o] :
( ! [X: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( plus_plus @ int @ X @ D2 ) ) ) )
=> ( ! [X: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( Q @ X )
=> ( Q @ ( plus_plus @ int @ X @ D2 ) ) ) )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ A3 )
=> ( X8
!= ( minus_minus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ( P @ X8 )
| ( Q @ X8 ) )
=> ( ( P @ ( plus_plus @ int @ X8 @ D2 ) )
| ( Q @ ( plus_plus @ int @ X8 @ D2 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_3128_aset_I10_J,axiom,
! [D3: int,D2: int,A3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ A3 )
=> ( X8
!= ( minus_minus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X8 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X8 @ D2 ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_3129_aset_I9_J,axiom,
! [D3: int,D2: int,A3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ A3 )
=> ( X8
!= ( minus_minus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X8 @ T2 ) )
=> ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X8 @ D2 ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_3130_bset_I10_J,axiom,
! [D3: int,D2: int,B2: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X8 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X8 @ D2 ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_3131_bset_I9_J,axiom,
! [D3: int,D2: int,B2: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X8 @ T2 ) )
=> ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X8 @ D2 ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_3132_periodic__finite__ex,axiom,
! [D3: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X: int,K: int] :
( ( P @ X )
= ( P @ ( minus_minus @ int @ X @ ( times_times @ int @ K @ D3 ) ) ) )
=> ( ( ? [X9: int] : ( P @ X9 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P @ X4 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_3133_aset_I7_J,axiom,
! [D2: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ A3 )
=> ( X8
!= ( minus_minus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X8 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X8 @ D2 ) ) ) ) ) ).
% aset(7)
thf(fact_3134_aset_I5_J,axiom,
! [D2: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ A3 )
=> ( X8
!= ( minus_minus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ord_less @ int @ X8 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X8 @ D2 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_3135_aset_I4_J,axiom,
! [D2: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ A3 )
=> ( X8
!= ( minus_minus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( X8 != T2 )
=> ( ( plus_plus @ int @ X8 @ D2 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_3136_aset_I3_J,axiom,
! [D2: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ A3 )
=> ( X8
!= ( minus_minus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( X8 = T2 )
=> ( ( plus_plus @ int @ X8 @ D2 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_3137_bset_I7_J,axiom,
! [D2: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( member @ int @ T2 @ B2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X8 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X8 @ D2 ) ) ) ) ) ) ).
% bset(7)
thf(fact_3138_bset_I5_J,axiom,
! [D2: int,B2: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ord_less @ int @ X8 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X8 @ D2 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_3139_bset_I4_J,axiom,
! [D2: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( member @ int @ T2 @ B2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( X8 != T2 )
=> ( ( minus_minus @ int @ X8 @ D2 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_3140_bset_I3_J,axiom,
! [D2: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( X8 = T2 )
=> ( ( minus_minus @ int @ X8 @ D2 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_3141_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A33: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A1 @ ( product_Pair @ nat @ A @ A22 @ A33 ) ) ) )
=> ( ! [F5: nat > A > A,A6: nat,B5: nat,Acc: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F5 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B5 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B5 @ A6 )
=> ( P @ F5 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B5 @ ( F5 @ A6 @ Acc ) ) )
=> ( P @ F5 @ A6 @ B5 @ Acc ) ) )
=> ( P @ A0 @ A1 @ A22 @ A33 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_3142_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I4: int,J2: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J2 @ I4 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert3 @ int @ I4 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J2 ) ) ) ) ) ).
% simp_from_to
thf(fact_3143_bset_I6_J,axiom,
! [D2: int,B2: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ord_less_eq @ int @ X8 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X8 @ D2 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_3144_bset_I8_J,axiom,
! [D2: int,T2: int,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ B2 )
=> ( X8
!= ( plus_plus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X8 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X8 @ D2 ) ) ) ) ) ) ).
% bset(8)
thf(fact_3145_aset_I6_J,axiom,
! [D2: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ A3 )
=> ( X8
!= ( minus_minus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ord_less_eq @ int @ X8 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X8 @ D2 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_3146_aset_I8_J,axiom,
! [D2: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ! [X8: int] :
( ! [Xa4: int] :
( ( member @ int @ Xa4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb4: int] :
( ( member @ int @ Xb4 @ A3 )
=> ( X8
!= ( minus_minus @ int @ Xb4 @ Xa4 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X8 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X8 @ D2 ) ) ) ) ) ).
% aset(8)
thf(fact_3147_cpmi,axiom,
! [D2: int,P: int > $o,P3: int > $o,B2: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ? [Z5: int] :
! [X: int] :
( ( ord_less @ int @ X @ Z5 )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ! [X: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B2 )
=> ( X
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( P @ X )
=> ( P @ ( minus_minus @ int @ X @ D2 ) ) ) )
=> ( ! [X: int,K: int] :
( ( P3 @ X )
= ( P3 @ ( minus_minus @ int @ X @ ( times_times @ int @ K @ D2 ) ) ) )
=> ( ( ? [X9: int] : ( P @ X9 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P3 @ X4 ) )
| ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ? [Y5: int] :
( ( member @ int @ Y5 @ B2 )
& ( P @ ( plus_plus @ int @ Y5 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_3148_mult__implies__one__step,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),M5: multiset @ A,N7: multiset @ A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M5 @ N7 ) @ ( mult @ A @ R2 ) )
=> ? [I8: multiset @ A,J5: multiset @ A] :
( ( N7
= ( plus_plus @ ( multiset @ A ) @ I8 @ J5 ) )
& ? [K5: multiset @ A] :
( ( M5
= ( plus_plus @ ( multiset @ A ) @ I8 @ K5 ) )
& ( J5
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X8: A] :
( ( member @ A @ X8 @ ( set_mset @ A @ K5 ) )
=> ? [Xa4: A] :
( ( member @ A @ Xa4 @ ( set_mset @ A @ J5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X8 @ Xa4 ) @ R2 ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_3149_map__tailrec__rev_Opelims,axiom,
! [A: $tType,B: $tType,X3: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X3 @ 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 ) ) @ X3 @ ( 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 ) ) @ X3 @ ( 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 @ X3 @ As @ ( cons @ B @ ( X3 @ 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_3150_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert3 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_3151_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set @ A,X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert3 @ A @ X3 @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_3152_less__eq,axiom,
! [M2: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% less_eq
thf(fact_3153_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_3154_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_3155_PowI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( member @ ( set @ A ) @ A3 @ ( pow @ A @ B2 ) ) ) ).
% PowI
thf(fact_3156_Pow__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A3 @ ( pow @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% Pow_iff
thf(fact_3157_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_3158_Pow__Int__eq,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( pow @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow @ A @ A3 ) @ ( pow @ A @ B2 ) ) ) ).
% Pow_Int_eq
thf(fact_3159_lenlex__transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( trans @ ( list @ A ) @ ( lenlex @ A @ R2 ) ) ) ).
% lenlex_transI
thf(fact_3160_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_3161_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_3162_Pow__singleton__iff,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A] :
( ( ( pow @ A @ X5 )
= ( insert3 @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
& ( Y6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_3163_Pow__empty,axiom,
! [A: $tType] :
( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert3 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_3164_card__Diff__insert,axiom,
! [A: $tType,A4: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ~ ( member @ A @ A4 @ B2 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ B2 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_3165_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A4: A,B3: A] :
( ( ( finite_card @ A @ ( insert3 @ A @ A4 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( A4 != B3 ) ) ).
% card_doubleton_eq_2_iff
thf(fact_3166_lexn__transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( trans @ A @ R2 )
=> ( trans @ ( list @ A ) @ ( lexn @ A @ R2 @ N ) ) ) ).
% lexn_transI
thf(fact_3167_Pow__top,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( set @ A ) @ A3 @ ( pow @ A @ A3 ) ) ).
% Pow_top
thf(fact_3168_listrel__trans,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( trans @ ( list @ A ) @ ( listrel @ A @ A @ R2 ) ) ) ).
% listrel_trans
thf(fact_3169_lex__transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( trans @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ).
% lex_transI
thf(fact_3170_lexord__transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( trans @ ( list @ A ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_transI
thf(fact_3171_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A,Y5: A,Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z6 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z6 ) @ R5 ) ) ) ) ) ).
% trans_def
thf(fact_3172_transI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X: A,Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R2 ) ) )
=> ( trans @ A @ R2 ) ) ).
% transI
thf(fact_3173_transE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A,Z2: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 ) ) ) ) ).
% transE
thf(fact_3174_transD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A,Z2: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 ) ) ) ) ).
% transD
thf(fact_3175_Pow__bottom,axiom,
! [A: $tType,B2: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B2 ) ) ).
% Pow_bottom
thf(fact_3176_PowD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A3 @ ( pow @ A @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% PowD
thf(fact_3177_Pow__mono,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A3 ) @ ( pow @ A @ B2 ) ) ) ).
% Pow_mono
thf(fact_3178_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_3179_trans__Int,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( trans @ A @ S3 )
=> ( trans @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ).
% trans_Int
thf(fact_3180_Un__Pow__subset,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow @ A @ A3 ) @ ( pow @ A @ B2 ) ) @ ( pow @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% Un_Pow_subset
thf(fact_3181_trans__Id,axiom,
! [A: $tType] : ( trans @ A @ ( id2 @ A ) ) ).
% trans_Id
thf(fact_3182_Pow__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( pow @ A @ A3 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_3183_card__insert__le,axiom,
! [A: $tType,A3: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( insert3 @ A @ X3 @ A3 ) ) ) ).
% card_insert_le
thf(fact_3184_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( P @ M ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_3185_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less_eq @ nat @ M @ N )
& ( P @ M ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_3186_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: A,As2: list @ A,Bs: list @ B] :
( ( map_tailrec_rev @ A @ B @ F3 @ ( cons @ A @ A4 @ As2 ) @ Bs )
= ( map_tailrec_rev @ A @ B @ F3 @ As2 @ ( cons @ B @ ( F3 @ A4 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_3187_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F3: A > B,Bs: list @ B] :
( ( map_tailrec_rev @ A @ B @ F3 @ ( nil @ A ) @ Bs )
= Bs ) ).
% map_tailrec_rev.simps(1)
thf(fact_3188_lexord__trans,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lexord @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lexord @ A @ R2 ) )
=> ( ( trans @ A @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Z2 ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_trans
thf(fact_3189_lenlex__trans,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( lenlex @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lenlex @ A @ R2 ) )
=> ( ( trans @ A @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Z2 ) @ ( lenlex @ A @ R2 ) ) ) ) ) ).
% lenlex_trans
thf(fact_3190_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( ( finite_card @ A @ A7 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_3191_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_3192_card__1__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( one_one @ nat ) )
=> ~ ! [X: A] :
( A3
!= ( insert3 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_3193_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_3194_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_3195_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert3 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_3196_atLeastAtMost__insertL,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( insert3 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_3197_atLeastAtMostSuc__conv,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) )
= ( insert3 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_3198_Icc__eq__insert__lb__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ N )
= ( insert3 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_3199_trans__singleton,axiom,
! [A: $tType,A4: A] : ( trans @ A @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_3200_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X3: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y != Xb ) )
=> ~ ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X3 @ As @ ( cons @ B @ ( X3 @ A6 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_3201_card__1__singleton__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X4: A] :
( A3
= ( insert3 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_3202_card__eq__SucD,axiom,
! [A: $tType,A3: set @ A,K3: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K3 ) )
=> ? [B5: A,B6: set @ A] :
( ( A3
= ( insert3 @ A @ B5 @ B6 ) )
& ~ ( member @ A @ B5 @ B6 )
& ( ( finite_card @ A @ B6 )
= K3 )
& ( ( K3
= ( zero_zero @ nat ) )
=> ( B6
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_3203_card__Suc__eq,axiom,
! [A: $tType,A3: set @ A,K3: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K3 ) )
= ( ? [B4: A,B7: set @ A] :
( ( A3
= ( insert3 @ A @ B4 @ B7 ) )
& ~ ( member @ A @ B4 @ B7 )
& ( ( finite_card @ A @ B7 )
= K3 )
& ( ( K3
= ( zero_zero @ nat ) )
=> ( B7
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_3204_card__Diff1__le,axiom,
! [A: $tType,A3: set @ A,X3: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ).
% card_Diff1_le
thf(fact_3205_mult__cancel__add__mset,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),Uu: A,X5: multiset @ A,Y6: multiset @ A] :
( ( trans @ A @ S3 )
=> ( ( irrefl @ A @ S3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( add_mset @ A @ Uu @ X5 ) @ ( add_mset @ A @ Uu @ Y6 ) ) @ ( mult @ A @ S3 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X5 @ Y6 ) @ ( mult @ A @ S3 ) ) ) ) ) ).
% mult_cancel_add_mset
thf(fact_3206_mult__cancel,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),X5: multiset @ A,Z7: multiset @ A,Y6: multiset @ A] :
( ( trans @ A @ S3 )
=> ( ( irrefl @ A @ S3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X5 @ Z7 ) @ ( plus_plus @ ( multiset @ A ) @ Y6 @ Z7 ) ) @ ( mult @ A @ S3 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X5 @ Y6 ) @ ( mult @ A @ S3 ) ) ) ) ) ).
% mult_cancel
thf(fact_3207_card__2__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X4: A,Y5: A] :
( ( S
= ( insert3 @ A @ X4 @ ( insert3 @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X4 != Y5 ) ) ) ) ).
% card_2_iff
thf(fact_3208_mult__cancel__max,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),X5: multiset @ A,Y6: multiset @ A] :
( ( trans @ A @ S3 )
=> ( ( irrefl @ A @ S3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X5 @ Y6 ) @ ( mult @ A @ S3 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X5 @ Y6 ) @ ( minus_minus @ ( multiset @ A ) @ Y6 @ X5 ) ) @ ( mult @ A @ S3 ) ) ) ) ) ).
% mult_cancel_max
thf(fact_3209_card__3__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X4: A,Y5: A,Z6: A] :
( ( S
= ( insert3 @ A @ X4 @ ( insert3 @ A @ Y5 @ ( insert3 @ A @ Z6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X4 != Y5 )
& ( Y5 != Z6 )
& ( X4 != Z6 ) ) ) ) ).
% card_3_iff
thf(fact_3210_odd__card__imp__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) )
=> ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_3211_card__Diff__singleton,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_3212_card__Diff__singleton__if,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_3213_multeqp__code__iff__reflcl__mult,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > A > $o,N7: multiset @ A,M5: multiset @ A] :
( ( irrefl @ A @ R )
=> ( ( trans @ A @ R )
=> ( ! [X: A,Y2: A] :
( ( P @ X @ Y2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R ) )
=> ( ( multeqp_code @ A @ P @ N7 @ M5 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ M5 ) @ ( sup_sup @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R ) @ ( id2 @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% multeqp_code_iff_reflcl_mult
thf(fact_3214_multp__code__iff__mult,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > A > $o,N7: multiset @ A,M5: multiset @ A] :
( ( irrefl @ A @ R )
=> ( ( trans @ A @ R )
=> ( ! [X: A,Y2: A] :
( ( P @ X @ Y2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R ) )
=> ( ( multp_code @ A @ P @ N7 @ M5 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ M5 ) @ ( mult @ A @ R ) ) ) ) ) ) ).
% multp_code_iff_mult
thf(fact_3215_map__tailrec__def,axiom,
! [B: $tType,A: $tType] :
( ( map_tailrec @ A @ B )
= ( ^ [F7: A > B,As4: list @ A] : ( rev @ B @ ( map_tailrec_rev @ A @ B @ F7 @ As4 @ ( nil @ B ) ) ) ) ) ).
% map_tailrec_def
thf(fact_3216_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_3217_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_3218_Nil__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_in_shuffles
thf(fact_3219_shuffles__commutes,axiom,
! [A: $tType] :
( ( shuffles @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] : ( shuffles @ A @ Ys3 @ Xs3 ) ) ) ).
% shuffles_commutes
thf(fact_3220_Cons__in__shuffles__rightI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z2: A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Z2 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_3221_Cons__in__shuffles__leftI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z2: A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs ) @ ( shuffles @ A @ ( cons @ A @ Z2 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_3222_Nil__in__shufflesI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( nil @ A ) )
=> ( ( Ys
= ( nil @ A ) )
=> ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_3223_shuffles_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( shuffles @ A @ ( nil @ A ) @ Ys )
= ( insert3 @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(1)
thf(fact_3224_shuffles_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert3 @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(2)
thf(fact_3225_shufflesE,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys
!= ( nil @ A ) ) )
=> ( ( ( Zs = Ys )
=> ( Xs
!= ( nil @ A ) ) )
=> ( ! [X: A,Xs5: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs5 ) )
=> ! [Z3: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z3 @ Zs4 ) )
=> ( ( X = Z3 )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs5 @ Ys ) ) ) ) )
=> ~ ! [Y2: A,Ys4: list @ A] :
( ( Ys
= ( cons @ A @ Y2 @ Ys4 ) )
=> ! [Z3: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z3 @ Zs4 ) )
=> ( ( Y2 = Z3 )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_3226_length__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( 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_3227_set__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( set2 @ A @ Zs )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_3228_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ A @ Ys )
=> ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( distinct @ A @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_3229_Cons__in__shuffles__iff,axiom,
! [A: $tType,Z2: A,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Z2 @ Zs ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( ( Xs
!= ( nil @ A ) )
& ( ( hd @ A @ Xs )
= Z2 )
& ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ ( tl @ A @ Xs ) @ Ys ) ) )
| ( ( Ys
!= ( nil @ A ) )
& ( ( hd @ A @ Ys )
= Z2 )
& ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ ( tl @ A @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_3230_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I4: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_3231_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert3 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_3232_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_3233_mult__cancel__max0,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),X5: multiset @ A,Y6: multiset @ A] :
( ( trans @ A @ S3 )
=> ( ( irrefl @ A @ S3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X5 @ Y6 ) @ ( mult @ A @ S3 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X5 @ ( inter_mset @ A @ X5 @ Y6 ) ) @ ( minus_minus @ ( multiset @ A ) @ Y6 @ ( inter_mset @ A @ X5 @ Y6 ) ) ) @ ( mult @ A @ S3 ) ) ) ) ) ).
% mult_cancel_max0
thf(fact_3234_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ A4 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) )
=> ( Phi @ A4 @ B3 ) )
=> ( ( ( A4 = B3 )
=> ( Phi @ A4 @ B3 ) )
=> ( Phi @ A4 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_3235_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_3236_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_3237_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_3238_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K3: A] :
( ( ord_less_eq @ A @ L @ K3 )
=> ( ( set_or5935395276787703475ssThan @ A @ K3 @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_3239_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_3240_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X3 ) )
= X3 ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_3241_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert3 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3242_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_3243_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ! [X: A] :
( ! [Y3: A] :
( ( ( Y3 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R2 ) )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A4 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_3244_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [H2: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_3245_wo__rel_OTOTALS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ! [X8: A] :
( ( member @ A @ X8 @ ( field2 @ A @ R2 ) )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X8 @ Xa3 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X8 ) @ R2 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_3246_atMost__Suc,axiom,
! [K3: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K3 ) )
= ( insert3 @ nat @ ( suc @ K3 ) @ ( set_ord_atMost @ nat @ K3 ) ) ) ).
% atMost_Suc
thf(fact_3247_well__order__induct__imp,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ! [X: A] :
( ! [Y3: A] :
( ( ( Y3 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R2 ) )
=> ( ( member @ A @ Y3 @ ( field2 @ A @ R2 ) )
=> ( P @ Y3 ) ) )
=> ( ( member @ A @ X @ ( field2 @ A @ R2 ) )
=> ( P @ X ) ) )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( P @ A4 ) ) ) ) ).
% well_order_induct_imp
thf(fact_3248_atMost__nat__numeral,axiom,
! [K3: num] :
( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K3 ) )
= ( insert3 @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K3 ) ) ) ) ).
% atMost_nat_numeral
thf(fact_3249_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less_eq @ A @ B3 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_3250_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_3251_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_3252_wo__rel_Ocases__Total,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ A4 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( Phi @ A4 @ B3 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R2 )
=> ( Phi @ A4 @ B3 ) )
=> ( Phi @ A4 @ B3 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_3253_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less_eq @ A @ B3 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_3254_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) @ ( insert3 @ A @ A4 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_3255_listset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( listset @ A @ ( nil @ ( set @ A ) ) )
= ( insert3 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listset.simps(1)
thf(fact_3256_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] : ( bNF_We4791949203932849705sMinim @ A @ R2 @ B2 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_3257_wo__rel_Ominim__inField,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) @ ( field2 @ A @ R2 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_3258_wo__rel_Ominim__in,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) @ B2 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_3259_wo__rel_Oequals__minim,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A4 @ B2 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B5 ) @ R2 ) )
=> ( A4
= ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_3260_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( bNF_We4791949203932849705sMinim @ A @ R2 @ B2 @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_3261_wo__rel_OisMinim__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( bNF_We4791949203932849705sMinim @ A @ R2 @ A3 @ B3 )
= ( ( member @ A @ B3 @ A3 )
& ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ X4 ) @ R2 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_3262_wo__rel_Ominim__least,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ B2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R2 @ B2 ) @ B3 ) @ R2 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_3263_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) ) @ R2 )
& ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) @ ( insert3 @ A @ A4 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_3264_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_3265_lists__empty,axiom,
! [A: $tType] :
( ( lists @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert3 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% lists_empty
thf(fact_3266_wo__rel_Omax2__among,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) @ ( insert3 @ A @ A4 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_3267_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X3: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X3 ) @ ( times_times @ A @ C2 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_3268_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,X3: B,A3: set @ B] :
( ( B3
= ( F3 @ X3 ) )
=> ( ( member @ B @ X3 @ A3 )
=> ( member @ A @ B3 @ ( image2 @ B @ A @ F3 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_3269_image__empty,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_3270_empty__is__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image2 @ B @ A @ F3 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_3271_image__is__empty,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( ( image2 @ B @ A @ F3 @ A3 )
= ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_3272_insert__image,axiom,
! [B: $tType,A: $tType,X3: A,A3: set @ A,F3: A > B] :
( ( member @ A @ X3 @ A3 )
=> ( ( insert3 @ B @ ( F3 @ X3 ) @ ( image2 @ A @ B @ F3 @ A3 ) )
= ( image2 @ A @ B @ F3 @ A3 ) ) ) ).
% insert_image
thf(fact_3273_image__insert,axiom,
! [A: $tType,B: $tType,F3: B > A,A4: B,B2: set @ B] :
( ( image2 @ B @ A @ F3 @ ( insert3 @ B @ A4 @ B2 ) )
= ( insert3 @ A @ ( F3 @ A4 ) @ ( image2 @ B @ A @ F3 @ B2 ) ) ) ).
% image_insert
thf(fact_3274_op__conc__empty__img__id,axiom,
! [A: $tType,L5: set @ ( list @ A )] :
( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( nil @ A ) ) @ L5 )
= L5 ) ).
% op_conc_empty_img_id
thf(fact_3275_Field__Union,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( field2 @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ A ) @ ( field2 @ A ) @ R ) ) ) ).
% Field_Union
thf(fact_3276_Cons__in__lists__iff,axiom,
! [A: $tType,X3: A,Xs: list @ A,A3: set @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( lists @ A @ A3 ) )
= ( ( member @ A @ X3 @ A3 )
& ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A3 ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_3277_in__listsI,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X @ A3 ) )
=> ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A3 ) ) ) ).
% in_listsI
thf(fact_3278_lists__Int__eq,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( lists @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
= ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ A3 ) @ ( lists @ A @ B2 ) ) ) ).
% lists_Int_eq
thf(fact_3279_append__in__lists__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,A3: set @ A] :
( ( member @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( lists @ A @ A3 ) )
= ( ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A3 ) )
& ( member @ ( list @ A ) @ Ys @ ( lists @ A @ A3 ) ) ) ) ).
% append_in_lists_conv
thf(fact_3280_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_3281_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D3: A,A4: A,B3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D3 @ B3 ) @ ( minus_minus @ A @ D3 @ A4 ) ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_3282_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_3283_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) )
= ( finite_card @ A @ A3 ) ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_3284_sorted__list__of__set__atMost__Suc,axiom,
! [K3: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K3 ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K3 ) ) @ ( cons @ nat @ ( suc @ K3 ) @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_3285_lists__UNIV,axiom,
! [A: $tType] :
( ( lists @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% lists_UNIV
thf(fact_3286_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A4 ) @ ( times_times @ A @ D3 @ B3 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_3287_rangeI,axiom,
! [A: $tType,B: $tType,F3: B > A,X3: B] : ( member @ A @ ( F3 @ X3 ) @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_3288_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,X3: B] :
( ( B3
= ( F3 @ X3 ) )
=> ( member @ A @ B3 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_3289_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F3: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F3 @ A3 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A3 )
& ( B2
= ( image2 @ B @ A @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_3290_image__subset__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A3 ) @ B2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( member @ A @ ( F3 @ X4 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_3291_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F3: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ~ ! [C9: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C9 @ A3 )
=> ( B2
!= ( image2 @ B @ A @ F3 @ C9 ) ) ) ) ).
% subset_imageE
thf(fact_3292_image__subsetI,axiom,
! [A: $tType,B: $tType,A3: set @ A,F3: A > B,B2: set @ B] :
( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( member @ B @ ( F3 @ X ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A3 ) @ B2 ) ) ).
% image_subsetI
thf(fact_3293_image__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A3 ) @ ( image2 @ A @ B @ F3 @ B2 ) ) ) ).
% image_mono
thf(fact_3294_image__Un,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,B2: set @ B] :
( ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A3 ) @ ( image2 @ B @ A @ F3 @ B2 ) ) ) ).
% image_Un
thf(fact_3295_Sup__SUP__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( A > $o ) )
= ( ^ [S2: set @ ( A > $o ),X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_3296_imageI,axiom,
! [B: $tType,A: $tType,X3: A,A3: set @ A,F3: A > B] :
( ( member @ A @ X3 @ A3 )
=> ( member @ B @ ( F3 @ X3 ) @ ( image2 @ A @ B @ F3 @ A3 ) ) ) ).
% imageI
thf(fact_3297_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F3: B > A,A3: set @ B] :
( ( member @ A @ Z2 @ ( image2 @ B @ A @ F3 @ A3 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( Z2
= ( F3 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_3298_bex__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P: A > $o] :
( ? [X8: A] :
( ( member @ A @ X8 @ ( image2 @ B @ A @ F3 @ A3 ) )
& ( P @ X8 ) )
=> ? [X: B] :
( ( member @ B @ X @ A3 )
& ( P @ ( F3 @ X ) ) ) ) ).
% bex_imageD
thf(fact_3299_image__cong,axiom,
! [B: $tType,A: $tType,M5: set @ A,N7: set @ A,F3: A > B,G: A > B] :
( ( M5 = N7 )
=> ( ! [X: A] :
( ( member @ A @ X @ N7 )
=> ( ( F3 @ X )
= ( G @ X ) ) )
=> ( ( image2 @ A @ B @ F3 @ M5 )
= ( image2 @ A @ B @ G @ N7 ) ) ) ) ).
% image_cong
thf(fact_3300_ball__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( P @ X ) )
=> ! [X8: B] :
( ( member @ B @ X8 @ A3 )
=> ( P @ ( F3 @ X8 ) ) ) ) ).
% ball_imageD
thf(fact_3301_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X3: A,A3: set @ A,B3: B,F3: A > B] :
( ( member @ A @ X3 @ A3 )
=> ( ( B3
= ( F3 @ X3 ) )
=> ( member @ B @ B3 @ ( image2 @ A @ B @ F3 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_3302_refl__on__UNION,axiom,
! [B: $tType,A: $tType,S: set @ A,A3: A > ( set @ B ),R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X: A] :
( ( member @ A @ X @ S )
=> ( refl_on @ B @ ( A3 @ X ) @ ( R2 @ X ) ) )
=> ( refl_on @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ S ) ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% refl_on_UNION
thf(fact_3303_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: set @ B,B2: set @ A] :
( ( ( image2 @ B @ A @ F3 @ A3 )
= B2 )
=> ( ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( pow @ B @ A3 ) )
= ( pow @ A @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_3304_lists_OCons,axiom,
! [A: $tType,A4: A,A3: set @ A,L: list @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( member @ ( list @ A ) @ L @ ( lists @ A @ A3 ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ A4 @ L ) @ ( lists @ A @ A3 ) ) ) ) ).
% lists.Cons
thf(fact_3305_listsE,axiom,
! [A: $tType,X3: A,L: list @ A,A3: set @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ X3 @ L ) @ ( lists @ A @ A3 ) )
=> ~ ( ( member @ A @ X3 @ A3 )
=> ~ ( member @ ( list @ A ) @ L @ ( lists @ A @ A3 ) ) ) ) ).
% listsE
thf(fact_3306_lists_ONil,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( list @ A ) @ ( nil @ A ) @ ( lists @ A @ A3 ) ) ).
% lists.Nil
thf(fact_3307_lists__mono,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( lists @ A @ A3 ) @ ( lists @ A @ B2 ) ) ) ).
% lists_mono
thf(fact_3308_in__listsD,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A3 ) )
=> ! [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X8 @ A3 ) ) ) ).
% in_listsD
thf(fact_3309_in__lists__conv__set,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X4 @ A3 ) ) ) ) ).
% in_lists_conv_set
thf(fact_3310_lists__IntI,axiom,
! [A: $tType,L: list @ A,A3: set @ A,B2: set @ A] :
( ( member @ ( list @ A ) @ L @ ( lists @ A @ A3 ) )
=> ( ( member @ ( list @ A ) @ L @ ( lists @ A @ B2 ) )
=> ( member @ ( list @ A ) @ L @ ( lists @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ).
% lists_IntI
thf(fact_3311_pairwise__imageI,axiom,
! [B: $tType,A: $tType,A3: set @ A,F3: A > B,P: B > B > $o] :
( ! [X: A,Y2: A] :
( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( ( X != Y2 )
=> ( ( ( F3 @ X )
!= ( F3 @ Y2 ) )
=> ( P @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ) ) )
=> ( pairwise @ B @ P @ ( image2 @ A @ B @ F3 @ A3 ) ) ) ).
% pairwise_imageI
thf(fact_3312_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A3 ) @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( pow @ B @ A3 ) ) @ ( pow @ A @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_3313_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( distinct @ A @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_3314_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I3: set @ B,F3: B > A,X3: A] :
( ( I3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ I3 )
=> ( ( F3 @ I5 )
= X3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ I3 ) )
= X3 ) ) ) ) ).
% SUP_eq_const
thf(fact_3315_subset__mset_OcSUP__least,axiom,
! [B: $tType,A: $tType,A3: set @ B,F3: B > ( multiset @ A ),M5: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( subseteq_mset @ A @ ( F3 @ X ) @ M5 ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) @ M5 ) ) ) ).
% subset_mset.cSUP_least
thf(fact_3316_range__subsetD,axiom,
! [B: $tType,A: $tType,F3: B > A,B2: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) @ B2 )
=> ( member @ A @ ( F3 @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_3317_image__Int__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A3 ) @ ( image2 @ B @ A @ F3 @ B2 ) ) ) ).
% image_Int_subset
thf(fact_3318_image__diff__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A3 ) @ ( image2 @ B @ A @ F3 @ B2 ) ) @ ( image2 @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_3319_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_3320_Cons__shuffles__subset1,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_3321_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( shuffles @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_3322_lists_Osimps,axiom,
! [A: $tType,A4: list @ A,A3: set @ A] :
( ( member @ ( list @ A ) @ A4 @ ( lists @ A @ A3 ) )
= ( ( A4
= ( nil @ A ) )
| ? [A5: A,L4: list @ A] :
( ( A4
= ( cons @ A @ A5 @ L4 ) )
& ( member @ A @ A5 @ A3 )
& ( member @ ( list @ A ) @ L4 @ ( lists @ A @ A3 ) ) ) ) ) ).
% lists.simps
thf(fact_3323_lists_Ocases,axiom,
! [A: $tType,A4: list @ A,A3: set @ A] :
( ( member @ ( list @ A ) @ A4 @ ( lists @ A @ A3 ) )
=> ( ( A4
!= ( nil @ A ) )
=> ~ ! [A6: A,L3: list @ A] :
( ( A4
= ( cons @ A @ A6 @ L3 ) )
=> ( ( member @ A @ A6 @ A3 )
=> ~ ( member @ ( list @ A ) @ L3 @ ( lists @ A @ A3 ) ) ) ) ) ) ).
% lists.cases
thf(fact_3324_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_3325_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A3: set @ A,F3: A > B,X3: A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A3 )
=> ( ( F3 @ Y2 )
= ( F3 @ X3 ) ) )
=> ( ( the_elem @ B @ ( image2 @ A @ B @ F3 @ A3 ) )
= ( F3 @ X3 ) ) ) ) ).
% the_elem_image_unique
thf(fact_3326_listrel__refl__on,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A3 @ R2 )
=> ( refl_on @ ( list @ A ) @ ( lists @ A @ A3 ) @ ( listrel @ A @ A @ R2 ) ) ) ).
% listrel_refl_on
thf(fact_3327_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_3328_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_3329_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X3: A,F3: B > A,A3: set @ B] :
( ( ord_less_eq @ A @ X3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) )
= ( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ Y5 @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_3330_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I3: set @ B,C2: A,F3: B > A] :
( ( I3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ I3 )
=> ( ord_less_eq @ A @ C2 @ ( F3 @ I5 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ I3 ) )
= C2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I3 )
=> ( ( F3 @ X4 )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3331_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,M5: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ M5 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ M5 ) ) ) ) ).
% cSUP_least
thf(fact_3332_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: A,X3: B] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F3 @ X3 )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_3333_in__image__insert__iff,axiom,
! [A: $tType,B2: set @ ( set @ A ),X3: A,A3: set @ A] :
( ! [C9: set @ A] :
( ( member @ ( set @ A ) @ C9 @ B2 )
=> ~ ( member @ A @ X3 @ C9 ) )
=> ( ( member @ ( set @ A ) @ A3 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert3 @ A @ X3 ) @ B2 ) )
= ( ( member @ A @ X3 @ A3 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_3334_wo__rel_Omax2__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 )
= B3 ) )
& ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 )
= A4 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_3335_Pow__insert,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( pow @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow @ A @ A3 ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert3 @ A @ A4 ) @ ( pow @ A @ A3 ) ) ) ) ).
% Pow_insert
thf(fact_3336_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list @ A,L2: list @ A,F3: A > B] :
( ( ( size_size @ ( list @ A ) @ L )
= ( size_size @ ( list @ A ) @ L2 ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F3 @ ( nth @ A @ L @ I5 ) )
= ( F3 @ ( nth @ A @ L2 @ I5 ) ) ) )
=> ( ( image2 @ A @ B @ F3 @ ( set2 @ A @ L ) )
= ( image2 @ A @ B @ F3 @ ( set2 @ A @ L2 ) ) ) ) ) ).
% set_image_eq_pointwiseI
thf(fact_3337_in__set__image__conv__nth,axiom,
! [B: $tType,A: $tType,F3: B > A,X3: B,L: list @ B] :
( ( member @ A @ ( F3 @ X3 ) @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ L ) ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ B ) @ L ) )
& ( ( F3 @ ( nth @ B @ L @ I4 ) )
= ( F3 @ X3 ) ) ) ) ) ).
% in_set_image_conv_nth
thf(fact_3338_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert3 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_3339_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N ) )
= ( insert3 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_3340_subset__subseqs,axiom,
! [A: $tType,X5: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( set2 @ A @ Xs ) )
=> ( member @ ( set @ A ) @ X5 @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_3341_wo__rel_Omax2__greater,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 ) ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_3342_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 )
= B3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_3343_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A4 @ B3 )
= A4 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R2 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_3344_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_3345_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_3346_shuffles_Oelims,axiom,
! [A: $tType,X3: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X3 @ Xa )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y
!= ( insert3 @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( insert3 @ ( list @ A ) @ X3 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y2: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ Y2 @ Ys5 ) )
=> ( Y
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y2 @ Ys5 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y2 ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs2 ) @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_3347_Union__image__insert,axiom,
! [A: $tType,B: $tType,F3: B > ( set @ A ),A4: B,B2: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( insert3 @ B @ A4 @ B2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( F3 @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ B2 ) ) ) ) ).
% Union_image_insert
thf(fact_3348_Union__image__empty,axiom,
! [B: $tType,A: $tType,A3: set @ A,F3: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= A3 ) ).
% Union_image_empty
thf(fact_3349_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,S3: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A4 ) @ ( minus_minus @ ( set @ A ) @ S3 @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A4 ) @ S3 ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A4 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_3350_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_3351_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A3 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A3 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A3 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_3352_finite__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( finite_finite2 @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% finite_shuffles
thf(fact_3353_finite__insert,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( finite_finite2 @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( finite_finite2 @ A @ A3 ) ) ).
% finite_insert
thf(fact_3354_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_3355_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_3356_finite__Diff2,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_Diff2
thf(fact_3357_finite__Diff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% finite_Diff
thf(fact_3358_listI__assn__finite,axiom,
! [B: $tType,A: $tType,I3: set @ nat,A3: A > B > assn,Xs: list @ A,Xsi: list @ B] :
( ~ ( finite_finite2 @ nat @ I3 )
=> ( ( vEBT_List_listI_assn @ A @ B @ I3 @ A3 @ Xs @ Xsi )
= ( bot_bot @ assn ) ) ) ).
% listI_assn_finite
thf(fact_3359_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% infinite_Icc_iff
thf(fact_3360_finite__Diff__insert,axiom,
! [A: $tType,A3: set @ A,A4: A,B2: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ B2 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_3361_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_3362_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K3: A] :
( ( ord_less_eq @ A @ L @ K3 )
=> ( ( set_or3652927894154168847AtMost @ A @ K3 @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_3363_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K3: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K3 @ L ) )
= ( ~ ( ord_less @ A @ K3 @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_3364_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K3: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K3 @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K3 @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_3365_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A4 @ B3 ) ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% infinite_Ioc_iff
thf(fact_3366_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% infinite_Ioo_iff
thf(fact_3367_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X3 ) )
= X3 ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_3368_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X3 @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_3369_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_3370_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A3 ) )
= A3 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_3371_card__0__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_3372_card__insert__disjoint,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_3373_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( linord4507533701916653071of_set @ A @ A3 )
= ( nil @ A ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_3374_finite__set__image,axiom,
! [A: $tType,A3: set @ ( list @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A3 ) )
=> ( ! [Xs2: list @ A] :
( ( member @ ( list @ A ) @ Xs2 @ A3 )
=> ( distinct @ A @ Xs2 ) )
=> ( finite_finite2 @ ( list @ A ) @ A3 ) ) ) ).
% finite_set_image
thf(fact_3375_finite_OinsertI,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ A @ ( insert3 @ A @ A4 @ A3 ) ) ) ).
% finite.insertI
thf(fact_3376_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_3377_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_3378_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A4 @ B3 ) ) ) ) ).
% infinite_Ioc
thf(fact_3379_finite__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( finite_finite2 @ A @ ( field2 @ A @ R2 ) ) ) ).
% finite_Field
thf(fact_3380_finite__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs2: list @ A] :
( ( set2 @ A @ Xs2 )
= A3 ) ) ).
% finite_list
thf(fact_3381_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N9: set @ nat] :
? [M: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N9 )
=> ( ord_less @ nat @ X4 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_3382_bounded__nat__set__is__finite,axiom,
! [N7: set @ nat,N: nat] :
( ! [X: nat] :
( ( member @ nat @ X @ N7 )
=> ( ord_less @ nat @ X @ N ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% bounded_nat_set_is_finite
thf(fact_3383_Diff__infinite__finite,axiom,
! [A: $tType,T4: set @ A,S: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_3384_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A3: set @ B,A4: B] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A3 ) ) )
=> ( ( member @ B @ A4 @ A3 )
=> ( finite_finite2 @ A @ ( B2 @ A4 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3385_finite__psubset__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [A14: set @ A] :
( ( finite_finite2 @ A @ A14 )
=> ( ! [B13: set @ A] :
( ( ord_less @ ( set @ A ) @ B13 @ A14 )
=> ( P @ B13 ) )
=> ( P @ A14 ) ) )
=> ( P @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_3386_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( ( linord4507533701916653071of_set @ A @ A3 )
= ( linord4507533701916653071of_set @ A @ B2 ) )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( A3 = B2 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_3387_infinite__UNIV__listI,axiom,
! [A: $tType] :
~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% infinite_UNIV_listI
thf(fact_3388_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X: A] :
( ( member @ A @ X @ A3 )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ A3 )
=> ( ( ord_less_eq @ A @ X @ Xa3 )
=> ( X = Xa3 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3389_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X: A] :
( ( member @ A @ X @ A3 )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ A3 )
=> ( ( ord_less_eq @ A @ Xa3 @ X )
=> ( X = Xa3 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3390_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X: A] :
( ( member @ A @ X @ S )
& ~ ? [Xa3: A] :
( ( member @ A @ Xa3 @ S )
& ( ord_less @ A @ Xa3 @ X ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_3391_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X5: set @ A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ X5 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ X5 )
& ( ord_less @ A @ X @ Xa3 ) ) )
=> ~ ( finite_finite2 @ A @ X5 ) ) ) ) ).
% infinite_growing
thf(fact_3392_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,X3: A,A4: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( member @ A @ X3 @ X5 )
=> ( ! [X: A] :
( ( member @ A @ X @ X5 )
=> ( ord_less @ A @ X @ A4 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X5 ) @ A4 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_3393_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
( ! [A14: set @ A] :
( ~ ( finite_finite2 @ A @ A14 )
=> ( P @ A14 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A,F9: set @ A] :
( ( finite_finite2 @ A @ F9 )
=> ( ~ ( member @ A @ X @ F9 )
=> ( ( P @ F9 )
=> ( P @ ( insert3 @ A @ X @ F9 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_3394_finite__ne__induct,axiom,
! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F )
=> ( ( F
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] : ( P @ ( insert3 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X: A,F9: set @ A] :
( ( finite_finite2 @ A @ F9 )
=> ( ( F9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X @ F9 )
=> ( ( P @ F9 )
=> ( P @ ( insert3 @ A @ X @ F9 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_3395_finite__induct,axiom,
! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A,F9: set @ A] :
( ( finite_finite2 @ A @ F9 )
=> ( ~ ( member @ A @ X @ F9 )
=> ( ( P @ F9 )
=> ( P @ ( insert3 @ A @ X @ F9 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_3396_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A5: set @ A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,B4: A] :
( ( A5
= ( insert3 @ A @ B4 @ A7 ) )
& ( finite_finite2 @ A @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_3397_finite_Ocases,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A14: set @ A] :
( ? [A6: A] :
( A4
= ( insert3 @ A @ A6 @ A14 ) )
=> ~ ( finite_finite2 @ A @ A14 ) ) ) ) ).
% finite.cases
thf(fact_3398_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) ) ) ) ).
% infinite_Icc
thf(fact_3399_finite__maxlen,axiom,
! [A: $tType,M5: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M5 )
=> ? [N5: nat] :
! [X8: list @ A] :
( ( member @ ( list @ A ) @ X8 @ M5 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X8 ) @ N5 ) ) ) ).
% finite_maxlen
thf(fact_3400_finite__distinct__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs2: list @ A] :
( ( ( set2 @ A @ Xs2 )
= A3 )
& ( distinct @ A @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_3401_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_3402_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) ) ) ) ).
% infinite_Ioo
thf(fact_3403_card__partition,axiom,
! [A: $tType,C3: set @ ( set @ A ),K3: nat] :
( ( finite_finite2 @ ( set @ A ) @ C3 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
=> ( ! [C5: set @ A] :
( ( member @ ( set @ A ) @ C5 @ C3 )
=> ( ( finite_card @ A @ C5 )
= K3 ) )
=> ( ! [C13: set @ A,C24: set @ A] :
( ( member @ ( set @ A ) @ C13 @ C3 )
=> ( ( member @ ( set @ A ) @ C24 @ C3 )
=> ( ( C13 != C24 )
=> ( ( inf_inf @ ( set @ A ) @ C13 @ C24 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K3 @ ( finite_card @ ( set @ A ) @ C3 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_3404_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S: set @ B,P: ( set @ B ) > $o,F3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X: B,S6: set @ B] :
( ( finite_finite2 @ B @ S6 )
=> ( ! [Y3: B] :
( ( member @ B @ Y3 @ S6 )
=> ( ord_less_eq @ A @ ( F3 @ Y3 ) @ ( F3 @ X ) ) )
=> ( ( P @ S6 )
=> ( P @ ( insert3 @ B @ X @ S6 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_3405_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A14: set @ A] :
( ( finite_finite2 @ A @ A14 )
=> ( ! [X8: A] :
( ( member @ A @ X8 @ A14 )
=> ( ord_less @ A @ B5 @ X8 ) )
=> ( ( P @ A14 )
=> ( P @ ( insert3 @ A @ B5 @ A14 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_3406_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A14: set @ A] :
( ( finite_finite2 @ A @ A14 )
=> ( ! [X8: A] :
( ( member @ A @ X8 @ A14 )
=> ( ord_less @ A @ X8 @ B5 ) )
=> ( ( P @ A14 )
=> ( P @ ( insert3 @ A @ B5 @ A14 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_3407_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,A4: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X5 ) @ A4 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X5 )
=> ( ord_less @ A @ X4 @ A4 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_3408_finite__subset__induct_H,axiom,
! [A: $tType,F: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ F @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F9: set @ A] :
( ( finite_finite2 @ A @ F9 )
=> ( ( member @ A @ A6 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ F9 @ A3 )
=> ( ~ ( member @ A @ A6 @ F9 )
=> ( ( P @ F9 )
=> ( P @ ( insert3 @ A @ A6 @ F9 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_3409_finite__subset__induct,axiom,
! [A: $tType,F: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ F @ A3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F9: set @ A] :
( ( finite_finite2 @ A @ F9 )
=> ( ( member @ A @ A6 @ A3 )
=> ( ~ ( member @ A @ A6 @ F9 )
=> ( ( P @ F9 )
=> ( P @ ( insert3 @ A @ A6 @ F9 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_3410_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A,Y2: A] :
( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( member @ A @ ( sup_sup @ A @ X @ Y2 ) @ A3 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_3411_finite__empty__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P @ A3 )
=> ( ! [A6: A,A14: set @ A] :
( ( finite_finite2 @ A @ A14 )
=> ( ( member @ A @ A6 @ A14 )
=> ( ( P @ A14 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A14 @ ( insert3 @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_3412_infinite__coinduct,axiom,
! [A: $tType,X5: ( set @ A ) > $o,A3: set @ A] :
( ( X5 @ A3 )
=> ( ! [A14: set @ A] :
( ( X5 @ A14 )
=> ? [X8: A] :
( ( member @ A @ X8 @ A14 )
& ( ( X5 @ ( minus_minus @ ( set @ A ) @ A14 @ ( insert3 @ A @ X8 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A14 @ ( insert3 @ A @ X8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_3413_infinite__remove,axiom,
! [A: $tType,S: set @ A,A4: A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_3414_card__eq__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% card_eq_0_iff
thf(fact_3415_card__ge__0__finite,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% card_ge_0_finite
thf(fact_3416_card__Suc__eq__finite,axiom,
! [A: $tType,A3: set @ A,K3: nat] :
( ( ( finite_card @ A @ A3 )
= ( suc @ K3 ) )
= ( ? [B4: A,B7: set @ A] :
( ( A3
= ( insert3 @ A @ B4 @ B7 ) )
& ~ ( member @ A @ B4 @ B7 )
& ( ( finite_card @ A @ B7 )
= K3 )
& ( finite_finite2 @ A @ B7 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_3417_card__insert__if,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( finite_card @ A @ A3 ) ) )
& ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( suc @ ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_3418_card__less__sym__Diff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_3419_card__le__sym__Diff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_3420_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A4 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B3 @ A4 )
| ( ord_less_eq @ A @ D3 @ C2 )
| ( ord_less_eq @ A @ B3 @ C2 )
| ( ord_less_eq @ A @ D3 @ A4 ) ) ) ) ).
% Ioc_disjoint
thf(fact_3421_psubset__card__mono,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_3422_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_3423_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_3424_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_3425_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_3426_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_3427_finite__remove__induct,axiom,
! [A: $tType,B2: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B2 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A14: set @ A] :
( ( finite_finite2 @ A @ A14 )
=> ( ( A14
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A14 @ B2 )
=> ( ! [X8: A] :
( ( member @ A @ X8 @ A14 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A14 @ ( insert3 @ A @ X8 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A14 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_3428_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B2: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B2 )
=> ( P @ B2 ) )
=> ( ! [A14: set @ A] :
( ( finite_finite2 @ A @ A14 )
=> ( ( A14
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A14 @ B2 )
=> ( ! [X8: A] :
( ( member @ A @ X8 @ A14 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A14 @ ( insert3 @ A @ X8 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A14 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_3429_card__gt__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
= ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A3 ) ) ) ).
% card_gt_0_iff
thf(fact_3430_finite__UNIV__card__ge__0,axiom,
! [A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_3431_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A3: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A3 ) )
= ( ? [A5: A,B7: set @ A] :
( ( A3
= ( insert3 @ A @ A5 @ B7 ) )
& ~ ( member @ A @ A5 @ B7 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B7 ) )
& ( finite_finite2 @ A @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_3432_card__1__singletonI,axiom,
! [A: $tType,S: set @ A,X3: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( finite_card @ A @ S )
= ( one_one @ nat ) )
=> ( ( member @ A @ X3 @ S )
=> ( S
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_1_singletonI
thf(fact_3433_finite__induct__select,axiom,
! [A: $tType,S: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T5: set @ A] :
( ( ord_less @ ( set @ A ) @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X8: A] :
( ( member @ A @ X8 @ ( minus_minus @ ( set @ A ) @ S @ T5 ) )
& ( P @ ( insert3 @ A @ X8 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_3434_card__Diff__subset,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_3435_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [X: list @ A] :
( ( ( set2 @ A @ X )
= A3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X )
& ( distinct @ A @ X )
& ! [Y3: list @ A] :
( ( ( ( set2 @ A @ Y3 )
= A3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y3 )
& ( distinct @ A @ Y3 ) )
=> ( Y3 = X ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_3436_card__psubset,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B2 ) )
=> ( ord_less @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_3437_diff__card__le__card__Diff,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B2 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_3438_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F3: A > ( set @ B )] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R2 )
=> ( ( finite_finite2 @ B @ ( Ub @ A6 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F3 @ B5 ) @ ( Ub @ A6 ) )
& ( ord_less @ ( set @ B ) @ ( F3 @ A6 ) @ ( F3 @ B5 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_set
thf(fact_3439_card__Diff__subset__Int,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_3440_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A4 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less_eq @ A @ B3 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_3441_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_3442_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A5: A,B4: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B4 ) @ ( insert3 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_3443_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less_eq @ A @ B3 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_3444_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_3445_Gcd__remove0__nat,axiom,
! [M5: set @ nat] :
( ( finite_finite2 @ nat @ M5 )
=> ( ( gcd_Gcd @ nat @ M5 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M5 @ ( insert3 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_3446_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_3447_card__Suc__Diff1,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_3448_card_Oinsert__remove,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_3449_card_Oremove,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( finite_card @ A @ A3 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_3450_card__insert__disjoint_H,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert3 @ A @ X3 @ A3 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_insert_disjoint'
thf(fact_3451_card__Diff1__less__iff,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) )
= ( ( finite_finite2 @ A @ A3 )
& ( member @ A @ X3 @ A3 ) ) ) ).
% card_Diff1_less_iff
thf(fact_3452_card__Diff2__less,axiom,
! [A: $tType,A3: set @ A,X3: A,Y: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert3 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_3453_card__Diff1__less,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff1_less
thf(fact_3454_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ~ ! [L3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L3 )
=> ( ( ( set2 @ A @ L3 )
= A3 )
=> ( ( size_size @ ( list @ A ) @ L3 )
!= ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_3455_card__Un__disjoint,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B2 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_3456_dvd__partition,axiom,
! [A: $tType,C3: set @ ( set @ A ),K3: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
=> ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ C3 )
=> ( dvd_dvd @ nat @ K3 @ ( finite_card @ A @ X ) ) )
=> ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ C3 )
=> ! [Xa4: set @ A] :
( ( member @ ( set @ A ) @ Xa4 @ C3 )
=> ( ( X != Xa4 )
=> ( ( inf_inf @ ( set @ A ) @ X @ Xa4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K3 @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_3457_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 ) @ ( insert3 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_3458_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 ) @ ( insert3 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_3459_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_3460_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X3 @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_3461_set__encode__insert,axiom,
! [A3: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ~ ( member @ nat @ N @ A3 )
=> ( ( nat_set_encode @ ( insert3 @ nat @ N @ A3 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% set_encode_insert
thf(fact_3462_in__finite__psubset,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B2 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
& ( finite_finite2 @ A @ B2 ) ) ) ).
% in_finite_psubset
thf(fact_3463_subset__mset_Osum__mono2,axiom,
! [A: $tType,B: $tType,B2: set @ B,A3: set @ B,F3: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B2 )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B2 @ A3 ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F3 @ B5 ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ A3 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ B2 ) ) ) ) ) ).
% subset_mset.sum_mono2
thf(fact_3464_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A3: B > ( set @ A ),I: B,B2: set @ A,J4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A3 @ I @ B2 ) @ J4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ B ) @ J4 @ ( insert3 @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I @ J4 ) @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_3465_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X8: A] :
( ( member @ A @ X8 @ S )
& ( ord_less @ B @ ( F3 @ X8 ) @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_3466_image__update,axiom,
! [B: $tType,A: $tType,X3: A,A3: set @ A,F3: A > B,N: B] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( image2 @ A @ B @ ( fun_upd @ A @ B @ F3 @ X3 @ N ) @ A3 )
= ( image2 @ A @ B @ F3 @ A3 ) ) ) ).
% image_update
thf(fact_3467_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_3468_fun__upd__image,axiom,
! [A: $tType,B: $tType,X3: B,A3: set @ B,F3: B > A,Y: A] :
( ( ( member @ B @ X3 @ A3 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F3 @ X3 @ Y ) @ A3 )
= ( insert3 @ A @ Y @ ( image2 @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X3 @ A3 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F3 @ X3 @ Y ) @ A3 )
= ( image2 @ B @ A @ F3 @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_3469_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S ) @ S ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_3470_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I4: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J2 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_3471_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S: set @ A,Y: A,F3: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S )
=> ( ord_less_eq @ B @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_3472_unbounded__k__infinite,axiom,
! [K3: nat,S: set @ nat] :
( ! [M6: nat] :
( ( ord_less @ nat @ K3 @ M6 )
=> ? [N8: nat] :
( ( ord_less @ nat @ M6 @ N8 )
& ( member @ nat @ N8 @ S ) ) )
=> ~ ( finite_finite2 @ nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_3473_infinite__nat__iff__unbounded,axiom,
! [S: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less @ nat @ M @ N2 )
& ( member @ nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_3474_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_3475_sorted__list__of__set__lessThan__Suc,axiom,
! [K3: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K3 ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K3 ) ) @ ( cons @ nat @ K3 @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_3476_finite__transitivity__chain,axiom,
! [A: $tType,A3: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X: A] :
~ ( R @ X @ X )
=> ( ! [X: A,Y2: A,Z3: A] :
( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A3 )
& ( R @ X @ Y3 ) ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_3477_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K3: A] :
( ( member @ A @ I @ ( set_ord_lessThan @ A @ K3 ) )
= ( ord_less @ A @ I @ K3 ) ) ) ).
% lessThan_iff
thf(fact_3478_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_3479_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( set_or7035219750837199246ssThan @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_3480_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A4 @ B3 ) )
= ( ~ ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_3481_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_3482_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A4 @ B3 ) ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% infinite_Ico_iff
thf(fact_3483_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X3 ) )
= X3 ) ) ) ).
% cSup_atLeastLessThan
thf(fact_3484_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X3 @ Y ) )
= Y ) ) ) ).
% Sup_atLeastLessThan
thf(fact_3485_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N: A,M2: A] :
( ( ord_less_eq @ A @ I @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M2 ) ) ) ) ).
% ivl_diff
thf(fact_3486_lessThan__minus__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: A,M2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_ord_lessThan @ A @ N ) @ ( set_ord_lessThan @ A @ M2 ) )
= ( set_or7035219750837199246ssThan @ A @ M2 @ N ) ) ) ).
% lessThan_minus_lessThan
thf(fact_3487_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_3488_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K3: A] :
( ( minus_minus @ ( set @ A ) @ ( insert3 @ A @ K3 @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K3 ) )
= ( insert3 @ A @ K3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_3489_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or7035219750837199246ssThan @ A @ A4 @ B3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C2 @ B3 ) @ ( minus_minus @ A @ C2 @ A4 ) ) ) ) ).
% image_diff_atLeastLessThan
thf(fact_3490_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A4 @ B3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C2 @ B3 ) @ ( minus_minus @ A @ C2 @ A4 ) ) ) ) ).
% image_minus_const_greaterThanAtMost
thf(fact_3491_atLeastLessThan__singleton,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ M2 ) )
= ( insert3 @ nat @ M2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_3492_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_3493_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A4 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( B3 = D3 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_3494_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A4 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( A4 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_3495_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A4 @ B3 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( A4 = C2 )
& ( B3 = D3 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_3496_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_3497_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
( ( set_ord_lessThan @ A @ X3 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_3498_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A4 @ B3 ) ) ) ) ).
% infinite_Ico
thf(fact_3499_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less @ nat @ M @ N )
& ( P @ M ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_3500_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less @ nat @ M @ N )
=> ( P @ M ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less_eq
thf(fact_3501_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_3502_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_3503_lessThan__Suc,axiom,
! [K3: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K3 ) )
= ( insert3 @ nat @ K3 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ).
% lessThan_Suc
thf(fact_3504_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_3505_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_3506_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M2 ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M2 @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_3507_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert3 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_3508_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_3509_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert3 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_3510_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_3511_lessThan__nat__numeral,axiom,
! [K3: num] :
( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K3 ) )
= ( insert3 @ nat @ ( pred_numeral @ K3 ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K3 ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_3512_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_3513_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A4 ) @ ( set_ord_lessThan @ A @ B3 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_3514_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A4 @ B3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less_eq @ A @ B3 @ D3 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_3515_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less @ A @ B3 @ D3 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_3516_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_3517_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A5: A,B4: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B4 ) @ ( insert3 @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_3518_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A4 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less @ A @ B3 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_3519_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
& ( ord_less_eq @ A @ B3 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_3520_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_3521_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N ) )
= ( insert3 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_3522_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_3523_atLeastLessThanSuc,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( insert3 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_3524_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert3 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_3525_listI__assn__conv,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,A3: A > B > assn,Xsi: list @ B] :
( ( N
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) @ A3 @ Xs @ Xsi )
= ( vEBT_List_list_assn @ A @ B @ A3 @ Xs @ Xsi ) ) ) ).
% listI_assn_conv
thf(fact_3526_list__assn__conv__idx,axiom,
! [B: $tType,A: $tType] :
( ( vEBT_List_list_assn @ A @ B )
= ( ^ [A7: A > B > assn,Xs3: list @ A] : ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ A7 @ Xs3 ) ) ) ).
% list_assn_conv_idx
thf(fact_3527_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,K3: A] :
( ( ( ord_less @ A @ X3 @ K3 )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K3 ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X3 @ K3 )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K3 ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_3528_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [U: A] :
( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert3 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_ord_atMost @ A @ U ) ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_3529_ivl__disj__un__one_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_3530_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 ) @ ( insert3 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_3531_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert3 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_3532_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_3533_atLeastLessThan__nat__numeral,axiom,
! [M2: nat,K3: num] :
( ( ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K3 ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K3 ) )
= ( insert3 @ nat @ ( pred_numeral @ K3 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( pred_numeral @ K3 ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K3 ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K3 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_3534_listI__assn__conv_H,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,A3: A > B > assn,Xsi: list @ B,F: assn] :
( ( N
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( times_times @ assn @ ( vEBT_List_listI_assn @ A @ B @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) @ A3 @ Xs @ Xsi ) @ F )
= ( times_times @ assn @ ( vEBT_List_list_assn @ A @ B @ A3 @ Xs @ Xsi ) @ F ) ) ) ).
% listI_assn_conv'
thf(fact_3535_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A] :
( ! [A6: A,B5: A,X: A] :
( ( member @ A @ A6 @ S )
=> ( ( member @ A @ B5 @ S )
=> ( ( ord_less_eq @ A @ A6 @ X )
=> ( ( ord_less_eq @ A @ X @ B5 )
=> ( member @ A @ X @ S ) ) ) ) )
=> ? [A6: A,B5: A] :
( ( S
= ( bot_bot @ ( set @ A ) ) )
| ( S
= ( top_top @ ( set @ A ) ) )
| ( S
= ( set_ord_lessThan @ A @ B5 ) )
| ( S
= ( set_ord_atMost @ A @ B5 ) )
| ( S
= ( set_ord_greaterThan @ A @ A6 ) )
| ( S
= ( set_ord_atLeast @ A @ A6 ) )
| ( S
= ( set_or5935395276787703475ssThan @ A @ A6 @ B5 ) )
| ( S
= ( set_or3652927894154168847AtMost @ A @ A6 @ B5 ) )
| ( S
= ( set_or7035219750837199246ssThan @ A @ A6 @ B5 ) )
| ( S
= ( set_or1337092689740270186AtMost @ A @ A6 @ B5 ) ) ) ) ) ).
% interval_cases
thf(fact_3536_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K3: nat,N: nat] :
( ( ord_less_eq @ nat @ K3 @ 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 @ K3 ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K3 ) ) ) ) ) ) ).
% fact_split
thf(fact_3537_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_3538_max__ext_Ocases,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X8: A] :
( ( member @ A @ X8 @ A1 )
=> ? [Xa4: A] :
( ( member @ A @ Xa4 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X8 @ Xa4 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_3539_max__ext_Osimps,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
= ( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A1 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_3540_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_3541_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Min_singleton
thf(fact_3542_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ A @ X3 @ X4 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_3543_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less @ A @ X3 @ X4 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_3544_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_3545_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_3546_image__minus__const__AtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atMost @ A @ B3 ) )
= ( set_ord_atLeast @ A @ ( minus_minus @ A @ C2 @ B3 ) ) ) ) ).
% image_minus_const_AtMost
thf(fact_3547_image__minus__const__atLeast,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A4: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atLeast @ A @ A4 ) )
= ( set_ord_atMost @ A @ ( minus_minus @ A @ C2 @ A4 ) ) ) ) ).
% image_minus_const_atLeast
thf(fact_3548_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min_insert
thf(fact_3549_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_3550_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_3551_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_3552_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A3 ) ) ) ) ).
% Min_in
thf(fact_3553_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ( set_ord_atLeast @ A @ X3 )
= ( top_top @ ( set @ A ) ) )
= ( X3
= ( bot_bot @ A ) ) ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_3554_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A4: B,C2: B,B3: B,D3: B,G: B > A,H2: B > A] :
( ( A4 = C2 )
=> ( ( B3 = D3 )
=> ( ! [X: B] :
( ( ord_less_eq @ B @ C2 @ X )
=> ( ( ord_less @ B @ X @ D3 )
=> ( ( G @ X )
= ( H2 @ X ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A4 @ B3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_3555_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_3556_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ X3 @ A6 ) )
=> ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_3557_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) )
=> ! [A15: A] :
( ( member @ A @ A15 @ A3 )
=> ( ord_less_eq @ A @ X3 @ A15 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_3558_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( ( member @ A @ M2 @ A3 )
& ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ A @ M2 @ X4 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_3559_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X3 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less_eq @ A @ X4 @ X3 ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_3560_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A3 )
= M2 )
= ( ( member @ A @ M2 @ A3 )
& ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ A @ M2 @ X4 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_3561_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ X3 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less @ A @ X4 @ X3 ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_3562_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A4: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A3 )
=> ( ord_less_eq @ A @ A4 @ B5 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert3 @ A @ A4 @ A3 ) )
= A4 ) ) ) ) ).
% Min_insert2
thf(fact_3563_max__ext__additive,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,R: set @ ( product_prod @ A @ A ),C3: set @ A,D2: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A3 @ B2 ) @ ( max_ext @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C3 @ D2 ) @ ( max_ext @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) @ ( sup_sup @ ( set @ A ) @ B2 @ D2 ) ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_ext_additive
thf(fact_3564_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_3565_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A4 ) @ ( set_ord_greaterThan @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_3566_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_3567_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_3568_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_3569_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_3570_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: nat,B3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A4 @ B3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A4 @ ( suc @ B3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A4 @ B3 ) ) @ ( G @ B3 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_3571_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_3572_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_3573_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_3574_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B2 ) @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_3575_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M5: set @ A,N7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M5 @ N7 )
=> ( ( M5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N7 ) @ ( lattic643756798350308766er_Min @ A @ M5 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_3576_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_3577_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N7: set @ A] :
( ! [X: A,Y2: A] :
( ( H2 @ ( ord_min @ A @ X @ Y2 ) )
= ( ord_min @ A @ ( H2 @ X ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798350308766er_Min @ A @ N7 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_3578_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B2 ) @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ).
% Min.subset
thf(fact_3579_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ A3 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_3580_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A,Y2: A] : ( member @ A @ ( ord_min @ A @ X @ Y2 ) @ ( insert3 @ A @ X @ ( insert3 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Min.closed
thf(fact_3581_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A3 ) @ ( lattic643756798350308766er_Min @ A @ B2 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_3582_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I4: int,J2: int] : ( set2 @ int @ ( upto @ I4 @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_3583_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N ) @ ( set_ord_atLeast @ A @ N ) )
= ( insert3 @ A @ N @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_3584_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A] :
( ( sup_sup @ ( set @ A ) @ ( insert3 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) )
= ( set_ord_atLeast @ A @ L ) ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_3585_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_3586_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_3587_ivl__disj__un__one_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_3588_atLeast__Suc,axiom,
! [K3: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K3 ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K3 ) @ ( insert3 @ nat @ K3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_3589_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_3590_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert3 @ A @ X3 @ A3 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( ord_min @ A @ X3 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_3591_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 ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ X5 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ Y6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Xa3 ) @ R ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X5 @ Y6 ) @ ( max_ext @ A @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_3592_prod__pos__nat__iff,axiom,
! [A: $tType,A3: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X4 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_3593_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X3 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod.insert
thf(fact_3594_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_3595_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A3: set @ B,B2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) )
=> ( ( F3 @ X )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_3596_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A3: set @ B,F3: B > A,A4: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( F3 @ A4 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A4 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( F3 @ A4 ) ) ) )
& ( ~ ( member @ B @ A4 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_3597_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F3: B > A] :
( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ X ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) ) ) ) ).
% prod_pos
thf(fact_3598_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S: set @ B,H2: B > A,G: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X12: A,Y12: A,X23: A,Y23: A] :
( ( ( R @ X12 @ X23 )
& ( R @ Y12 @ Y23 ) )
=> ( R @ ( times_times @ A @ X12 @ Y12 ) @ ( times_times @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( R @ ( H2 @ X ) @ ( G @ X ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S ) ) ) ) ) ) ) ).
% prod.related
thf(fact_3599_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X3 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X3 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_3600_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S4: set @ B,T6: set @ C,S: set @ B,I: C > B,J: B > C,T4: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S @ S4 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S )
=> ( ( H2 @ ( J @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_3601_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I3: set @ A,I: A,F3: A > B] :
( ( finite_finite2 @ A @ I3 )
=> ( ( member @ A @ I @ I3 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I ) )
=> ( ! [I5: A] :
( ( member @ A @ I5 @ I3 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ I5 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I3 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_3602_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I3 )
=> ( ( I3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I5: A] :
( ( member @ A @ I5 @ I3 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I5 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I3 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_3603_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B2: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B2 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_3604_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: set @ B,A3: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A3 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C3 @ B2 ) )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B2 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C3 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_3605_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: set @ B,A3: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A3 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C3 @ B2 ) )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C3 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B2 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_3606_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_3607_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_3608_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ X )
= ( one_one @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_3609_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( one_one @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_3610_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_3611_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,B2: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_3612_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ I5 )
= ( one_one @ A ) ) )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ ( minus_minus @ ( set @ B ) @ S @ T4 ) )
=> ( ( G @ I5 )
= ( one_one @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ ( inf_inf @ ( set @ B ) @ S @ T4 ) )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_3613_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F3: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I5 ) )
& ( ord_less @ A @ ( F3 @ I5 ) @ ( G @ I5 ) ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_3614_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,X3: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_3615_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X3 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( G @ X3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_3616_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) )
=> ( ( G @ X )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B2 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_3617_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B2 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_3618_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B2 @ A3 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_3619_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B2: set @ A,A3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ B5 ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ A6 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A3 ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ B2 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_3620_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K4: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K4 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_3621_f__arg__min__list__f,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs: list @ A,F3: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( F3 @ ( arg_min_list @ A @ B @ F3 @ Xs ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F3 @ ( set2 @ A @ Xs ) ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_3622_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_3623_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_3624_image__ident,axiom,
! [A: $tType,Y6: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : X4
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_3625_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_3626_singleton__conv,axiom,
! [A: $tType,A4: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A4 )
= ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_3627_singleton__conv2,axiom,
! [A: $tType,A4: A] :
( ( collect @ A
@ ( ^ [Y4: A,Z4: A] : Y4 = Z4
@ A4 ) )
= ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_3628_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S7: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S7: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_3629_finite__Collect__less__nat,axiom,
! [K3: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less @ nat @ N2 @ K3 ) ) ) ).
% finite_Collect_less_nat
thf(fact_3630_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Max_singleton
thf(fact_3631_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A4 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_3632_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( bot_bot @ A )
@ A3 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_3633_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B2 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B2 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_3634_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: B > A,A3: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B2 @ A3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B2 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_3635_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% cSUP_const
thf(fact_3636_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F3: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A3 ) )
= F3 ) ) ) ).
% SUP_const
thf(fact_3637_foldl__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I4: nat,X4: A] : ( suc @ I4 )
@ ( zero_zero @ nat )
@ L )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldl_length
thf(fact_3638_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D3: A,A4: A,B3: A] :
( ( image2 @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A4 @ D3 ) @ ( minus_minus @ A @ B3 @ D3 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_3639_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F3: B > A,G: B > A,S: set @ B] :
( ( image2 @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ S )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ S @ ( collect @ B @ P ) ) )
@ ( image2 @ B @ A @ G
@ ( inf_inf @ ( set @ B ) @ S
@ ( collect @ B
@ ^ [X4: B] :
~ ( P @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_3640_UN__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : C2
@ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_3641_sorted__wrt__rev__linord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ L ) ) ) ) ).
% sorted_wrt_rev_linord
thf(fact_3642_subset__mset_OcSUP__const,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X4: B] : C2
@ A3 ) )
= C2 ) ) ).
% subset_mset.cSUP_const
thf(fact_3643_precise__extr__pure_I1_J,axiom,
! [B: $tType,A: $tType,P: $o,R: A > B > assn] :
( ( precise @ A @ B
@ ^ [X4: A,Y5: B] : ( times_times @ assn @ ( pure_assn @ P ) @ ( R @ X4 @ Y5 ) ) )
= ( P
=> ( precise @ A @ B @ R ) ) ) ).
% precise_extr_pure(1)
thf(fact_3644_precise__extr__pure_I2_J,axiom,
! [B: $tType,A: $tType,R: A > B > assn,P: $o] :
( ( precise @ A @ B
@ ^ [X4: A,Y5: B] : ( times_times @ assn @ ( R @ X4 @ Y5 ) @ ( pure_assn @ P ) ) )
= ( P
=> ( precise @ A @ B @ R ) ) ) ).
% precise_extr_pure(2)
thf(fact_3645_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ A @ X4 @ X3 ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_3646_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less @ A @ X4 @ X3 ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_3647_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ X4 @ ( top_top @ A ) )
=> ? [Y5: B] :
( ( member @ B @ Y5 @ A3 )
& ( ord_less @ A @ X4 @ ( F3 @ Y5 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_3648_range__constant,axiom,
! [B: $tType,A: $tType,X3: A] :
( ( image2 @ B @ A
@ ^ [Uu3: B] : X3
@ ( top_top @ ( set @ B ) ) )
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_3649_UN__singleton,axiom,
! [A: $tType,A3: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X4: A] : ( insert3 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) )
= A3 ) ).
% UN_singleton
thf(fact_3650_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C3: set @ B,A4: A,B2: B > ( set @ A )] :
( ( ( C3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert3 @ A @ A4 @ ( B2 @ X4 ) )
@ C3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert3 @ A @ A4 @ ( B2 @ X4 ) )
@ C3 ) )
= ( insert3 @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C3 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_3651_UN__simps_I3_J,axiom,
! [E: $tType,F6: $tType,C3: set @ F6,A3: set @ E,B2: F6 > ( set @ E )] :
( ( ( C3
= ( bot_bot @ ( set @ F6 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F6 @ ( set @ E )
@ ^ [X4: F6] : ( sup_sup @ ( set @ E ) @ A3 @ ( B2 @ X4 ) )
@ C3 ) )
= ( bot_bot @ ( set @ E ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ F6 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F6 @ ( set @ E )
@ ^ [X4: F6] : ( sup_sup @ ( set @ E ) @ A3 @ ( B2 @ X4 ) )
@ C3 ) )
= ( sup_sup @ ( set @ E ) @ A3 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F6 @ ( set @ E ) @ B2 @ C3 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_3652_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C3: set @ C,A3: C > ( set @ D ),B2: set @ D] :
( ( ( C3
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X4 ) @ B2 )
@ C3 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X4 ) @ B2 )
@ C3 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A3 @ C3 ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_3653_UN__insert,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A4: B,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( insert3 @ B @ A4 @ A3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( B2 @ A4 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A3 ) ) ) ) ).
% UN_insert
thf(fact_3654_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,C2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% Max_const
thf(fact_3655_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ B,C2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% Min_const
thf(fact_3656_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A
@ ^ [C6: A] : ( divide_divide @ A @ C6 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A4 @ D3 ) @ ( divide_divide @ A @ B3 @ D3 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_3657_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_3658_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_3659_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( 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 ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_3660_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H2: B > A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( G @ X4 ) @ ( H2 @ X4 ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A3 ) ) ) ) ).
% prod.distrib
thf(fact_3661_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I3: set @ B,X3: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I3 )
& ( ( X3 @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I3 )
& ( ( Y @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I3 )
& ( ( times_times @ A @ ( X3 @ I4 ) @ ( Y @ I4 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3662_image__constant__conv,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A3 )
= ( insert3 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_3663_image__constant,axiom,
! [A: $tType,B: $tType,X3: A,A3: set @ A,C2: B] :
( ( member @ A @ X3 @ A3 )
=> ( ( image2 @ A @ B
@ ^ [X4: A] : C2
@ A3 )
= ( insert3 @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_3664_filter__preserves__multiset,axiom,
! [A: $tType,M5: A > nat,P: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M5 @ X4 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( P @ X4 ) @ ( M5 @ X4 ) @ ( zero_zero @ nat ) ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_3665_card__less__Suc2,axiom,
! [M5: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ ( suc @ K4 ) @ M5 )
& ( ord_less @ nat @ K4 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ K4 @ M5 )
& ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_3666_card__less__Suc,axiom,
! [M5: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ ( suc @ K4 ) @ M5 )
& ( ord_less @ nat @ K4 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ K4 @ M5 )
& ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_3667_card__less,axiom,
! [M5: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ K4 @ M5 )
& ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_3668_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,S3: set @ A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A4 )
@ ( inf_inf @ ( set @ A ) @ S3 @ T2 ) )
= ( inf_inf @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A4 )
@ S3 )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A4 )
@ T2 ) ) ) ) ).
% translation_subtract_Int
thf(fact_3669_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,S3: set @ A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A4 )
@ ( minus_minus @ ( set @ A ) @ S3 @ T2 ) )
= ( minus_minus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A4 )
@ S3 )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A4 )
@ T2 ) ) ) ) ).
% translation_subtract_diff
thf(fact_3670_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I4: set @ ( product_prod @ A @ B ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ I4 )
@ S ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_3671_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F3: B > A,K3: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ K3 )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F3 @ S ) ) @ K3 ) ) ) ) ) ).
% Max_add_commute
thf(fact_3672_pairwise__image,axiom,
! [A: $tType,B: $tType,R2: A > A > $o,F3: B > A,S3: set @ B] :
( ( pairwise @ A @ R2 @ ( image2 @ B @ A @ F3 @ S3 ) )
= ( pairwise @ B
@ ^ [X4: B,Y5: B] :
( ( ( F3 @ X4 )
!= ( F3 @ Y5 ) )
=> ( R2 @ ( F3 @ X4 ) @ ( F3 @ Y5 ) ) )
@ S3 ) ) ).
% pairwise_image
thf(fact_3673_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image2 @ B @ A @ F3 @ A3 ) )
& ( P @ X4 ) ) )
= ( image2 @ B @ A @ F3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( P @ ( F3 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_3674_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G: C > B,A3: set @ C] :
( ( image2 @ B @ A @ F3 @ ( image2 @ C @ B @ G @ A3 ) )
= ( image2 @ C @ A
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
@ A3 ) ) ).
% image_image
thf(fact_3675_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A,A3: set @ B] :
( ( member @ A @ B3 @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ~ ! [X: B] :
( ( B3
= ( F3 @ X ) )
=> ~ ( member @ B @ X @ A3 ) ) ) ).
% imageE
thf(fact_3676_SUP__Sup__eq,axiom,
! [A: $tType,S: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I4: set @ A,X4: A] : ( member @ A @ X4 @ I4 )
@ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_3677_lex__def,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: 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 @ R5 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% lex_def
thf(fact_3678_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,G: B > C] :
( ( image2 @ B @ A
@ ^ [X4: B] : ( F3 @ ( G @ X4 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image2 @ C @ A @ F3 @ ( image2 @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_3679_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F3: B > A] :
( ( member @ A @ B3 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X: B] :
( B3
!= ( F3 @ X ) ) ) ).
% rangeE
thf(fact_3680_insert__def,axiom,
! [A: $tType] :
( ( insert3 @ A )
= ( ^ [A5: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] : X4 = A5 ) ) ) ) ).
% insert_def
thf(fact_3681_min__list_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A] :
( ( min_list @ A @ ( cons @ A @ X3 @ Xs ) )
= ( case_list @ A @ A @ X3
@ ^ [A5: A,List2: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs ) )
@ Xs ) ) ) ).
% min_list.simps
thf(fact_3682_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 )
@ ^ [Z6: A,Zs3: list @ A] : ( append @ A @ Zs3 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_3683_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_3684_Collect__subset,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( P @ X4 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_3685_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R )
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_3686_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,X3: A,Y: A,Zs: list @ A] :
( ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_3687_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ A4 ) )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ B3 ) ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% subset_divisors_dvd
thf(fact_3688_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_3689_sup__Un__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R )
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_3690_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
| ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_3691_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_3692_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_3693_inf__Int__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R )
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ R @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_3694_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% Int_def
thf(fact_3695_Int__Collect,axiom,
! [A: $tType,X3: A,A3: set @ A,P: A > $o] :
( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_3696_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% inf_set_def
thf(fact_3697_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_3698_Pow__def,axiom,
! [A: $tType] :
( ( pow @ A )
= ( ^ [A7: set @ A] :
( collect @ ( set @ A )
@ ^ [B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% Pow_def
thf(fact_3699_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_3700_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_3701_tl__def,axiom,
! [A: $tType] :
( ( tl @ A )
= ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [X213: A,X224: list @ A] : X224 ) ) ).
% tl_def
thf(fact_3702_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $true ) ) ).
% UNIV_def
thf(fact_3703_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ A4 ) )
@ ( collect @ A
@ ^ [C6: A] : ( dvd_dvd @ A @ C6 @ B3 ) ) )
= ( ( dvd_dvd @ A @ A4 @ B3 )
& ~ ( dvd_dvd @ A @ B3 @ A4 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_3704_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A7 ) ) ) ) ).
% Compl_eq
thf(fact_3705_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
~ ( P @ X4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_3706_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 ) ) ) ) ) ).
% uminus_set_def
thf(fact_3707_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
& ~ ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_3708_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_3709_pairwise__trivial,axiom,
! [A: $tType,I3: set @ A] :
( pairwise @ A
@ ^ [I4: A,J2: A] : J2 != I4
@ I3 ) ).
% pairwise_trivial
thf(fact_3710_sngr__prec,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( precise @ A @ ( ref @ A )
@ ^ [X4: A,P7: ref @ A] : ( sngr_assn @ A @ P7 @ X4 ) ) ) ).
% sngr_prec
thf(fact_3711_snga__prec,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( precise @ ( list @ A ) @ ( array @ A )
@ ^ [X4: list @ A,P7: array @ A] : ( snga_assn @ A @ P7 @ X4 ) ) ) ).
% snga_prec
thf(fact_3712_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H2: B > C,F1: B,F23: A > ( list @ A ) > B,List: list @ A] :
( ( H2 @ ( case_list @ B @ A @ F1 @ F23 @ List ) )
= ( case_list @ C @ A @ ( H2 @ F1 )
@ ^ [X13: A,X24: list @ A] : ( H2 @ ( F23 @ X13 @ X24 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_3713_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ).
% less_set_def
thf(fact_3714_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X4: A] : X4 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_3715_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H3: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_3716_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L4: A] : ( collect @ A @ ( ord_less @ A @ L4 ) ) ) ) ) ).
% greaterThan_def
thf(fact_3717_sorted__wrt__true,axiom,
! [A: $tType,Xs: list @ A] :
( sorted_wrt @ A
@ ^ [Uu3: A,Uv: A] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_3718_sorted__wrt__rev,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( sorted_wrt @ A @ P @ ( rev @ A @ Xs ) )
= ( sorted_wrt @ A
@ ^ [X4: A,Y5: A] : ( P @ Y5 @ X4 )
@ Xs ) ) ).
% sorted_wrt_rev
thf(fact_3719_insert__Collect,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( insert3 @ A @ A4 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U3: A] :
( ( U3 != A4 )
=> ( P @ U3 ) ) ) ) ).
% insert_Collect
thf(fact_3720_insert__compr,axiom,
! [A: $tType] :
( ( insert3 @ A )
= ( ^ [A5: A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A5 )
| ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_3721_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% Set.empty_def
thf(fact_3722_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A4: A] :
( ( ( P @ A4 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A4 )
& ( P @ X4 ) ) )
= ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A4 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A4 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_3723_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A4: A] :
( ( ( P @ A4 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A4 = X4 )
& ( P @ X4 ) ) )
= ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A4 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A4 = X4 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_3724_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_3725_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C6: ( product_prod @ A @ B ) > C,X4: A,Y5: B] : ( C6 @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) ) ) ).
% curry_def
thf(fact_3726_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_3727_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_3728_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_3729_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_3730_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 )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_3731_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( P @ K4 )
& ( ord_less @ nat @ K4 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_3732_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: A,F3: B > A,A3: set @ B] :
( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ A4 @ ( F3 @ X4 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_3733_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A3: set @ B,Y: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ Y )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ ( F3 @ I ) @ Y ) ) ) ) ).
% SUP_lessD
thf(fact_3734_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U3: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ U3 ) ) ) ) ) ).
% lessThan_def
thf(fact_3735_foldl__length__aux,axiom,
! [A: $tType,A4: nat,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I4: nat,X4: A] : ( suc @ I4 )
@ A4
@ L )
= ( plus_plus @ nat @ A4 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldl_length_aux
thf(fact_3736_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A3: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B2 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_3737_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B2 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_3738_UN__empty,axiom,
! [B: $tType,A: $tType,B2: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_3739_UN__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( bot_bot @ ( set @ A ) )
@ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_3740_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A3: set @ A,A4: B,B2: A > ( set @ B )] :
( ( member @ A @ U @ A3 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( insert3 @ B @ A4 @ ( B2 @ X4 ) )
@ A3 ) )
= ( insert3 @ B @ A4 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_3741_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A4 )
@ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A4 )
@ T2 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_3742_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_3743_UN__extend__simps_I6_J,axiom,
! [L6: $tType,K7: $tType,A3: K7 > ( set @ L6 ),C3: set @ K7,B2: set @ L6] :
( ( minus_minus @ ( set @ L6 ) @ ( complete_Sup_Sup @ ( set @ L6 ) @ ( image2 @ K7 @ ( set @ L6 ) @ A3 @ C3 ) ) @ B2 )
= ( complete_Sup_Sup @ ( set @ L6 )
@ ( image2 @ K7 @ ( set @ L6 )
@ ^ [X4: K7] : ( minus_minus @ ( set @ L6 ) @ ( A3 @ X4 ) @ B2 )
@ C3 ) ) ) ).
% UN_extend_simps(6)
thf(fact_3744_nat__seg__image__imp__finite,axiom,
! [A: $tType,A3: set @ A,F3: nat > A,N: nat] :
( ( A3
= ( image2 @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% nat_seg_image_imp_finite
thf(fact_3745_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] :
? [N2: nat,F7: nat > A] :
( A7
= ( image2 @ nat @ A @ F7
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_3746_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_3747_reflcl__set__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R2 )
@ ^ [Y4: A,Z4: A] : Y4 = Z4 )
= ( ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_3748_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 )
@ ^ [S7: set @ A] :
( ( minus_minus @ ( set @ A ) @ S7 @ S )
= ( minus_minus @ ( set @ A ) @ S5 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_3749_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_3750_distinct__finite__set,axiom,
! [A: $tType,X3: set @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( set2 @ A @ Ys3 )
= X3 )
& ( distinct @ A @ Ys3 ) ) ) ) ).
% distinct_finite_set
thf(fact_3751_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R2: B > ( set @ A ),S: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I4: B,X4: A] : ( member @ A @ X4 @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_3752_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y5: B] : C2
@ A3 ) )
= ( bot_bot @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y5: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% SUP_constant
thf(fact_3753_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_3754_add__mset__in__multiset,axiom,
! [A: $tType,M5: A > nat,A4: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M5 @ X4 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( X4 = A4 ) @ ( suc @ ( M5 @ X4 ) ) @ ( M5 @ X4 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_3755_power__numeral__even,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z2: A,W2: num] :
( ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ ( bit0 @ W2 ) ) )
= ( times_times @ A @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% power_numeral_even
thf(fact_3756_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A4: B,A3: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( insert3 @ B @ A4 @ A3 ) ) )
= ( sup_sup @ A @ ( F3 @ A4 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) ) ) ) ).
% SUP_insert
thf(fact_3757_diff__preserves__multiset,axiom,
! [A: $tType,M5: A > nat,N7: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M5 @ X4 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( M5 @ X4 ) @ ( N7 @ X4 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_3758_power__numeral__odd,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z2: A,W2: num] :
( ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ ( bit1 @ W2 ) ) )
= ( times_times @ A @ ( times_times @ A @ Z2 @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W2 ) ) ) @ ( power_power @ A @ Z2 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% power_numeral_odd
thf(fact_3759_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X4: B] :
( ( G @ X4 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_3760_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C3: set @ B,A4: A,B2: B > ( set @ A )] :
( ( ( C3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert3 @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C3 ) ) )
= ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert3 @ A @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ C3 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert3 @ A @ A4 @ ( B2 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_3761_UN__extend__simps_I3_J,axiom,
! [E: $tType,F6: $tType,C3: set @ F6,A3: set @ E,B2: F6 > ( set @ E )] :
( ( ( C3
= ( bot_bot @ ( set @ F6 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A3 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F6 @ ( set @ E ) @ B2 @ C3 ) ) )
= A3 ) )
& ( ( C3
!= ( bot_bot @ ( set @ F6 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A3 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F6 @ ( set @ E ) @ B2 @ C3 ) ) )
= ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F6 @ ( set @ E )
@ ^ [X4: F6] : ( sup_sup @ ( set @ E ) @ A3 @ ( B2 @ X4 ) )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3762_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C3: set @ C,A3: C > ( set @ D ),B2: set @ D] :
( ( ( C3
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A3 @ C3 ) ) @ B2 )
= B2 ) )
& ( ( C3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A3 @ C3 ) ) @ B2 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X4 ) @ B2 )
@ C3 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3763_finite__divisors__nat,axiom,
! [M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D6: nat] : ( dvd_dvd @ nat @ D6 @ M2 ) ) ) ) ).
% finite_divisors_nat
thf(fact_3764_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs: list @ B,F3: B > ( list @ A )] :
( ( set2 @ A @ ( bind @ B @ A @ Xs @ F3 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( set2 @ A @ ( F3 @ X4 ) )
@ ( set2 @ B @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_3765_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ).
% Max_in
thf(fact_3766_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert3 @ A @ ( F3 @ X4 ) @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) )
= ( image2 @ B @ A @ F3 @ A3 ) ) ).
% UNION_singleton_eq_range
thf(fact_3767_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F3: B > A,K3: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ K3 )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F3 @ S ) ) @ K3 ) ) ) ) ) ).
% Min_add_commute
thf(fact_3768_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
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_3769_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,P: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( H2 @ X4 ) @ ( G @ X4 ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_3770_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
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_3771_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
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3772_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_3773_finite__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3774_wf__finite__segments,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( ( trans @ A @ R2 )
=> ( ! [X: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X ) @ R2 ) ) )
=> ( wf @ A @ R2 ) ) ) ) ).
% wf_finite_segments
thf(fact_3775_slice__prepend,axiom,
! [A: $tType,I: nat,K3: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ I @ K3 )
=> ( ( ord_less_eq @ nat @ K3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( slice @ A @ I @ K3 @ Xs )
= ( slice @ A @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( plus_plus @ nat @ K3 @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( append @ A @ Ys @ Xs ) ) ) ) ) ).
% slice_prepend
thf(fact_3776_distinct__finite__subset,axiom,
! [A: $tType,X3: set @ A] :
( ( finite_finite2 @ A @ X3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ X3 )
& ( distinct @ A @ Ys3 ) ) ) ) ) ).
% distinct_finite_subset
thf(fact_3777_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F3: nat > A,A4: nat,B3: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A4 @ B3 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( times_times @ A @ ( F3 @ A5 ) )
@ A4
@ B3
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_3778_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image2 @ nat @ nat
@ ^ [M: nat] : ( modulo_modulo @ nat @ M @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_3779_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 ) )
@ ^ [Z6: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z6 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_3780_zip__Cons1,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X3 @ Xs ) @ Ys )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y5: B,Ys3: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ ( zip @ A @ B @ Xs @ Ys3 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_3781_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_3782_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A4: B,B3: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C2 @ K4 ) )
@ S )
= ( times_times @ A @ ( B3 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C2 @ K4 ) )
@ S )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3783_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I3: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I3 )
=> ( ! [X: B] :
( ( member @ B @ X @ I3 )
=> ( finite_finite2 @ C @ ( A3 @ X ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ I3 )
=> ! [Xa4: B] :
( ( member @ B @ Xa4 @ I3 )
=> ( ( X != Xa4 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X ) @ ( A3 @ Xa4 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A3 @ I3 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A3 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_3784_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
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_3785_finite__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A3 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3786_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_3787_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_3788_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 )
@ ^ [X4: set @ A] : ( member @ ( set @ A ) @ X4 @ ( set2 @ ( set @ A ) @ L ) ) ) ) ) ).
% foldl_set
thf(fact_3789_Pow__set_I2_J,axiom,
! [B: $tType,X3: B,Xs: list @ B] :
( ( pow @ B @ ( set2 @ B @ ( cons @ B @ X3 @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow @ B @ ( set2 @ B @ Xs ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert3 @ B @ X3 ) @ ( pow @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_3790_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,Y: A,C2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X3 @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X3 @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X3 @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_3791_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A4 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B3 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A4 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_3792_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A4 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B3 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A4 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_3793_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A4 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A4 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_3794_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A4 @ B3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A4 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B3 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A4 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_3795_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A4: B,B3: B > A,C2: A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ C2 )
@ S )
= ( times_times @ A @ ( B3 @ A4 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ C2 )
@ S )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_3796_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F3: nat > ( set @ A ),S: set @ A] :
( ! [I5: nat] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ I5 ) @ S )
=> ( ( finite_finite2 @ A @ S )
=> ( ? [N10: nat] :
( ! [N5: nat] :
( ( ord_less_eq @ nat @ N5 @ N10 )
=> ! [M6: nat] :
( ( ord_less_eq @ nat @ M6 @ N10 )
=> ( ( ord_less @ nat @ M6 @ N5 )
=> ( ord_less @ ( set @ A ) @ ( F3 @ M6 ) @ ( F3 @ N5 ) ) ) ) )
& ! [N5: nat] :
( ( ord_less_eq @ nat @ N10 @ N5 )
=> ( ( F3 @ N10 )
= ( F3 @ N5 ) ) ) )
=> ( ( F3 @ ( finite_card @ A @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_3797_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ A6 @ X3 ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 ) ) ) ) ) ).
% Max.boundedI
thf(fact_3798_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ X3 )
=> ! [A15: A] :
( ( member @ A @ A15 @ A3 )
=> ( ord_less_eq @ A @ A15 @ X3 ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_3799_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ( member @ A @ M2 @ A3 )
& ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ A @ X4 @ M2 ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_3800_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less_eq @ A @ X3 @ X4 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_3801_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,M2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A3 )
= M2 )
= ( ( member @ A @ M2 @ A3 )
& ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ A @ X4 @ M2 ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_3802_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less @ A @ X3 @ X4 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_3803_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,A4: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A3 )
=> ( ord_less_eq @ A @ B5 @ A4 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert3 @ A @ A4 @ A3 ) )
= A4 ) ) ) ) ).
% Max_insert2
thf(fact_3804_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_3805_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Max_Sup
thf(fact_3806_card__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A3 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_3807_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X3: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X3 @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y )
=> ( ( ( ord_less @ nat @ X3 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) )
= ( insert3 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X3 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X3 @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_3808_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
@ ^ [I4: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_3809_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
@ ^ [I4: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_3810_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K4 @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K4 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_3811_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,K3: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A4 @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_3812_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K3: nat,A4: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K3 ) @ ( gbinomial @ A @ A4 @ K3 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_3813_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K4 ) )
@ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_3814_arg__min__list_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,X3: A] :
( ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= X3 ) ) ).
% arg_min_list.simps(1)
thf(fact_3815_arg__min__list__in,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs: list @ A,F3: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( arg_min_list @ A @ B @ F3 @ Xs ) @ ( set2 @ A @ Xs ) ) ) ) ).
% arg_min_list_in
thf(fact_3816_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X9: set @ nat] :
( if @ nat
@ ( X9
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X9 ) ) ) ) ).
% Sup_nat_def
thf(fact_3817_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: nat,K3: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K3 @ P4 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K3 ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K3 ) @ ( one_one @ A ) @ ( H2 @ ( 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 @ K3 ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_3818_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A4: A,K3: nat] :
( ( gbinomial @ A @ A4 @ ( suc @ K3 ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A4 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K3 ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_3819_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_3820_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M5: set @ A,N7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M5 @ N7 )
=> ( ( M5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M5 ) @ ( lattic643756798349783984er_Max @ A @ N7 ) ) ) ) ) ) ).
% Max_mono
thf(fact_3821_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_3822_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K4: nat] :
( if @ A
@ ( K4
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L4: nat] : ( times_times @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ L4 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_3823_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K3: nat,A3: set @ A] :
( ( ord_less @ nat @ K3 @ ( finite_card @ A @ A3 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K3 )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K3 ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_3824_card__lists__distinct__length__eq,axiom,
! [A: $tType,A3: set @ A,K3: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ nat @ K3 @ ( finite_card @ A @ A3 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K3 )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K3 ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_3825_set__Cons__sing__Nil,axiom,
! [A: $tType,A3: set @ A] :
( ( set_Cons @ A @ A3 @ ( insert3 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image2 @ A @ ( list @ A )
@ ^ [X4: A] : ( cons @ A @ X4 @ ( nil @ A ) )
@ A3 ) ) ).
% set_Cons_sing_Nil
thf(fact_3826_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
@ ^ [X4: A] : ( transitive_trancl @ B @ ( R @ X4 ) ) ) ) ).
% same_fst_trancl
thf(fact_3827_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
@ ^ [I4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_3828_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
@ ^ [I4: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_3829_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X: A,Y2: B] :
( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_3830_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_3831_top2I,axiom,
! [A: $tType,B: $tType,X3: A,Y: B] : ( top_top @ ( A > B > $o ) @ X3 @ Y ) ).
% top2I
thf(fact_3832_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_3833_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum.insert
thf(fact_3834_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y: A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : Y
@ A3 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ Y ) ) ) ).
% sum_constant
thf(fact_3835_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,P: B > $o,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X4 ) ) @ ( F3 @ X4 ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_3836_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,F3: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X4 ) ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_3837_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_3838_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_3839_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A3 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_3840_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: set @ nat,C2: nat > A,D3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A3 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D3 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_3841_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X3: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) ) ).
% predicate1D
thf(fact_3842_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X3: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X3 @ Y )
=> ( Q @ X3 @ Y ) ) ) ).
% predicate2D
thf(fact_3843_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X3: A,Q: A > $o] :
( ( P @ X3 )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X3 ) ) ) ).
% rev_predicate1D
thf(fact_3844_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X3: A,Y: B,Q: A > B > $o] :
( ( P @ X3 @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X3 @ Y ) ) ) ).
% rev_predicate2D
thf(fact_3845_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R2: A,F3: B > A,A3: set @ B] :
( ( times_times @ A @ R2 @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( times_times @ A @ R2 @ ( F3 @ N2 ) )
@ A3 ) ) ) ).
% sum_distrib_left
thf(fact_3846_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F3: B > A,A3: set @ B,R2: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ R2 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( times_times @ A @ ( F3 @ N2 ) @ R2 )
@ A3 ) ) ) ).
% sum_distrib_right
thf(fact_3847_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F3: A > B,A3: set @ A,G: C > B,B2: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B2 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J2: C] : ( times_times @ B @ ( F3 @ I4 ) @ ( G @ J2 ) )
@ B2 )
@ A3 ) ) ) ).
% sum_product
thf(fact_3848_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F3: B > A,G: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ A3 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum_subtractf
thf(fact_3849_sum__strict__mono__ex1,axiom,
! [A: $tType,I9: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: set @ I9,F3: I9 > A,G: I9 > A] :
( ( finite_finite2 @ I9 @ A3 )
=> ( ! [X: I9] :
( ( member @ I9 @ X @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( G @ X ) ) )
=> ( ? [X8: I9] :
( ( member @ I9 @ X8 @ A3 )
& ( ord_less @ A @ ( F3 @ X8 ) @ ( G @ X8 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I9 @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ I9 @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_3850_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: set @ B,F3: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ord_less @ A @ ( F3 @ X ) @ ( G @ X ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_3851_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_3852_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A4: B,C2: B,B3: B,D3: B,G: B > A,H2: B > A] :
( ( A4 = C2 )
=> ( ( B3 = D3 )
=> ( ! [X: B] :
( ( ord_less_eq @ B @ C2 @ X )
=> ( ( ord_less @ B @ X @ D3 )
=> ( ( G @ X )
= ( H2 @ X ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A4 @ B3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_3853_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S4: set @ B,T6: set @ C,S: set @ B,I: C > B,J: B > C,T4: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S @ S4 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S )
=> ( ( H2 @ ( J @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_3854_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,P4: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P4 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P4 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P4 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_3855_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X4: B] :
( ( G @ X4 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_3856_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,M2: nat,I3: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X3 @ ( plus_plus @ nat @ M2 @ I4 ) )
@ I3 )
= ( times_times @ A @ ( power_power @ A @ X3 @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ I3 ) ) ) ) ).
% sum_power_add
thf(fact_3857_distinct__sum__list__conv__Sum,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( groups8242544230860333062m_list @ A @ Xs )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : X4
@ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_3858_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I3: set @ B,I: B,F3: B > A] :
( ( finite_finite2 @ B @ I3 )
=> ( ( member @ B @ I @ I3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ I3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I5 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I3 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_3859_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I3: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ I3 )
=> ( ( I3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ I3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I5 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I3 ) ) ) ) ) ) ).
% sum_pos
thf(fact_3860_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( zero_zero @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_3861_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ X )
= ( zero_zero @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_3862_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_3863_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_3864_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C3: set @ B,A3: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A3 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C3 @ B2 ) )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C3 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B2 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_3865_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C3: set @ B,A3: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A3 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C3 @ B2 ) )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B2 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C3 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_3866_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B2: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B2 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_3867_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,K6: A,F3: B > A] :
( ! [I5: B] :
( ( member @ B @ I5 @ A3 )
=> ( ord_less_eq @ A @ K6 @ ( F3 @ I5 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) ) ) ) ).
% sum_bounded_below
thf(fact_3868_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F3: B > A,K6: A] :
( ! [I5: B] :
( ( member @ B @ I5 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ I5 ) @ K6 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K6 ) ) ) ) ).
% sum_bounded_above
thf(fact_3869_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,B2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B2 ) ) ) ) ) ) ).
% sum_diff
thf(fact_3870_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ I5 )
= ( zero_zero @ A ) ) )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ ( minus_minus @ ( set @ B ) @ S @ T4 ) )
=> ( ( G @ I5 )
= ( zero_zero @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ ( inf_inf @ ( set @ B ) @ S @ T4 ) )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_3871_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,B2: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_3872_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_3873_listset_Osimps_I2_J,axiom,
! [A: $tType,A3: set @ A,As5: list @ ( set @ A )] :
( ( listset @ A @ ( cons @ ( set @ A ) @ A3 @ As5 ) )
= ( set_Cons @ A @ A3 @ ( listset @ A @ As5 ) ) ) ).
% listset.simps(2)
thf(fact_3874_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M2 ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_3875_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ M2 ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_3876_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_3877_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F3 @ M2 ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_3878_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ M2 ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_3879_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B2: set @ B,A3: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B2 )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B2 @ A3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B2 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_3880_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,X3: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_3881_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X3: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X3 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( G @ X3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_3882_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,A4: B,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ A4 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( F3 @ A4 ) ) ) )
& ( ~ ( member @ B @ A4 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_3883_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,B2: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_3884_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B2 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_3885_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A3: set @ A,B2: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_3886_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B2 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_3887_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_3888_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A4: B,B3: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C2 @ K4 ) )
@ S )
= ( plus_plus @ A @ ( B3 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C2 @ K4 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_3889_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I3: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I3 )
=> ( ! [X: B] :
( ( member @ B @ X @ I3 )
=> ( finite_finite2 @ C @ ( A3 @ X ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ I3 )
=> ! [Xa4: B] :
( ( member @ B @ Xa4 @ I3 )
=> ( ( X != Xa4 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X ) @ ( A3 @ Xa4 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A3 @ I3 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A3 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_3890_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X3 @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_3891_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_3892_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,N: nat] :
( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X3 @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_3893_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs3: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_3894_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B2: set @ A,A3: set @ A,B3: A,F3: A > B] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F3 @ B3 ) )
=> ( ! [X: A] :
( ( member @ A @ X @ B2 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ B2 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_3895_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A3: set @ C,F3: C > B] :
( ( member @ C @ I @ A3 )
=> ( ! [X: C] :
( ( member @ C @ X @ ( minus_minus @ ( set @ C ) @ A3 @ ( insert3 @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X ) ) )
=> ( ( finite_finite2 @ C @ A3 )
=> ( ord_less_eq @ B @ ( F3 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F3 @ A3 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_3896_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A3: set @ B,F3: B > A,K6: A] :
( ! [I5: B] :
( ( member @ B @ I5 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ I5 ) @ ( divide_divide @ A @ K6 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ K6 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_3897_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F3: B > A,K6: A] :
( ! [I5: B] :
( ( member @ B @ I5 @ A3 )
=> ( ord_less @ A @ ( F3 @ I5 ) @ K6 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A3 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K6 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_3898_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I3: set @ A,X3: A > B,A4: A > B,B3: B,Delta: B] :
( ! [I5: A] :
( ( member @ A @ I5 @ I3 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X3 @ I5 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X3 @ I3 )
= ( one_one @ B ) )
=> ( ! [I5: A] :
( ( member @ A @ I5 @ I3 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A4 @ I5 ) @ B3 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( times_times @ B @ ( A4 @ I4 ) @ ( X3 @ I4 ) )
@ I3 )
@ B3 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_3899_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S: set @ A,R: set @ B,G: A > B,F3: 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
@ ^ [X4: A] : ( F3 @ ( G @ X4 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y5: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S )
& ( ( G @ X4 )
= Y5 ) ) ) ) )
@ ( F3 @ Y5 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_3900_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_3901_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( minus_minus @ A @ ( F3 @ K4 ) @ ( F3 @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F3 @ M2 ) @ ( F3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( minus_minus @ A @ ( F3 @ K4 ) @ ( F3 @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_3902_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X3 @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P7: nat] : ( times_times @ A @ ( power_power @ A @ X3 @ P7 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P7 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_3903_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X3 @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) ) @ ( power_power @ A @ X3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_3904_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_3905_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( minus_minus @ A @ ( F3 @ K4 ) @ ( F3 @ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M2 ) ) ) ) ) ).
% sum_telescope''
thf(fact_3906_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X3: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ X3 @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_3907_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q8: nat] : ( ord_less @ nat @ Q8 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_3908_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X3: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X3 @ M2 ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_3909_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_3910_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A4 @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ M2 ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_3911_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,B3: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A4 @ B3 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( power_power @ A @ A4 @ K4 ) ) @ ( power_power @ A @ B3 @ ( minus_minus @ nat @ N @ K4 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_3912_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A4: A,B3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A4 @ B3 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ A4 @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ B3 @ ( minus_minus @ nat @ N @ K4 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_3913_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F7: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F7 @ ( nth @ B @ Xs3 @ N2 ) ) @ ( power_power @ A @ A5 @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_3914_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: nat,K3: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K3 @ P4 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K3 ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K3 ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K3 ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_3915_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q8: nat] : ( ord_less @ nat @ Q8 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_3916_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A4: A,X3: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A4 ) @ K4 ) @ ( power_power @ A @ X3 @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A4 ) @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X3 ) @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_3917_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: A,D3: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) ) ) ).
% double_arith_series
thf(fact_3918_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_3919_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,N: nat] :
( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ).
% sum_gp0
thf(fact_3920_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A4: A,X3: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A4 ) @ K4 ) @ ( power_power @ A @ X3 @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A4 ) @ ( one_one @ A ) ) @ K4 ) @ ( power_power @ A @ X3 @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_3921_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_3922_card__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A3 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A3 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_3923_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_3924_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_3925_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A,D3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A4 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_3926_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X3: A,M2: nat,N: nat] :
( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X3 @ M2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_3927_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_3928_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_3929_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R2: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ R2 @ K4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R2 @ ( suc @ M2 ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_3930_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M2: nat,X3: A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ( X3
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) )
& ( ( X3
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X3 @ M2 ) @ ( power_power @ A @ X3 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X3 ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_3931_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_3932_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A4 @ K4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A4 @ ( plus_plus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_3933_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X3: A,Y7: B,Y: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X3 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R @ X3 ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y7 ) @ ( product_Pair @ A @ B @ X3 @ Y ) ) @ ( same_fst @ A @ B @ P @ R ) ) ) ) ).
% same_fstI
thf(fact_3934_trancl__finite__eq__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_trancl @ A @ R )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_3935_relpow__finite__bounded1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),K3: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K3 @ R )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_3936_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q8: A,R5: A] : ( product_Pair @ A @ A @ Q8 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_3937_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A4: A,B3: B,A3: set @ ( product_prod @ A @ B ),F3: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ A3 )
=> ( member @ C @ ( F3 @ A4 @ B3 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_3938_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A4: B,B3: C] :
( ( product_case_prod @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A4 @ B3 ) )
= ( F3 @ A4 @ B3 ) ) ).
% case_prod_conv
thf(fact_3939_finite__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),N: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% finite_relpow
thf(fact_3940_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q8: A,R5: A] : ( product_Pair @ A @ A @ Q8 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_3941_relpow__Suc__D2_H,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A ),X8: A,Y3: A,Z5: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X8 @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z5 ) @ R ) )
=> ? [W: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X8 @ W ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W @ Z5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_3942_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q4: product_prod @ A @ B,F3: A > B > C,G: A > B > C,P4: product_prod @ A @ B] :
( ! [X: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X @ Y2 )
= Q4 )
=> ( ( F3 @ X @ Y2 )
= ( G @ X @ Y2 ) ) )
=> ( ( P4 = Q4 )
=> ( ( product_case_prod @ A @ B @ C @ F3 @ P4 )
= ( product_case_prod @ A @ B @ C @ G @ Q4 ) ) ) ) ).
% split_cong
thf(fact_3943_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F3 @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_3944_nested__case__prod__simp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F7: B > C > D > A,X4: product_prod @ B @ C,Y5: D] :
( product_case_prod @ B @ C @ A
@ ^ [A5: B,B4: C] : ( F7 @ A5 @ B4 @ Y5 )
@ X4 ) ) ) ).
% nested_case_prod_simp
thf(fact_3945_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
=> ~ ! [X: B,Y2: C] :
( ( Z2
= ( product_Pair @ B @ C @ X @ Y2 ) )
=> ~ ( Q @ ( P @ X @ Y2 ) ) ) ) ).
% case_prodE2
thf(fact_3946_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y5: B] : ( F3 @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_3947_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X: A,Y2: B] :
( ( F3 @ X @ Y2 )
= ( G @ ( product_Pair @ A @ B @ X @ Y2 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F3 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_3948_uncurry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( uncurry @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% uncurry_def
thf(fact_3949_relpow__0__E,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) )
=> ( X3 = Y ) ) ).
% relpow_0_E
thf(fact_3950_relpow__0__I,axiom,
! [A: $tType,X3: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) ) ).
% relpow_0_I
thf(fact_3951_relpow__Suc__E,axiom,
! [A: $tType,X3: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R ) ) ) ).
% relpow_Suc_E
thf(fact_3952_relpow__Suc__I,axiom,
! [A: $tType,X3: A,Y: A,N: nat,R: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) ) ) ) ).
% relpow_Suc_I
thf(fact_3953_relpow__Suc__D2,axiom,
! [A: $tType,X3: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ? [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_D2
thf(fact_3954_relpow__Suc__E2,axiom,
! [A: $tType,X3: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_E2
thf(fact_3955_relpow__Suc__I2,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) ) ) ) ).
% relpow_Suc_I2
thf(fact_3956_sum__SucD,axiom,
! [A: $tType,F3: A > nat,A3: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A3 )
= ( suc @ N ) )
=> ? [X: A] :
( ( member @ A @ X @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X ) ) ) ) ).
% sum_SucD
thf(fact_3957_relpow__E,axiom,
! [A: $tType,X3: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X3 != Z2 ) )
=> ~ ! [Y2: A,M6: nat] :
( ( N
= ( suc @ M6 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M6 @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R ) ) ) ) ) ).
% relpow_E
thf(fact_3958_relpow__E2,axiom,
! [A: $tType,X3: A,Z2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X3 != Z2 ) )
=> ~ ! [Y2: A,M6: nat] :
( ( N
= ( suc @ M6 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M6 @ R ) ) ) ) ) ) ).
% relpow_E2
thf(fact_3959_trancl__power,axiom,
! [A: $tType,P4: product_prod @ A @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P4 @ ( transitive_trancl @ A @ R ) )
= ( ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( member @ ( product_prod @ A @ A ) @ P4 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ) ).
% trancl_power
thf(fact_3960_relpow__empty,axiom,
! [A: $tType,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% relpow_empty
thf(fact_3961_sum__diff__nat,axiom,
! [A: $tType,B2: set @ A,A3: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_3962_sum__diff1__nat,axiom,
! [A: $tType,A4: A,A3: set @ A,F3: A > nat] :
( ( ( member @ A @ A4 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A3 ) @ ( F3 @ A4 ) ) ) )
& ( ~ ( member @ A @ A4 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A3 ) ) ) ) ).
% sum_diff1_nat
thf(fact_3963_relpow__fun__conv,axiom,
! [A: $tType,A4: A,B3: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
= ( ? [F7: nat > A] :
( ( ( F7 @ ( zero_zero @ nat ) )
= A4 )
& ( ( F7 @ N )
= B3 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) @ R ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_3964_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys: list @ A,Xs: list @ B,F3: C > A > C,B3: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldl @ C @ ( product_prod @ B @ A )
@ ^ [B4: C] :
( product_case_prod @ B @ A @ C
@ ^ [X4: B] : ( F3 @ B4 ) )
@ B3
@ ( zip @ B @ A @ Xs @ Ys ) )
= ( foldl @ C @ A @ F3 @ B3 @ Ys ) ) ) ).
% foldl_snd_zip
thf(fact_3965_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_3966_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I3: set @ A,A3: A > ( set @ B )] :
( ( finite_finite2 @ A @ I3 )
=> ( ! [X: A] :
( ( member @ A @ X @ I3 )
=> ( finite_finite2 @ B @ ( A3 @ X ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ I3 )
=> ! [Xa4: A] :
( ( member @ A @ Xa4 @ I3 )
=> ( ( X != Xa4 )
=> ( ( inf_inf @ ( set @ B ) @ ( A3 @ X ) @ ( A3 @ Xa4 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ I3 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A3 @ I4 ) )
@ I3 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_3967_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M: nat,N2: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N2
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M @ N2 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q8: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q8 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_3968_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L4: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q8: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L4 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L4 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q8 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_3969_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L4: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q8: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L4 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L4 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q8 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_3970_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N2: nat,R4: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I4 @ R4 )
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I4 )
& ( ord_less_eq @ nat @ I4 @ ( suc @ N2 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_3971_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L4: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q8: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L4 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L4 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q8 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_3972_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F3: nat > B,Ns: list @ nat] :
( ! [X: nat,Y2: nat] :
( ( ord_less_eq @ nat @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F3 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_3973_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_3974_Pow__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( pow @ A @ A3 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A7: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A7 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert3 @ A @ X4 ) @ A7 ) )
@ ( insert3 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A3 ) ) ) ).
% Pow_fold
thf(fact_3975_listI__assn__def,axiom,
! [A: $tType,B: $tType] :
( ( vEBT_List_listI_assn @ A @ B )
= ( ^ [I10: set @ nat,A7: A > B > assn,Xs3: list @ A,Xsi3: list @ B] :
( times_times @ assn
@ ( pure_assn
@ ( ( ( size_size @ ( list @ B ) @ Xsi3 )
= ( size_size @ ( list @ A ) @ Xs3 ) )
& ( ord_less_eq @ ( set @ nat ) @ I10 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) )
@ ( finite_fold @ nat @ assn
@ ^ [I4: nat,A5: assn] : ( times_times @ assn @ A5 @ ( A7 @ ( nth @ A @ Xs3 @ I4 ) @ ( nth @ B @ Xsi3 @ I4 ) ) )
@ ( one_one @ assn )
@ I10 ) ) ) ) ).
% listI_assn_def
thf(fact_3976_case__prodI2,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,C2: A > B > $o] :
( ! [A6: A,B5: B] :
( ( P4
= ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( C2 @ A6 @ B5 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) ) ).
% case_prodI2
thf(fact_3977_case__prodI,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A4: A,B3: B] :
( ( F3 @ A4 @ B3 )
=> ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A4 @ B3 ) ) ) ).
% case_prodI
thf(fact_3978_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P4: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A6: A,B5: B] :
( ( P4
= ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( member @ C @ Z2 @ ( C2 @ A6 @ B5 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_3979_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A4: B,B3: C] :
( ( member @ A @ Z2 @ ( C2 @ A4 @ B3 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A4 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_3980_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P4: product_prod @ A @ B,C2: A > B > C > $o,X3: C] :
( ! [A6: A,B5: B] :
( ( ( product_Pair @ A @ B @ A6 @ B5 )
= P4 )
=> ( C2 @ A6 @ B5 @ X3 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ X3 ) ) ).
% case_prodI2'
thf(fact_3981_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X4: A] : X4 )
= ( ^ [Xs3: list @ A] : Xs3 ) ) ).
% map_ident
thf(fact_3982_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: list @ A] :
( ( ( map @ A @ B @ F3 @ A4 )
= ( nil @ B ) )
= ( A4
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_3983_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F3 @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_3984_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_3985_map__eq__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,G: B > A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ B @ A @ G @ Xs ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs ) )
=> ( ( F3 @ X4 )
= ( G @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_3986_length__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_3987_map__append,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F3 @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F3 @ Xs ) @ ( map @ B @ A @ F3 @ Ys ) ) ) ).
% map_append
thf(fact_3988_fold__empty,axiom,
! [B: $tType,A: $tType,F3: B > A > A,Z2: A] :
( ( finite_fold @ B @ A @ F3 @ Z2 @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ).
% fold_empty
thf(fact_3989_remdups__eq__nil__right__iff,axiom,
! [A: $tType,X3: list @ A] :
( ( ( nil @ A )
= ( remdups @ A @ X3 ) )
= ( X3
= ( nil @ A ) ) ) ).
% remdups_eq_nil_right_iff
thf(fact_3990_remdups__eq__nil__iff,axiom,
! [A: $tType,X3: list @ A] :
( ( ( remdups @ A @ X3 )
= ( nil @ A ) )
= ( X3
= ( nil @ A ) ) ) ).
% remdups_eq_nil_iff
thf(fact_3991_set__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_remdups
thf(fact_3992_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_3993_remdups__id__iff__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups @ A @ Xs )
= Xs )
= ( distinct @ A @ Xs ) ) ).
% remdups_id_iff_distinct
thf(fact_3994_distinct__remdups,axiom,
! [A: $tType,Xs: list @ A] : ( distinct @ A @ ( remdups @ A @ Xs ) ) ).
% distinct_remdups
thf(fact_3995_list_Oset__map,axiom,
! [B: $tType,A: $tType,F3: A > B,V2: list @ A] :
( ( set2 @ B @ ( map @ A @ B @ F3 @ V2 ) )
= ( image2 @ A @ B @ F3 @ ( set2 @ A @ V2 ) ) ) ).
% list.set_map
thf(fact_3996_map__fun__upd,axiom,
! [B: $tType,A: $tType,Y: A,Xs: list @ A,F3: A > B,V2: B] :
( ~ ( member @ A @ Y @ ( set2 @ A @ Xs ) )
=> ( ( map @ A @ B @ ( fun_upd @ A @ B @ F3 @ Y @ V2 ) @ Xs )
= ( map @ A @ B @ F3 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_3997_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_3998_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F3: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F3 @ Xs ) @ N )
= ( F3 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_3999_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( concat @ A
@ ( map @ B @ ( list @ A )
@ ^ [X4: B] : ( cons @ A @ ( F3 @ X4 ) @ ( nil @ A ) )
@ Xs ) )
= ( map @ B @ A @ F3 @ Xs ) ) ).
% concat_map_singleton
thf(fact_4000_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P2: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X10: A,Y8: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y5: B] :
( ( X10 = X4 )
& ( P2 @ X4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y5 ) @ ( R4 @ X4 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_4001_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P4 )
=> ~ ! [X: A,Y2: B] :
( ( P4
= ( product_Pair @ A @ B @ X @ Y2 ) )
=> ~ ( C2 @ X @ Y2 ) ) ) ).
% case_prodE
thf(fact_4002_case__prodD,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A4: A,B3: B] :
( ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( F3 @ A4 @ B3 ) ) ).
% case_prodD
thf(fact_4003_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_4004_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W2: list @ A,F3: B > A,Ww: list @ B,A4: B] :
( ( W2
= ( map @ B @ A @ F3 @ Ww ) )
=> ( ( cons @ A @ ( F3 @ A4 ) @ W2 )
= ( map @ B @ A @ F3 @ ( cons @ B @ A4 @ Ww ) ) ) ) ).
% map_consI(1)
thf(fact_4005_map__eq__consE,axiom,
! [B: $tType,A: $tType,F3: B > A,Ls3: list @ B,Fa: A,Fl: list @ A] :
( ( ( map @ B @ A @ F3 @ Ls3 )
= ( cons @ A @ Fa @ Fl ) )
=> ~ ! [A6: B,L3: list @ B] :
( ( Ls3
= ( cons @ B @ A6 @ L3 ) )
=> ( ( ( F3 @ A6 )
= Fa )
=> ( ( map @ B @ A @ F3 @ L3 )
!= Fl ) ) ) ) ).
% map_eq_consE
thf(fact_4006_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F3: A > B,X21: A,X222: list @ A] :
( ( map @ A @ B @ F3 @ ( cons @ A @ X21 @ X222 ) )
= ( cons @ B @ ( F3 @ X21 ) @ ( map @ A @ B @ F3 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_4007_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,F3: B > A,Ys: list @ B] :
( ( ( cons @ A @ X3 @ Xs )
= ( map @ B @ A @ F3 @ Ys ) )
=> ? [Z3: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z3 @ Zs2 ) )
& ( X3
= ( F3 @ Z3 ) )
& ( Xs
= ( map @ B @ A @ F3 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_4008_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F3: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z3: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z3 @ Zs2 ) )
& ( ( F3 @ Z3 )
= Y )
& ( ( map @ B @ A @ F3 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_4009_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,F3: B > A,Ys: list @ B] :
( ( ( cons @ A @ X3 @ Xs )
= ( map @ B @ A @ F3 @ Ys ) )
= ( ? [Z6: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z6 @ Zs3 ) )
& ( X3
= ( F3 @ Z6 ) )
& ( Xs
= ( map @ B @ A @ F3 @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_4010_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ? [Z6: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z6 @ Zs3 ) )
& ( ( F3 @ Z6 )
= Y )
& ( ( map @ B @ A @ F3 @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_4011_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P4: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P4 ) )
=> ~ ! [X: B,Y2: C] :
( ( P4
= ( product_Pair @ B @ C @ X @ Y2 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X @ Y2 ) ) ) ) ).
% mem_case_prodE
thf(fact_4012_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [A5: A,B4: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A10: A,B14: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A10 ) @ Ra )
| ( ( A5 = A10 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B14 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_4013_map2__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,H2: B > C > A,F3: D > B,Xs: list @ D,G: D > C] :
( ( map @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ H2 ) @ ( zip @ B @ C @ ( map @ D @ B @ F3 @ Xs ) @ ( map @ D @ C @ G @ Xs ) ) )
= ( map @ D @ A
@ ^ [X4: D] : ( H2 @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_4014_list_Omap__ident,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A
@ ^ [X4: A] : X4
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_4015_rotate1__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( rotate1 @ A @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( rotate1 @ B @ Xs ) ) ) ).
% rotate1_map
thf(fact_4016_remdups_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups.simps(1)
thf(fact_4017_rotate__map,axiom,
! [A: $tType,B: $tType,N: nat,F3: B > A,Xs: list @ B] :
( ( rotate @ A @ N @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( rotate @ B @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_4018_ex__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ B,F3: A > B] :
( ( ? [Xs3: list @ A] :
( Ys
= ( map @ A @ B @ F3 @ Xs3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Ys ) )
=> ? [Y5: A] :
( X4
= ( F3 @ Y5 ) ) ) ) ) ).
% ex_map_conv
thf(fact_4019_map__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F3: A > B,G: A > B] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ( F3 @ X )
= ( G @ X ) ) )
=> ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_4020_map__idI,axiom,
! [A: $tType,Xs: list @ A,F3: A > A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( F3 @ X )
= X ) )
=> ( ( map @ A @ A @ F3 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_4021_map__ext,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B,G: A > B] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( F3 @ X )
= ( G @ X ) ) )
=> ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ G @ Xs ) ) ) ).
% map_ext
thf(fact_4022_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X3: list @ A,Xa: list @ A,F3: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) )
=> ( ( member @ A @ Za @ ( set2 @ A @ Xa ) )
=> ( ( ( F3 @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map @ A @ B @ F3 @ X3 )
= ( map @ A @ B @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_4023_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X3: list @ A,F3: A > B,G: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) )
=> ( ( F3 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map @ A @ B @ F3 @ X3 )
= ( map @ A @ B @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_4024_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X3: list @ A,Ya: list @ A,F3: A > B,G: A > B] :
( ( X3 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ Ya ) )
=> ( ( F3 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map @ A @ B @ F3 @ X3 )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_4025_map__tl,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( map @ B @ A @ F3 @ ( tl @ B @ Xs ) )
= ( tl @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ).
% map_tl
thf(fact_4026_drop__map,axiom,
! [A: $tType,B: $tType,N: nat,F3: B > A,Xs: list @ B] :
( ( drop @ A @ N @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( drop @ B @ N @ Xs ) ) ) ).
% drop_map
thf(fact_4027_map__update,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,K3: nat,Y: B] :
( ( map @ B @ A @ F3 @ ( list_update @ B @ Xs @ K3 @ Y ) )
= ( list_update @ A @ ( map @ B @ A @ F3 @ Xs ) @ K3 @ ( F3 @ Y ) ) ) ).
% map_update
thf(fact_4028_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,Xs: list @ B,G: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ C @ A @ G @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_4029_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,L: list @ B,L2: list @ B,I: nat] :
( ( ( map @ B @ A @ F3 @ L )
= ( map @ B @ A @ F3 @ L2 ) )
=> ( ( F3 @ ( nth @ B @ L @ I ) )
= ( F3 @ ( nth @ B @ L2 @ I ) ) ) ) ).
% map_eq_nth_eq
thf(fact_4030_take__map,axiom,
! [A: $tType,B: $tType,N: nat,F3: B > A,Xs: list @ B] :
( ( take @ A @ N @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( take @ B @ N @ Xs ) ) ) ).
% take_map
thf(fact_4031_distinct__mapI,axiom,
! [A: $tType,B: $tType,F3: B > A,L: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ L ) )
=> ( distinct @ B @ L ) ) ).
% distinct_mapI
thf(fact_4032_remdups__map__remdups,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( remdups @ A @ ( map @ B @ A @ F3 @ ( remdups @ B @ Xs ) ) )
= ( remdups @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_4033_remdups__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups @ A @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ Xs ) ) ).
% remdups_remdups
thf(fact_4034_map__concat,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ ( list @ B )] :
( ( map @ B @ A @ F3 @ ( concat @ B @ Xs ) )
= ( concat @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F3 ) @ Xs ) ) ) ).
% map_concat
thf(fact_4035_rev__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( rev @ A @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( rev @ B @ Xs ) ) ) ).
% rev_map
thf(fact_4036_map__butlast,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( map @ B @ A @ F3 @ ( butlast @ B @ Xs ) )
= ( butlast @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ).
% map_butlast
thf(fact_4037_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F3: A > B] :
( ( map @ A @ B @ F3 @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_4038_List_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys: list @ A,Zs: list @ A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ? [Us2: list @ B,Vs3: list @ B] :
( ( Xs
= ( append @ B @ Us2 @ Vs3 ) )
& ( Ys
= ( map @ B @ A @ F3 @ Us2 ) )
& ( Zs
= ( map @ B @ A @ F3 @ Vs3 ) ) ) ) ) ).
% List.map_eq_append_conv
thf(fact_4039_List_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ A,Zs: list @ A,F3: B > A,Xs: list @ B] :
( ( ( append @ A @ Ys @ Zs )
= ( map @ B @ A @ F3 @ Xs ) )
= ( ? [Us2: list @ B,Vs3: list @ B] :
( ( Xs
= ( append @ B @ Us2 @ Vs3 ) )
& ( Ys
= ( map @ B @ A @ F3 @ Us2 ) )
& ( Zs
= ( map @ B @ A @ F3 @ Vs3 ) ) ) ) ) ).
% List.append_eq_map_conv
thf(fact_4040_Misc_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Ls3: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F3 @ Ls3 )
= ( append @ A @ Fl @ Fl2 ) )
= ( ? [L4: list @ B,L7: list @ B] :
( ( Ls3
= ( append @ B @ L4 @ L7 ) )
& ( ( map @ B @ A @ F3 @ L4 )
= Fl )
& ( ( map @ B @ A @ F3 @ L7 )
= Fl2 ) ) ) ) ).
% Misc.map_eq_append_conv
thf(fact_4041_Misc_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Fl: list @ A,Fl2: list @ A,F3: B > A,Ls3: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F3 @ Ls3 ) )
= ( ? [L4: list @ B,L7: list @ B] :
( ( Ls3
= ( append @ B @ L4 @ L7 ) )
& ( ( map @ B @ A @ F3 @ L4 )
= Fl )
& ( ( map @ B @ A @ F3 @ L7 )
= Fl2 ) ) ) ) ).
% Misc.append_eq_map_conv
thf(fact_4042_map__eq__appendE,axiom,
! [B: $tType,A: $tType,F3: B > A,Ls3: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F3 @ Ls3 )
= ( append @ A @ Fl @ Fl2 ) )
=> ~ ! [L3: list @ B,L8: list @ B] :
( ( Ls3
= ( append @ B @ L3 @ L8 ) )
=> ( ( ( map @ B @ A @ F3 @ L3 )
= Fl )
=> ( ( map @ B @ A @ F3 @ L8 )
!= Fl2 ) ) ) ) ).
% map_eq_appendE
thf(fact_4043_append__eq__mapE,axiom,
! [B: $tType,A: $tType,Fl: list @ A,Fl2: list @ A,F3: B > A,Ls3: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F3 @ Ls3 ) )
=> ~ ! [L3: list @ B,L8: list @ B] :
( ( Ls3
= ( append @ B @ L3 @ L8 ) )
=> ( ( ( map @ B @ A @ F3 @ L3 )
= Fl )
=> ( ( map @ B @ A @ F3 @ L8 )
!= Fl2 ) ) ) ) ).
% append_eq_mapE
thf(fact_4044_distinct__remdups__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups @ A @ Xs )
= Xs ) ) ).
% distinct_remdups_id
thf(fact_4045_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_4046_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R: A > A > $o,F3: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ R @ ( map @ B @ A @ F3 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X4: B,Y5: B] : ( R @ ( F3 @ X4 ) @ ( F3 @ Y5 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_4047_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P4: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ Z2 )
=> ~ ! [X: A,Y2: B] :
( ( P4
= ( product_Pair @ A @ B @ X @ Y2 ) )
=> ~ ( C2 @ X @ Y2 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_4048_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A4: A,B3: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A4 @ B3 ) @ C2 )
=> ( R @ A4 @ B3 @ C2 ) ) ).
% case_prodD'
thf(fact_4049_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: ( product_prod @ B @ C ) > A,Xs: list @ B,G: D > C,Ys: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F3 @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G @ Ys ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X4: B,Y5: D] : ( F3 @ ( product_Pair @ B @ C @ X4 @ ( G @ Y5 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys ) ) ) ).
% map_zip_map2
thf(fact_4050_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: ( product_prod @ B @ C ) > A,G: D > B,Xs: list @ D,Ys: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F3 @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs ) @ Ys ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X4: D,Y5: C] : ( F3 @ ( product_Pair @ B @ C @ ( G @ X4 ) @ Y5 ) ) )
@ ( zip @ D @ C @ Xs @ Ys ) ) ) ).
% map_zip_map
thf(fact_4051_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_4052_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G: A > B > A,A4: A,F3: C > B,Xs: list @ C] :
( ( foldl @ A @ B @ G @ A4 @ ( map @ C @ B @ F3 @ Xs ) )
= ( foldl @ A @ C
@ ^ [A5: A,X4: C] : ( G @ A5 @ ( F3 @ X4 ) )
@ A4
@ Xs ) ) ).
% foldl_map
thf(fact_4053_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F3: C > B,Ys: list @ C] :
( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F3 @ Ys ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X4: A,Y5: C] : ( product_Pair @ A @ B @ X4 @ ( F3 @ Y5 ) ) )
@ ( zip @ A @ C @ Xs @ Ys ) ) ) ).
% zip_map2
thf(fact_4054_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,Xs: list @ C,Ys: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F3 @ Xs ) @ Ys )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F3 @ X4 ) ) )
@ ( zip @ C @ B @ Xs @ Ys ) ) ) ).
% zip_map1
thf(fact_4055_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: 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 )
@ ^ [X4: C,Y5: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G @ Y5 ) ) )
@ ( zip @ C @ D @ Xs @ Ys ) )
= ( zip @ A @ B @ ( map @ C @ A @ F3 @ Xs ) @ ( map @ D @ B @ G @ Ys ) ) ) ).
% map_prod_fun_zip
thf(fact_4056_insert__remdups,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( insert @ A @ X3 @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ ( insert @ A @ X3 @ Xs ) ) ) ).
% insert_remdups
thf(fact_4057_image__set,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs ) )
= ( set2 @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ).
% image_set
thf(fact_4058_map__consI_I2_J,axiom,
! [B: $tType,A: $tType,W2: list @ A,L: list @ A,F3: B > A,Ww: list @ B,A4: B] :
( ( ( append @ A @ W2 @ L )
= ( append @ A @ ( map @ B @ A @ F3 @ Ww ) @ L ) )
=> ( ( cons @ A @ ( F3 @ A4 ) @ ( append @ A @ W2 @ L ) )
= ( append @ A @ ( map @ B @ A @ F3 @ ( cons @ B @ A4 @ Ww ) ) @ L ) ) ) ).
% map_consI(2)
thf(fact_4059_distinct__map__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,L: list @ B,X3: B,Y: B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ L ) )
=> ( ( ( F3 @ X3 )
= ( F3 @ Y ) )
=> ( ( member @ B @ X3 @ ( set2 @ B @ L ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ L ) )
=> ( X3 = Y ) ) ) ) ) ).
% distinct_map_eq
thf(fact_4060_hd__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F3 @ Xs ) )
= ( F3 @ ( hd @ A @ Xs ) ) ) ) ).
% hd_map
thf(fact_4061_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A4: list @ A,F3: A > B] :
( ( A4
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F3 @ A4 ) )
= ( F3 @ ( hd @ A @ A4 ) ) ) ) ).
% list.map_sel(1)
thf(fact_4062_list_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A4: list @ A,F3: A > B] :
( ( A4
!= ( nil @ A ) )
=> ( ( tl @ B @ ( map @ A @ B @ F3 @ A4 ) )
= ( map @ A @ B @ F3 @ ( tl @ A @ A4 ) ) ) ) ).
% list.map_sel(2)
thf(fact_4063_last__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ B @ ( map @ A @ B @ F3 @ Xs ) )
= ( F3 @ ( last @ A @ Xs ) ) ) ) ).
% last_map
thf(fact_4064_lists__image__witness,axiom,
! [A: $tType,B: $tType,X3: list @ A,F3: B > A,Q: set @ B] :
( ( member @ ( list @ A ) @ X3 @ ( lists @ A @ ( image2 @ B @ A @ F3 @ Q ) ) )
=> ~ ! [Xo: list @ B] :
( ( member @ ( list @ B ) @ Xo @ ( lists @ B @ Q ) )
=> ( X3
!= ( map @ B @ A @ F3 @ Xo ) ) ) ) ).
% lists_image_witness
thf(fact_4065_map__eq__map__tailrec,axiom,
! [B: $tType,A: $tType] :
( ( map @ A @ B )
= ( map_tailrec @ A @ B ) ) ).
% map_eq_map_tailrec
thf(fact_4066_wf,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ).
% wf
thf(fact_4067_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X4: B,Y5: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y5 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_4068_wf__if__measure,axiom,
! [A: $tType,P: A > $o,F3: A > nat,G: A > A] :
( ! [X: A] :
( ( P @ X )
=> ( ord_less @ nat @ ( F3 @ ( G @ X ) ) @ ( F3 @ X ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,X4: A] :
( ( P @ X4 )
& ( Y5
= ( G @ X4 ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_4069_remdups_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X3 @ Xs ) )
= ( remdups @ A @ Xs ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( remdups @ A @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_4070_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( ( semiring_0 @ A )
=> ! [F3: B > A,C2: A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C2 )
@ Xs ) )
= ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) @ C2 ) ) ) ).
% sum_list_mult_const
thf(fact_4071_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F3: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C2 @ ( F3 @ X4 ) )
@ Xs ) )
= ( times_times @ A @ C2 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ) ).
% sum_list_const_mult
thf(fact_4072_sum__list__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F3: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ Xs ) )
= ( minus_minus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_subtractf
thf(fact_4073_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_4074_set__eq__iff__mset__remdups__eq,axiom,
! [A: $tType,X3: list @ A,Y: list @ A] :
( ( ( set2 @ A @ X3 )
= ( set2 @ A @ Y ) )
= ( ( mset @ A @ ( remdups @ A @ X3 ) )
= ( mset @ A @ ( remdups @ A @ Y ) ) ) ) ).
% set_eq_iff_mset_remdups_eq
thf(fact_4075_remove1__remdups,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X3 @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ ( remove1 @ A @ X3 @ Xs ) ) ) ) ).
% remove1_remdups
thf(fact_4076_union__fold__insert,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= ( finite_fold @ A @ ( set @ A ) @ ( insert3 @ A ) @ B2 @ A3 ) ) ) ).
% union_fold_insert
thf(fact_4077_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,X3: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( remove1 @ B @ X3 @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_4078_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J2 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus @ nat @ K4 @ I4 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_4079_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J2 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus @ nat @ K4 @ I4 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_4080_finite__subset__wf,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X9: set @ A,Y9: set @ A] :
( ( ord_less @ ( set @ A ) @ X9 @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ Y9 @ A3 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_4081_wf__bounded__supset,axiom,
! [A: $tType,S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [Q9: set @ A,Q2: set @ A] :
( ( ord_less @ ( set @ A ) @ Q2 @ Q9 )
& ( ord_less_eq @ ( set @ A ) @ Q9 @ S ) ) ) ) ) ) ).
% wf_bounded_supset
thf(fact_4082_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G: A > B] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_4083_lists__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( lists @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
= ( image2 @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F3 ) @ ( lists @ B @ A3 ) ) ) ).
% lists_image
thf(fact_4084_map__tailrec__rev,axiom,
! [A: $tType,B: $tType] :
( ( map_tailrec_rev @ B @ A )
= ( ^ [F7: B > A,As4: list @ B] : ( append @ A @ ( rev @ A @ ( map @ B @ A @ F7 @ As4 ) ) ) ) ) ).
% map_tailrec_rev
thf(fact_4085_minus__fold__remove,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ B2 @ A3 )
= ( finite_fold @ A @ ( set @ A ) @ ( remove @ A ) @ B2 @ A3 ) ) ) ).
% minus_fold_remove
thf(fact_4086_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_4087_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_4088_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S2: set @ ( A > B > $o ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S2 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_4089_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F3 @ X ) @ ( G @ X ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_4090_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A7: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
& ( finite_finite2 @ A @ B7 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_4091_distinct__idx,axiom,
! [B: $tType,A: $tType,F3: B > A,L: list @ B,I: nat,J: nat] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ L ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ( F3 @ ( nth @ B @ L @ I ) )
= ( F3 @ ( nth @ B @ L @ J ) ) )
=> ( I = J ) ) ) ) ) ).
% distinct_idx
thf(fact_4092_map__upd__eq,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F3: A > B,X3: A] :
( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F3 @ ( nth @ A @ L @ I ) )
= ( F3 @ X3 ) ) )
=> ( ( map @ A @ B @ F3 @ ( list_update @ A @ L @ I @ X3 ) )
= ( map @ A @ B @ F3 @ L ) ) ) ).
% map_upd_eq
thf(fact_4093_sum__list__distinct__conv__sum__set,axiom,
! [C: $tType,B: $tType] :
( ( comm_monoid_add @ C )
=> ! [Xs: list @ B,F3: B > C] :
( ( distinct @ B @ Xs )
=> ( ( groups8242544230860333062m_list @ C @ ( map @ B @ C @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ B @ C @ F3 @ ( set2 @ B @ Xs ) ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_4094_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_4095_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X3: B,Xs: list @ B,F3: B > A] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) )
= ( plus_plus @ A @ ( F3 @ X3 ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( remove1 @ B @ X3 @ Xs ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_4096_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( complete_Sup_Sup @ A @ A3 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A3 ) ) ) ) ).
% Sup_fold_sup
thf(fact_4097_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R2: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X4: C] : R2
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R2 ) ) ) ).
% sum_list_triv
thf(fact_4098_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X3 @ A3 ) ) ) ) ).
% Min.eq_fold
thf(fact_4099_image__fold__insert,axiom,
! [B: $tType,A: $tType,A3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( image2 @ A @ B @ F3 @ A3 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K4: A] : ( insert3 @ B @ ( F3 @ K4 ) )
@ ( bot_bot @ ( set @ B ) )
@ A3 ) ) ) ).
% image_fold_insert
thf(fact_4100_trancl__insert2,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y5: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A4 ) @ ( transitive_trancl @ A @ R2 ) )
| ( X4 = A4 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ Y5 ) @ ( transitive_trancl @ A @ R2 ) )
| ( Y5 = B3 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_4101_map__by__foldl,axiom,
! [B: $tType,A: $tType,F3: A > B,L: list @ A] :
( ( foldl @ ( list @ B ) @ A
@ ^ [L4: list @ B,X4: A] : ( append @ B @ L4 @ ( cons @ B @ ( F3 @ X4 ) @ ( nil @ B ) ) )
@ ( nil @ B )
@ L )
= ( map @ A @ B @ F3 @ L ) ) ).
% map_by_foldl
thf(fact_4102_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_4103_map__distinct__upd__conv,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F3: A > B,X3: B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( distinct @ A @ L )
=> ( ( list_update @ B @ ( map @ A @ B @ F3 @ L ) @ I @ X3 )
= ( map @ A @ B @ ( fun_upd @ A @ B @ F3 @ ( nth @ A @ L @ I ) @ X3 ) @ L ) ) ) ) ).
% map_distinct_upd_conv
thf(fact_4104_aux,axiom,
! [B: $tType,A: $tType,P: A > B > assn,A4: A,As2: list @ A,C2: B,Cs: list @ B] :
( ( finite_fold @ nat @ assn
@ ^ [I4: nat,Aa3: assn] : ( times_times @ assn @ Aa3 @ ( P @ ( nth @ A @ ( cons @ A @ A4 @ As2 ) @ I4 ) @ ( nth @ B @ ( cons @ B @ C2 @ Cs ) @ I4 ) ) )
@ ( one_one @ assn )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ ( size_size @ ( list @ A ) @ As2 ) ) ) )
= ( times_times @ assn @ ( P @ A4 @ C2 )
@ ( finite_fold @ nat @ assn
@ ^ [I4: nat,Aa3: assn] : ( times_times @ assn @ Aa3 @ ( P @ ( nth @ A @ As2 @ I4 ) @ ( nth @ B @ Cs @ I4 ) ) )
@ ( one_one @ assn )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ As2 ) ) ) ) ) ).
% aux
thf(fact_4105_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R5: A > A > $o,Xs3: list @ A,Ys3: list @ A] : ( member @ ( 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 @ R5 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_4106_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R5: A > A > $o,Xs3: list @ A,Ys3: list @ A] : ( member @ ( 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 @ R5 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_4107_mult1__lessE,axiom,
! [A: $tType,N7: multiset @ A,M5: multiset @ A,R2: A > A > $o] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ M5 ) @ ( mult1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R2 ) ) ) )
=> ( ( asymp @ A @ R2 )
=> ~ ! [A6: A,M0: multiset @ A] :
( ( M5
= ( add_mset @ A @ A6 @ M0 ) )
=> ! [K5: multiset @ A] :
( ( N7
= ( plus_plus @ ( multiset @ A ) @ M0 @ K5 ) )
=> ( ~ ( member @ A @ A6 @ ( set_mset @ A @ K5 ) )
=> ~ ! [B10: A] :
( ( member @ A @ B10 @ ( set_mset @ A @ K5 ) )
=> ( R2 @ B10 @ A6 ) ) ) ) ) ) ) ).
% mult1_lessE
thf(fact_4108_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S )
=> ( ( finite_finite2 @ B @ A3 )
=> ~ ! [L3: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L3 ) )
=> ( ( ( set2 @ B @ L3 )
= A3 )
=> ( ( size_size @ ( list @ B ) @ L3 )
!= ( finite_card @ B @ A3 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_4109_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 ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys3 )
@ Xs3 ) ) ) ) ).
% product_concat_map
thf(fact_4110_asymp_Ocases,axiom,
! [A: $tType,A4: A > A > $o] :
( ( asymp @ A @ A4 )
=> ! [A15: A,B10: A] :
( ( A4 @ A15 @ B10 )
=> ~ ( A4 @ B10 @ A15 ) ) ) ).
% asymp.cases
thf(fact_4111_asymp_Osimps,axiom,
! [A: $tType] :
( ( asymp @ A )
= ( ^ [A5: A > A > $o] :
? [R4: A > A > $o] :
( ( A5 = R4 )
& ! [X4: A,Y5: A] :
( ( R4 @ X4 @ Y5 )
=> ~ ( R4 @ Y5 @ X4 ) ) ) ) ) ).
% asymp.simps
thf(fact_4112_asymp_Ointros,axiom,
! [A: $tType,R: A > A > $o] :
( ! [A6: A,B5: A] :
( ( R @ A6 @ B5 )
=> ~ ( R @ B5 @ A6 ) )
=> ( asymp @ A @ R ) ) ).
% asymp.intros
thf(fact_4113_asympD,axiom,
! [A: $tType,R: A > A > $o,X3: A,Y: A] :
( ( asymp @ A @ R )
=> ( ( R @ X3 @ Y )
=> ~ ( R @ Y @ X3 ) ) ) ).
% asympD
thf(fact_4114_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 )
@ ^ [Y5: A] : ( cons @ A @ Y5 @ Ys3 )
@ Xs )
@ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_4115_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 ) )
@ ^ [X4: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 ) @ ( product_lists @ A @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_4116_asymp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( asymp @ A @ ( ord_less @ A ) ) ) ).
% asymp_less
thf(fact_4117_length__concat,axiom,
! [B: $tType,Xss2: list @ ( list @ B )] :
( ( size_size @ ( list @ B ) @ ( concat @ B @ Xss2 ) )
= ( groups8242544230860333062m_list @ nat @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss2 ) ) ) ).
% length_concat
thf(fact_4118_zip__same__conv__map,axiom,
! [A: $tType,Xs: list @ A] :
( ( zip @ A @ A @ Xs @ Xs )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Xs ) ) ).
% zip_same_conv_map
thf(fact_4119_List_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind @ A @ B )
= ( ^ [Xs3: list @ A,F7: A > ( list @ B )] : ( concat @ B @ ( map @ A @ ( list @ B ) @ F7 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_4120_asymp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( asymp @ A
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ) ).
% asymp_greater
thf(fact_4121_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
=> ( distinct @ B @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_4122_sum__list__Suc,axiom,
! [A: $tType,F3: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X4: A] : ( suc @ ( F3 @ X4 ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_4123_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_4124_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 ) )
@ ^ [Y5: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X4: A,Z6: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y5 @ Z6 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_4125_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 ) ) )
@ ^ [X4: A,Y5: B,Z6: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y5 @ Z6 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_4126_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 )
@ ^ [X4: B,Y5: A] : ( product_Pair @ A @ B @ Y5 @ X4 ) )
@ ( zip @ B @ A @ Ys3 @ Xs3 ) ) ) ) ).
% zip_commute
thf(fact_4127_sum__set__upto__conv__sum__list__int,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: int > A,I: int,J: int] :
( ( groups7311177749621191930dd_sum @ int @ A @ F3 @ ( set2 @ int @ ( upto @ I @ J ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ int @ A @ F3 @ ( upto @ I @ J ) ) ) ) ) ).
% sum_set_upto_conv_sum_list_int
thf(fact_4128_interv__sum__list__conv__sum__set__int,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ! [F3: int > B,K3: int,L: int] :
( ( groups8242544230860333062m_list @ B @ ( map @ int @ B @ F3 @ ( upto @ K3 @ L ) ) )
= ( groups7311177749621191930dd_sum @ int @ B @ F3 @ ( set2 @ int @ ( upto @ K3 @ L ) ) ) ) ) ).
% interv_sum_list_conv_sum_set_int
thf(fact_4129_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_4130_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X3 @ Xs ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( subseqs @ A @ Xs ) ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_4131_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X3 @ Xs ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_4132_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 ) )
@ ^ [X4: A] : X4 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_4133_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F3: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X4 ) @ ( F3 @ X4 ) )
@ ( set2 @ A @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_4134_mergesort__remdups__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( mergesort_remdups @ A )
= ( ^ [Xs3: list @ A] :
( merge_list @ A @ ( nil @ ( list @ A ) )
@ ( map @ A @ ( list @ A )
@ ^ [X4: A] : ( cons @ A @ X4 @ ( nil @ A ) )
@ Xs3 ) ) ) ) ) ).
% mergesort_remdups_def
thf(fact_4135_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X5: set @ A,F3: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X5 )
=> ( ( finite_finite2 @ A @ X5 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X4 ) @ ( F3 @ X4 ) )
@ X5 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_4136_fold__union__pair,axiom,
! [B: $tType,A: $tType,B2: set @ A,X3: B,A3: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B2 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y5: A] : ( insert3 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y5 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B2 ) )
@ A3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y5: A] : ( insert3 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y5 ) )
@ A3
@ B2 ) ) ) ).
% fold_union_pair
thf(fact_4137_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 ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_4138_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [A13: A,A23: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B1: A,B22: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ A13 @ ( insert3 @ A @ A23 @ ( insert3 @ A @ B1 @ ( insert3 @ A @ B22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R5 ) )
& ( ( ( A13 = B1 )
& ( A23 = B22 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A13 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B1 @ B22 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A13 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B1 @ B22 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A13 @ B1 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A13 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B1 @ B22 ) )
& ( A13 = B1 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B22 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_4139_natLeq__on__wo__rel,axiom,
! [N: nat] :
( bNF_Wellorder_wo_rel @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_4140_Field__natLeq__on,axiom,
! [N: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) )
= ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_4141_transpose_Osimps_I3_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Xss2: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Xss2 ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X3
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H3: A,T3: list @ A] : ( cons @ A @ H3 @ ( 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 ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_4142_transpose_Oelims,axiom,
! [A: $tType,X3: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X3 )
= Y )
=> ( ( ( X3
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( ( X3
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ( Y
!= ( transpose @ A @ Xss ) ) )
=> ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( X3
= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) )
=> ( Y
!= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H3: A,T3: list @ A] : ( cons @ A @ H3 @ ( 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 ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_4143_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),As6: A > B] :
! [I4: A,J2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I4 @ J2 ) @ R5 )
=> ( ord_less_eq @ B @ ( As6 @ I4 ) @ ( As6 @ J2 ) ) ) ) ) ) ).
% relChain_def
thf(fact_4144_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_4145_transpose_Osimps_I1_J,axiom,
! [A: $tType] :
( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ).
% transpose.simps(1)
thf(fact_4146_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_4147_transpose__map__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ ( list @ B )] :
( ( transpose @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F3 ) @ Xs ) )
= ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F3 ) @ ( transpose @ B @ Xs ) ) ) ).
% transpose_map_map
thf(fact_4148_transpose__empty,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( ( transpose @ A @ Xs )
= ( nil @ ( list @ A ) ) )
= ( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( X4
= ( nil @ A ) ) ) ) ) ).
% transpose_empty
thf(fact_4149_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_4150_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_4151_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,F3: B > A,A3: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L ) )
& ( ( set2 @ B @ L )
= A3 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A3 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_4152_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_4153_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,F3: B > A,X3: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert3 @ B @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_4154_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_4155_filter__filter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs: list @ A] :
( ( filter2 @ A @ P @ ( filter2 @ A @ Q @ Xs ) )
= ( filter2 @ A
@ ^ [X4: A] :
( ( Q @ X4 )
& ( P @ X4 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_4156_filter__True,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) )
=> ( ( filter2 @ A @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_4157_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_4158_remove1__filter__not,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ~ ( P @ X3 )
=> ( ( remove1 @ A @ X3 @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% remove1_filter_not
thf(fact_4159_removeAll__filter__not,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ~ ( P @ X3 )
=> ( ( removeAll @ A @ X3 @ ( filter2 @ A @ P @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% removeAll_filter_not
thf(fact_4160_set__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P @ Xs ) )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% set_filter
thf(fact_4161_filter__False,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X ) )
=> ( ( filter2 @ A @ P @ Xs )
= ( nil @ A ) ) ) ).
% filter_False
thf(fact_4162_partition__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( member @ ( list @ A ) @ Xs
@ ( shuffles @ A @ ( filter2 @ A @ P @ Xs )
@ ( filter2 @ A
@ ^ [X4: A] :
~ ( P @ X4 )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_4163_filter_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( ( P @ X3 )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( filter2 @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X3 )
=> ( ( filter2 @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( filter2 @ A @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_4164_filter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Xs )
= Xs )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) ) ) ) ).
% filter_id_conv
thf(fact_4165_filter__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( filter2 @ A @ P @ Xs )
= ( filter2 @ A @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_4166_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_4167_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_4168_sorted__wrt__filter,axiom,
! [A: $tType,F3: A > A > $o,Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ F3 @ Xs )
=> ( sorted_wrt @ A @ F3 @ ( filter2 @ A @ P @ Xs ) ) ) ).
% sorted_wrt_filter
thf(fact_4169_filter__remove1,axiom,
! [A: $tType,Q: A > $o,X3: A,Xs: list @ A] :
( ( filter2 @ A @ Q @ ( remove1 @ A @ X3 @ Xs ) )
= ( remove1 @ A @ X3 @ ( filter2 @ A @ Q @ Xs ) ) ) ).
% filter_remove1
thf(fact_4170_filter_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( filter2 @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% filter.simps(1)
thf(fact_4171_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_4172_distinct__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( filter2 @ A @ P @ Xs ) ) ) ).
% distinct_filter
thf(fact_4173_removeAll__filter__not__eq,axiom,
! [A: $tType] :
( ( removeAll @ A )
= ( ^ [X4: A] :
( filter2 @ A
@ ^ [Y5: A] : X4 != Y5 ) ) ) ).
% removeAll_filter_not_eq
thf(fact_4174_empty__filter__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( nil @ A )
= ( filter2 @ A @ P @ Xs ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X4 ) ) ) ) ).
% empty_filter_conv
thf(fact_4175_filter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Xs )
= ( nil @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ~ ( P @ X4 ) ) ) ) ).
% filter_empty_conv
thf(fact_4176_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_4177_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_4178_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_4179_distinct__map__filter,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,P: B > $o] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
=> ( distinct @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_4180_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_4181_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
@ ^ [X4: A] :
~ ( P @ X4 )
@ Xs ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_4182_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_4183_inter__set__filter,axiom,
! [A: $tType,A3: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( set2 @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_4184_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X4: A] :
( X4
= ( G @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_4185_mset__eq__length__filter,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Z2: A] :
( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) )
=> ( ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y4: A,Z4: A] : Y4 = Z4
@ Z2 )
@ Xs ) )
= ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y4: A,Z4: A] : Y4 = Z4
@ Z2 )
@ Ys ) ) ) ) ).
% mset_eq_length_filter
thf(fact_4186_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_4187_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_4188_union__coset__filter,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( coset @ A @ Xs ) @ A3 )
= ( coset @ A
@ ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A3 )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_4189_length__filter__less,axiom,
! [A: $tType,X3: A,Xs: list @ A,P: A > $o] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_4190_filter__eq__Cons__iff,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X3: A,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X3 @ Xs ) )
= ( ? [Us2: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X3 @ Vs3 ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Us2 ) )
=> ~ ( P @ X4 ) )
& ( P @ X3 )
& ( Xs
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_4191_Cons__eq__filter__iff,axiom,
! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X3 @ Xs )
= ( filter2 @ A @ P @ Ys ) )
= ( ? [Us2: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X3 @ Vs3 ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Us2 ) )
=> ~ ( P @ X4 ) )
& ( P @ X3 )
& ( Xs
= ( filter2 @ A @ P @ Vs3 ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_4192_filter__eq__ConsD,axiom,
! [A: $tType,P: A > $o,Ys: list @ A,X3: A,Xs: list @ A] :
( ( ( filter2 @ A @ P @ Ys )
= ( cons @ A @ X3 @ Xs ) )
=> ? [Us3: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X3 @ Vs2 ) ) )
& ! [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X8 ) )
& ( P @ X3 )
& ( Xs
= ( filter2 @ A @ P @ Vs2 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_4193_Cons__eq__filterD,axiom,
! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X3 @ Xs )
= ( filter2 @ A @ P @ Ys ) )
=> ? [Us3: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X3 @ Vs2 ) ) )
& ! [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Us3 ) )
=> ~ ( P @ X8 ) )
& ( P @ X3 )
& ( Xs
= ( filter2 @ A @ P @ Vs2 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_4194_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,P: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ).
% sorted_filter
thf(fact_4195_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,G: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F3
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= ( G @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_4196_minus__coset__filter,axiom,
! [A: $tType,A3: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( coset @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_4197_filter__eq__snocD,axiom,
! [A: $tType,P: A > $o,L: list @ A,L2: list @ A,X3: A] :
( ( ( filter2 @ A @ P @ L )
= ( append @ A @ L2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ L ) )
& ( P @ X3 ) ) ) ).
% filter_eq_snocD
thf(fact_4198_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P: B > $o,F3: B > A] :
( ! [X: B] :
( ( member @ B @ X @ ( set2 @ B @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( F3 @ X )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_4199_set__minus__filter__out,axiom,
! [A: $tType,Xs: list @ A,Y: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert3 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : X4 != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_4200_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_4201_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_4202_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_4203_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_4204_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
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P4 @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_4205_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,F3: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( nil @ B ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
thf(fact_4206_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,F3: B > A,A3: set @ B,B2: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ S )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 )
= ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ B2 ) )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( A3 = B2 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
thf(fact_4207_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 ) ) )
=> ? [M6: nat] :
( ( ord_less_eq @ nat @ N @ M6 )
& ( ord_less @ nat @ M6 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ ( filter2 @ A @ P @ Xs ) @ N )
= ( nth @ A @ Xs @ M6 ) )
& ( ( filter2 @ A @ P @ ( take @ A @ M6 @ Xs ) )
= ( take @ A @ N @ ( filter2 @ A @ P @ Xs ) ) ) ) ) ).
% filter_nth_ex_nth
thf(fact_4208_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_4209_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,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) )
= A3 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
thf(fact_4210_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,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) )
= ( finite_card @ B @ A3 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_4211_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,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S )
=> ( distinct @ A @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_4212_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,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S )
=> ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_4213_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,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S )
=> ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_4214_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_4215_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,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 )
= ( nil @ B ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4216_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 )
@ ^ [H3: A,T3: list @ A] : ( cons @ A @ H3 @ ( 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_4217_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_4218_distinct__concat_H,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xs ) )
=> ( ! [Ys5: list @ A] :
( ( member @ ( list @ A ) @ Ys5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys5 ) )
=> ( ! [Ys5: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys5 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys5 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat'
thf(fact_4219_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( set2 @ B @ Xs ) @ S )
=> ( ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( set2 @ B @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_4220_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 ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( 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_4221_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,F3: B > A,X3: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert3 @ B @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert3 @ B @ X3 @ A3 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_4222_remove__rev__alt__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X4: A,Xs3: list @ A] :
( filter2 @ A
@ ^ [Y5: A] : Y5 != X4
@ ( rev @ A @ Xs3 ) ) ) ) ).
% remove_rev_alt_def
thf(fact_4223_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,F3: B > A,X3: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert3 @ B @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X3 @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert3 @ B @ X3 @ A3 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_4224_sort__by__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs )
= ( append @ A
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ ( filter2 @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ ( nth @ A @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ Xs ) )
@ ( append @ A
@ ( filter2 @ A
@ ^ [X4: A] :
( X4
= ( nth @ A @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ Xs )
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ ( filter2 @ A @ ( ord_less @ A @ ( nth @ A @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Xs ) ) ) ) ) ) ).
% sort_by_quicksort
thf(fact_4225_sort__upto,axiom,
! [I: int,J: int] :
( ( linorder_sort_key @ int @ int
@ ^ [X4: int] : X4
@ ( upto @ I @ J ) )
= ( upto @ I @ J ) ) ).
% sort_upto
thf(fact_4226_sort__key__simps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A] :
( ( linorder_sort_key @ B @ A @ F3 @ ( nil @ B ) )
= ( nil @ B ) ) ) ).
% sort_key_simps(1)
thf(fact_4227_set__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( set2 @ B @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) )
= ( set2 @ B @ Xs ) ) ) ).
% set_sort
thf(fact_4228_length__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ) ).
% length_sort
thf(fact_4229_distinct__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( distinct @ B @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) )
= ( distinct @ B @ Xs ) ) ) ).
% distinct_sort
thf(fact_4230_filter__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F3: B > A,Xs: list @ B] :
( ( filter2 @ B @ P @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) )
= ( linorder_sort_key @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs ) ) ) ) ).
% filter_sort
thf(fact_4231_linorder_Oinsort__key_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( insort_key @ A @ B )
= ( insort_key @ A @ B ) ) ).
% linorder.insort_key.cong
thf(fact_4232_sort__key__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [C2: B,Xs: list @ A] :
( ( linorder_sort_key @ A @ B
@ ^ [X4: A] : C2
@ Xs )
= Xs ) ) ).
% sort_key_const
thf(fact_4233_sort__key__stable,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,K3: B,Xs: list @ A] :
( ( filter2 @ A
@ ^ [Y5: A] :
( ( F3 @ Y5 )
= K3 )
@ ( linorder_sort_key @ A @ B @ F3 @ Xs ) )
= ( filter2 @ A
@ ^ [Y5: A] :
( ( F3 @ Y5 )
= K3 )
@ Xs ) ) ) ).
% sort_key_stable
thf(fact_4234_sorted__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs ) ) ) ).
% sorted_sort
thf(fact_4235_sorted__sort__id,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs )
= Xs ) ) ) ).
% sorted_sort_id
thf(fact_4236_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) ) ) ) ).
% sorted_sort_key
thf(fact_4237_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
@ ^ [X4: A] : X4
@ ( remdups @ A @ Xs ) ) ) ) ).
% sorted_list_of_set_sort_remdups
thf(fact_4238_properties__for__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Ys: list @ B,Xs: list @ B,F3: B > A] :
( ( ( mset @ B @ Ys )
= ( mset @ B @ Xs ) )
=> ( ! [K: B] :
( ( member @ B @ K @ ( set2 @ B @ Ys ) )
=> ( ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ K )
= ( F3 @ X4 ) )
@ Ys )
= ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ K )
= ( F3 @ X4 ) )
@ Xs ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Ys ) )
=> ( ( linorder_sort_key @ B @ A @ F3 @ Xs )
= Ys ) ) ) ) ) ).
% properties_for_sort_key
thf(fact_4239_sort__key__by__quicksort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F7: B > A,Xs3: list @ B] :
( append @ B
@ ( linorder_sort_key @ B @ A @ F7
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F7 @ X4 ) @ ( F7 @ ( nth @ B @ Xs3 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs3 ) )
@ ( append @ B
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F7 @ X4 )
= ( F7 @ ( nth @ B @ Xs3 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs3 )
@ ( linorder_sort_key @ B @ A @ F7
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F7 @ ( nth @ B @ Xs3 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( F7 @ X4 ) )
@ Xs3 ) ) ) ) ) ) ) ).
% sort_key_by_quicksort
thf(fact_4240_sort__key__by__quicksort__code,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F7: B > A,Xs3: list @ B] :
( case_list @ ( list @ B ) @ B @ ( nil @ B )
@ ^ [X4: B] :
( case_list @ ( list @ B ) @ B @ Xs3
@ ^ [Y5: B] :
( case_list @ ( list @ B ) @ B @ ( if @ ( list @ B ) @ ( ord_less_eq @ A @ ( F7 @ X4 ) @ ( F7 @ Y5 ) ) @ Xs3 @ ( cons @ B @ Y5 @ ( cons @ B @ X4 @ ( 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 @ F7 @ Lts ) @ ( append @ B @ Eqs @ ( linorder_sort_key @ B @ A @ F7 @ Gts ) ) ) )
@ ( linorder_part @ B @ A @ F7 @ ( F7 @ ( 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_4241_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I5 ) )
= N ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I4: nat] :
( map @ nat @ A
@ ^ [J2: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J2 ) @ I4 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_4242_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F7: A > nat,R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y5: A] :
( ( ord_less @ nat @ ( F7 @ X4 ) @ ( F7 @ Y5 ) )
| ( ( ord_less_eq @ nat @ ( F7 @ X4 ) @ ( F7 @ Y5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R4 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_4243_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 )
@ ^ [X4: A,Xs3: list @ A] : X4
@ ( filter2 @ A @ P
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs ) ) ) ) ) ).
% Bleast_code
thf(fact_4244_remdups__upt,axiom,
! [M2: nat,N: nat] :
( ( remdups @ nat @ ( upt @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% remdups_upt
thf(fact_4245_sort__upt,axiom,
! [M2: nat,N: nat] :
( ( linorder_sort_key @ nat @ nat
@ ^ [X4: nat] : X4
@ ( upt @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% sort_upt
thf(fact_4246_tl__upt,axiom,
! [M2: nat,N: nat] :
( ( tl @ nat @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ N ) ) ).
% tl_upt
thf(fact_4247_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( hd @ nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_4248_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_4249_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_4250_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_4251_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_4252_upt__merge,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
& ( ord_less_eq @ nat @ J @ K3 ) )
=> ( ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ K3 ) )
= ( upt @ I @ K3 ) ) ) ).
% upt_merge
thf(fact_4253_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_4254_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ I ) ) ).
% length_upt
thf(fact_4255_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_4256_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_4257_nth__upt,axiom,
! [I: nat,K3: nat,J: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K3 ) @ J )
=> ( ( nth @ nat @ ( upt @ I @ J ) @ K3 )
= ( plus_plus @ nat @ I @ K3 ) ) ) ).
% nth_upt
thf(fact_4258_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_4259_atLeastLessThan__upt,axiom,
( ( set_or7035219750837199246ssThan @ nat )
= ( ^ [I4: nat,J2: nat] : ( set2 @ nat @ ( upt @ I4 @ J2 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_4260_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct @ nat @ ( upt @ I @ J ) ) ).
% distinct_upt
thf(fact_4261_map__add__upt_H,axiom,
! [Ofs: nat,A4: nat,B3: nat] :
( ( map @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ I4 @ Ofs )
@ ( upt @ A4 @ B3 ) )
= ( upt @ ( plus_plus @ nat @ A4 @ Ofs ) @ ( plus_plus @ nat @ B3 @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_4262_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_4263_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N2: nat,M: nat] : ( set2 @ nat @ ( upt @ N2 @ ( suc @ M ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_4264_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_4265_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N2: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_4266_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list @ nat,Q4: nat] :
( ( ( cons @ nat @ M2 @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M2 @ Q4 ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M2 ) @ Q4 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_4267_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_4268_sorted__wrt__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M2 @ N ) ) ).
% sorted_wrt_upt
thf(fact_4269_map__add__upt,axiom,
! [N: nat,M2: nat] :
( ( map @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ I4 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M2 ) )
= ( upt @ N @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% map_add_upt
thf(fact_4270_sorted__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M2 @ N ) ) ).
% sorted_upt
thf(fact_4271_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N2: nat,M: nat] : ( set2 @ nat @ ( upt @ ( suc @ N2 ) @ ( suc @ M ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_4272_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 ) )
= ( ? [K4: nat] :
( ( ord_less_eq @ nat @ I @ K4 )
& ( ord_less_eq @ nat @ K4 @ J )
& ( ( upt @ I @ K4 )
= Xs )
& ( ( upt @ K4 @ J )
= Ys ) ) ) ) ) ).
% upt_eq_append_conv
thf(fact_4273_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N2: nat,M: nat] : ( set2 @ nat @ ( upt @ ( suc @ N2 ) @ M ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_4274_filter__upt__take__conv,axiom,
! [A: $tType,P: A > $o,M2: nat,L: list @ A,N: nat] :
( ( filter2 @ nat
@ ^ [I4: nat] : ( P @ ( nth @ A @ ( take @ A @ M2 @ L ) @ I4 ) )
@ ( upt @ N @ M2 ) )
= ( filter2 @ nat
@ ^ [I4: nat] : ( P @ ( nth @ A @ L @ I4 ) )
@ ( upt @ N @ M2 ) ) ) ).
% filter_upt_take_conv
thf(fact_4275_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N2: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) ) ) ).
% atMost_upto
thf(fact_4276_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_4277_enumerate__eq__zip,axiom,
! [A: $tType] :
( ( enumerate @ A )
= ( ^ [N2: nat,Xs3: list @ A] : ( zip @ nat @ A @ ( upt @ N2 @ ( plus_plus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) @ Xs3 ) ) ) ).
% enumerate_eq_zip
thf(fact_4278_upt__filter__extend,axiom,
! [U: nat,U4: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ U @ U4 )
=> ( ! [I5: nat] :
( ( ( ord_less_eq @ nat @ U @ I5 )
& ( ord_less @ nat @ I5 @ U4 ) )
=> ~ ( P @ I5 ) )
=> ( ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
= ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U4 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_4279_map__decr__upt,axiom,
! [M2: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( upt @ M2 @ N ) ) ).
% map_decr_upt
thf(fact_4280_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_4281_interv__sum__list__conv__sum__set__nat,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ! [F3: nat > B,M2: nat,N: nat] :
( ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F3 @ ( upt @ M2 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ B @ F3 @ ( set2 @ nat @ ( upt @ M2 @ N ) ) ) ) ) ).
% interv_sum_list_conv_sum_set_nat
thf(fact_4282_sum__set__upt__conv__sum__list__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set2 @ nat @ ( upt @ M2 @ N ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ nat @ A @ F3 @ ( upt @ M2 @ N ) ) ) ) ) ).
% sum_set_upt_conv_sum_list_nat
thf(fact_4283_upt__add__eq__append,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus @ nat @ J @ K3 ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K3 ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_4284_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F3: nat > A,M2: nat] :
( ( enumerate @ A @ N @ ( map @ nat @ A @ F3 @ ( upt @ N @ M2 ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K4: nat] : ( product_Pair @ nat @ A @ K4 @ ( F3 @ K4 ) )
@ ( upt @ N @ M2 ) ) ) ).
% enumerate_map_upt
thf(fact_4285_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X3: nat,Xs: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X3 @ Xs ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X3 )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_4286_upt__rec,axiom,
( upt
= ( ^ [I4: nat,J2: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I4 @ J2 ) @ ( cons @ nat @ I4 @ ( upt @ ( suc @ I4 ) @ J2 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_4287_map__upt__Suc,axiom,
! [A: $tType,F3: nat > A,N: nat] :
( ( map @ nat @ A @ F3 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( cons @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I4: nat] : ( F3 @ ( suc @ I4 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_4288_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_4289_mlex__less,axiom,
! [A: $tType,F3: A > nat,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F3 @ R ) ) ) ).
% mlex_less
thf(fact_4290_mlex__iff,axiom,
! [A: $tType,X3: A,Y: A,F3: A > nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F3 @ R ) )
= ( ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X3 )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R ) ) ) ) ).
% mlex_iff
thf(fact_4291_mlex__leq,axiom,
! [A: $tType,F3: A > nat,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( mlex_prod @ A @ F3 @ R ) ) ) ) ).
% mlex_leq
thf(fact_4292_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M2: nat,F3: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M2 ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F3 @ ( upt @ M2 @ N ) ) @ I )
= ( F3 @ ( plus_plus @ nat @ M2 @ I ) ) ) ) ).
% nth_map_upt
thf(fact_4293_part__code_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Pivot: A] :
( ( linorder_part @ B @ A @ F3 @ 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_4294_upt__eq__lel__conv,axiom,
! [L: nat,H2: nat,Is1: list @ nat,I: nat,Is2: list @ nat] :
( ( ( upt @ L @ H2 )
= ( append @ nat @ Is1 @ ( cons @ nat @ I @ Is2 ) ) )
= ( ( Is1
= ( upt @ L @ I ) )
& ( Is2
= ( upt @ ( suc @ I ) @ H2 ) )
& ( ord_less_eq @ nat @ L @ I )
& ( ord_less @ nat @ I @ H2 ) ) ) ).
% upt_eq_lel_conv
thf(fact_4295_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_4296_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_4297_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,M2: nat,F3: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N @ M2 ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I5 )
= ( F3 @ ( plus_plus @ nat @ M2 @ I5 ) ) ) )
=> ( ( map @ nat @ A @ F3 @ ( upt @ M2 @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_4298_map__nth__upt__drop__take__conv,axiom,
! [A: $tType,N7: nat,L: list @ A,M5: nat] :
( ( ord_less_eq @ nat @ N7 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( map @ nat @ A @ ( nth @ A @ L ) @ ( upt @ M5 @ N7 ) )
= ( drop @ A @ M5 @ ( take @ A @ N7 @ L ) ) ) ) ).
% map_nth_upt_drop_take_conv
thf(fact_4299_filter__upt__last,axiom,
! [A: $tType,P: A > $o,L: list @ A,Js2: list @ nat,J: nat,I: nat] :
( ( ( filter2 @ nat
@ ^ [K4: nat] : ( P @ ( nth @ A @ L @ K4 ) )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( append @ nat @ Js2 @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
=> ( ( ord_less @ nat @ J @ I )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% filter_upt_last
thf(fact_4300_part__code_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Pivot: A,X3: B,Xs: list @ B] :
( ( linorder_part @ B @ A @ F3 @ Pivot @ ( cons @ B @ X3 @ 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 @ ( F3 @ X3 ) @ Pivot ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( cons @ B @ X3 @ Lts ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ Gts ) ) @ ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ Pivot @ ( F3 @ X3 ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ ( cons @ B @ X3 @ Gts ) ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( cons @ B @ X3 @ Eqs ) @ Gts ) ) ) ) )
@ ( linorder_part @ B @ A @ F3 @ Pivot @ Xs ) ) ) ) ).
% part_code(2)
thf(fact_4301_part__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_part @ B @ A )
= ( ^ [F7: B > A,Pivot2: A,Xs3: list @ B] :
( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) )
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F7 @ X4 ) @ Pivot2 )
@ Xs3 )
@ ( product_Pair @ ( list @ B ) @ ( list @ B )
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F7 @ X4 )
= Pivot2 )
@ Xs3 )
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ Pivot2 @ ( F7 @ X4 ) )
@ Xs3 ) ) ) ) ) ) ).
% part_def
thf(fact_4302_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 )
@ ^ [X4: list @ A] :
( X4
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_4303_Set__filter__fold,axiom,
! [A: $tType,A3: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( filter3 @ A @ P @ A3 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X4: A,A16: set @ A] : ( if @ ( set @ A ) @ ( P @ X4 ) @ ( insert3 @ A @ X4 @ A16 ) @ A16 )
@ ( bot_bot @ ( set @ A ) )
@ A3 ) ) ) ).
% Set_filter_fold
thf(fact_4304_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Xss2 ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ Xss2 ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X3
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H3: A,T3: list @ A] : ( cons @ A @ H3 @ ( 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 ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_4305_transpose_Opelims,axiom,
! [A: $tType,X3: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X3 )
= Y )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X3 )
=> ( ( ( X3
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( ( X3
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ( ( Y
= ( transpose @ A @ Xss ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) ) ) )
=> ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( X3
= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) )
=> ( ( Y
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H3: A,T3: list @ A] : ( cons @ A @ H3 @ ( 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 ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_4306_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_4307_member__filter,axiom,
! [A: $tType,X3: A,P: A > $o,A3: set @ A] :
( ( member @ A @ X3 @ ( filter3 @ A @ P @ A3 ) )
= ( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% member_filter
thf(fact_4308_takeWhile__eq__all__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P @ Xs )
= Xs )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) ) ) ) ).
% takeWhile_eq_all_conv
thf(fact_4309_takeWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_4310_takeWhile__append1,axiom,
! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% takeWhile_append1
thf(fact_4311_distinct__takeWhile,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( takeWhile @ A @ P @ Xs ) ) ) ).
% distinct_takeWhile
thf(fact_4312_takeWhile_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( takeWhile @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% takeWhile.simps(1)
thf(fact_4313_takeWhile__cong,axiom,
! [A: $tType,L: list @ A,K3: list @ A,P: A > $o,Q: A > $o] :
( ( L = K3 )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ L ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( takeWhile @ A @ P @ L )
= ( takeWhile @ A @ Q @ K3 ) ) ) ) ).
% takeWhile_cong
thf(fact_4314_set__takeWhileD,axiom,
! [A: $tType,X3: A,P: A > $o,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( takeWhile @ A @ P @ Xs ) ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ).
% set_takeWhileD
thf(fact_4315_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P2: A > $o,A7: set @ A] :
( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A7 )
& ( P2 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_4316_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_4317_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_4318_takeWhile_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( takeWhile @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( nil @ A ) ) ) ) ).
% takeWhile.simps(2)
thf(fact_4319_takeWhile__tail,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A,L: list @ A] :
( ~ ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ ( cons @ A @ X3 @ L ) ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ).
% takeWhile_tail
thf(fact_4320_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_4321_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_4322_takeWhile__eq__take,axiom,
! [A: $tType] :
( ( takeWhile @ A )
= ( ^ [P2: A > $o,Xs3: list @ A] : ( take @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs3 ) ) @ Xs3 ) ) ) ).
% takeWhile_eq_take
thf(fact_4323_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_4324_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_4325_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_4326_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_4327_takeWhile__not__last,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( takeWhile @ A
@ ^ [Y5: A] :
( Y5
!= ( last @ A @ Xs ) )
@ Xs )
= ( butlast @ A @ Xs ) ) ) ).
% takeWhile_not_last
thf(fact_4328_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ J )
=> ( P @ ( nth @ A @ Xs @ I5 ) ) )
=> ( ( 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_4329_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_4330_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_4331_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_4332_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ N )
=> ( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I5 ) ) ) )
=> ( ( ( 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_4333_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 ) ) ) )
=> ( ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ 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 ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) )
=> ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_4334_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F3 @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ T2 @ ( F3 @ X4 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X4: B] : ( ord_less @ A @ T2 @ ( F3 @ X4 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_4335_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_4336_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R5 ) )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A7 )
& ( X4 != Y5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% cofinal_def
thf(fact_4337_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_4338_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F: set @ A,I3: set @ A,F3: A > B,I: A] :
( ( finite_finite2 @ A @ F )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I3 )
& ( ( F3 @ I4 )
!= ( zero_zero @ B ) ) ) )
@ F )
=> ( ( ( member @ A @ I @ I3 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I3 @ ( insert3 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I3 ) @ ( F3 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I3 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I3 @ ( insert3 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I3 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_4339_foldr__append,axiom,
! [B: $tType,A: $tType,F3: B > A > A,Xs: list @ B,Ys: list @ B,A4: A] :
( ( foldr @ B @ A @ F3 @ ( append @ B @ Xs @ Ys ) @ A4 )
= ( foldr @ B @ A @ F3 @ Xs @ ( foldr @ B @ A @ F3 @ Ys @ A4 ) ) ) ).
% foldr_append
thf(fact_4340_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_4341_foldr__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldr @ A @ nat
@ ^ [X4: A] : suc
@ L
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldr_length
thf(fact_4342_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I3: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I3 )
& ( ( P4 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I3 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert3 @ B @ I @ I3 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I3 ) ) )
& ( ~ ( member @ B @ I @ I3 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert3 @ B @ I @ I3 ) )
= ( plus_plus @ A @ ( P4 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I3 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_4343_foldr__cong,axiom,
! [B: $tType,A: $tType,A4: A,B3: A,L: list @ B,K3: list @ B,F3: B > A > A,G: B > A > A] :
( ( A4 = B3 )
=> ( ( L = K3 )
=> ( ! [A6: A,X: B] :
( ( member @ B @ X @ ( set2 @ B @ L ) )
=> ( ( F3 @ X @ A6 )
= ( G @ X @ A6 ) ) )
=> ( ( foldr @ B @ A @ F3 @ L @ A4 )
= ( foldr @ B @ A @ G @ K3 @ B3 ) ) ) ) ) ).
% foldr_cong
thf(fact_4344_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys: list @ A,Xs: list @ B,F3: A > C > C,B3: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldr @ ( product_prod @ B @ A ) @ C
@ ( product_case_prod @ B @ A @ ( C > C )
@ ^ [X4: B] : F3 )
@ ( zip @ B @ A @ Xs @ Ys )
@ B3 )
= ( foldr @ A @ C @ F3 @ Ys @ B3 ) ) ) ).
% foldr_snd_zip
thf(fact_4345_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F7: B > A > A,Xs3: list @ B,A5: A] :
( foldl @ A @ B
@ ^ [X4: A,Y5: B] : ( F7 @ Y5 @ X4 )
@ A5
@ ( rev @ B @ Xs3 ) ) ) ) ).
% foldr_conv_foldl
thf(fact_4346_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F7: A > B > A,A5: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X4: B,Y5: A] : ( F7 @ Y5 @ X4 )
@ ( rev @ B @ Xs3 )
@ A5 ) ) ) ).
% foldl_conv_foldr
thf(fact_4347_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( zero_zero @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_4348_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ I5 )
= ( zero_zero @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_4349_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_4350_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T4 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_4351_foldr__length__aux,axiom,
! [A: $tType,L: list @ A,A4: nat] :
( ( foldr @ A @ nat
@ ^ [X4: A] : suc
@ L
@ A4 )
= ( plus_plus @ nat @ A4 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldr_length_aux
thf(fact_4352_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F7: B > A,A5: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X4: B,B4: A] : ( plus_plus @ A @ ( F7 @ X4 ) @ ( times_times @ A @ A5 @ B4 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_4353_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I3: set @ A,F3: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I3 )
& ( ( F3 @ I4 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I3 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I3 @ ( insert3 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I3 ) @ ( F3 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I3 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I3 @ ( insert3 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I3 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_4354_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L4: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q8: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L4 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q8 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L4 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q8 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_4355_Id__on__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( id_on @ A @ A3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A3 ) ) ) ).
% Id_on_fold
thf(fact_4356_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ( max_extp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R4 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_4357_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A7: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A7 ) ) ) ) ).
% Id_on_def
thf(fact_4358_Id__onI,axiom,
! [A: $tType,A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( id_on @ A @ A3 ) ) ) ).
% Id_onI
thf(fact_4359_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_4360_Id__onE,axiom,
! [A: $tType,C2: product_prod @ A @ A,A3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A3 ) )
=> ~ ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( C2
!= ( product_Pair @ A @ A @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_4361_Id__on__eqI,axiom,
! [A: $tType,A4: A,B3: A,A3: set @ A] :
( ( A4 = B3 )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( id_on @ A @ A3 ) ) ) ) ).
% Id_on_eqI
thf(fact_4362_Id__on__iff,axiom,
! [A: $tType,X3: A,Y: A,A3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( id_on @ A @ A3 ) )
= ( ( X3 = Y )
& ( member @ A @ X3 @ A3 ) ) ) ).
% Id_on_iff
thf(fact_4363_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_4364_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ L @ ( plus_plus @ code_integer @ U @ ( one_one @ code_integer ) ) )
= ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_4365_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or1337092689740270186AtMost @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_4366_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] :
( ( image2 @ code_integer @ code_integer
@ ^ [X4: code_integer] : ( plus_plus @ code_integer @ X4 @ L )
@ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% image_add_integer_atLeastLessThan
thf(fact_4367_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M: num,N2: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M ) @ ( numeral_numeral @ code_integer @ N2 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M ) @ ( numeral_numeral @ code_integer @ N2 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_4368_trans__Id__on,axiom,
! [A: $tType,A3: set @ A] : ( trans @ A @ ( id_on @ A @ A3 ) ) ).
% trans_Id_on
thf(fact_4369_refl__on__Id__on,axiom,
! [A: $tType,A3: set @ A] : ( refl_on @ A @ A3 @ ( id_on @ A @ A3 ) ) ).
% refl_on_Id_on
thf(fact_4370_Id__on__def_H,axiom,
! [A: $tType,A3: A > $o] :
( ( id_on @ A @ ( collect @ A @ A3 ) )
= ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y5: A] :
( ( X4 = Y5 )
& ( A3 @ X4 ) ) ) ) ) ).
% Id_on_def'
thf(fact_4371_Misc_Ofoldr__Cons,axiom,
! [A: $tType,Xs: list @ A] :
( ( foldr @ A @ ( list @ A ) @ ( cons @ A ) @ Xs @ ( nil @ A ) )
= Xs ) ).
% Misc.foldr_Cons
thf(fact_4372_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_4373_filter__conv__foldr,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P2: A > $o,Xs3: list @ A] :
( foldr @ A @ ( list @ A )
@ ^ [X4: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P2 @ X4 ) @ ( cons @ A @ X4 @ Xt ) @ Xt )
@ Xs3
@ ( nil @ A ) ) ) ) ).
% filter_conv_foldr
thf(fact_4374_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 )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_4375_max__extp__max__ext__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) )
= ( ^ [X4: set @ A,Y5: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y5 ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_4376_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R5: A > A > $o,X4: set @ A,Y5: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y5 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_4377_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A4: B,Xs: list @ B,F3: B > A] :
( ( member @ B @ A4 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ A4 )
= ( F3 @ X4 ) )
@ Xs ) )
= A4 )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A4 @ ( remove1 @ B @ A4 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_4378_finite__enumerate,axiom,
! [S: set @ nat] :
( ( finite_finite2 @ nat @ S )
=> ? [R3: nat > nat] :
( ( strict_mono_on @ nat @ nat @ R3 @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S ) ) )
& ! [N8: nat] :
( ( ord_less @ nat @ N8 @ ( finite_card @ nat @ S ) )
=> ( member @ nat @ ( R3 @ N8 ) @ S ) ) ) ) ).
% finite_enumerate
thf(fact_4379_lexn_Osimps_I2_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( lexn @ A @ R2 @ ( 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 ) @ R2 @ ( lexn @ A @ R2 @ 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_4380_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] :
? [I4: nat] :
( ( Uu3
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I4 ) @ ( nth @ B @ Ys @ I4 ) ) )
& ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_4381_finite__atLeastLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% finite_atLeastLessThan_integer
thf(fact_4382_finite__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% finite_atLeastAtMost_integer
thf(fact_4383_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G: D > B,A4: C,B3: D] :
( ( product_map_prod @ C @ A @ D @ B @ F3 @ G @ ( product_Pair @ C @ D @ A4 @ B3 ) )
= ( product_Pair @ A @ B @ ( F3 @ A4 ) @ ( G @ B3 ) ) ) ).
% map_prod_simp
thf(fact_4384_finite__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% finite_greaterThanLessThan_integer
thf(fact_4385_finite__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% finite_greaterThanAtMost_integer
thf(fact_4386_remove1__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [X3: B,F3: B > A,Xs: list @ B] :
( ( remove1 @ B @ X3 @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs ) )
= Xs ) ) ).
% remove1_insort_key
thf(fact_4387_eq__or__mem__image__simp,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: B,B2: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [L4: B] :
( ( Uu3
= ( F3 @ L4 ) )
& ( ( L4 = A4 )
| ( member @ B @ L4 @ B2 ) ) ) )
= ( insert3 @ A @ ( F3 @ A4 )
@ ( collect @ A
@ ^ [Uu3: A] :
? [L4: B] :
( ( Uu3
= ( F3 @ L4 ) )
& ( member @ B @ L4 @ B2 ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_4388_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% length_insort
thf(fact_4389_sort__key__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs: list @ B] :
( ( linorder_sort_key @ B @ A @ F3 @ ( cons @ B @ X3 @ Xs ) )
= ( linorder_insort_key @ B @ A @ F3 @ X3 @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) ) ) ) ).
% sort_key_simps(2)
thf(fact_4390_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A4: A,B3: B,R: set @ ( product_prod @ A @ B ),F3: A > C,G: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A4 ) @ ( G @ B3 ) ) @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G ) @ R ) ) ) ).
% map_prod_imageI
thf(fact_4391_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_4392_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_4393_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F3: C > A,G: D > B,R: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G ) @ R ) )
=> ~ ! [X: C,Y2: D] :
( ( C2
= ( product_Pair @ A @ B @ ( F3 @ X ) @ ( G @ Y2 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X @ Y2 ) @ R ) ) ) ).
% prod_fun_imageE
thf(fact_4394_filter__insort__triv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,X3: B,F3: B > A,Xs: list @ B] :
( ~ ( P @ X3 )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs ) )
= ( filter2 @ B @ P @ Xs ) ) ) ) ).
% filter_insort_triv
thf(fact_4395_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X4: B] :
( ( Uu3
= ( F3 @ X4 ) )
& ( P @ X4 ) ) )
= ( image2 @ B @ A @ F3 @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_4396_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X4: B] :
( ( Uu3
= ( F3 @ X4 ) )
& ( member @ B @ X4 @ A3 ) ) )
= ( image2 @ B @ A @ F3 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_4397_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A7: set @ A,XS: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z6: list @ A] :
? [X4: A,Xs3: list @ A] :
( ( Z6
= ( cons @ A @ X4 @ Xs3 ) )
& ( member @ A @ X4 @ A7 )
& ( member @ ( list @ A ) @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_4398_insort__left__comm,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Xs: list @ A] :
( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y
@ Xs ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs ) ) ) ) ).
% insort_left_comm
thf(fact_4399_insort__key__left__comm,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Y: B,Xs: list @ B] :
( ( ( F3 @ X3 )
!= ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ Y @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs ) )
= ( linorder_insort_key @ B @ A @ F3 @ X3 @ ( linorder_insort_key @ B @ A @ F3 @ Y @ Xs ) ) ) ) ) ).
% insort_key_left_comm
thf(fact_4400_insort__not__Nil,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,A4: B,Xs: list @ B] :
( ( linorder_insort_key @ B @ A @ F3 @ A4 @ Xs )
!= ( nil @ B ) ) ) ).
% insort_not_Nil
thf(fact_4401_finite__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U ) ) ).
% finite_atLeastZeroLessThan_integer
thf(fact_4402_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Y: B,Ys: list @ B] :
( ( ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X3 @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ X3 @ ( cons @ B @ Y @ Ys ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X3 @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Ys ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_4403_insort__key_Osimps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B] :
( ( linorder_insort_key @ B @ A @ F3 @ X3 @ ( nil @ B ) )
= ( cons @ B @ X3 @ ( nil @ B ) ) ) ) ).
% insort_key.simps(1)
thf(fact_4404_set__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs: list @ B] :
( ( set2 @ B @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs ) )
= ( insert3 @ B @ X3 @ ( set2 @ B @ Xs ) ) ) ) ).
% set_insort_key
thf(fact_4405_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F3: A > B,A3: set @ A,R2: A,S3: A] :
( ( strict_mono_on @ A @ B @ F3 @ A3 )
=> ( ( member @ A @ R2 @ A3 )
=> ( ( member @ A @ S3 @ A3 )
=> ( ( ord_less @ A @ R2 @ S3 )
=> ( ord_less @ B @ ( F3 @ R2 ) @ ( F3 @ S3 ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_4406_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A3: set @ A,F3: A > B] :
( ! [R3: A,S8: A] :
( ( member @ A @ R3 @ A3 )
=> ( ( member @ A @ S8 @ A3 )
=> ( ( ord_less @ A @ R3 @ S8 )
=> ( ord_less @ B @ ( F3 @ R3 ) @ ( F3 @ S8 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F3 @ A3 ) ) ) ).
% strict_mono_onI
thf(fact_4407_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F7: A > B,A7: set @ A] :
! [R5: A,S7: A] :
( ( ( member @ A @ R5 @ A7 )
& ( member @ A @ S7 @ A7 )
& ( ord_less @ A @ R5 @ S7 ) )
=> ( ord_less @ B @ ( F7 @ R5 ) @ ( F7 @ S7 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_4408_distinct__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs: list @ B] :
( ( distinct @ B @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs ) )
= ( ~ ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
& ( distinct @ B @ Xs ) ) ) ) ).
% distinct_insort
thf(fact_4409_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( collect @ A
@ ^ [U3: A] :
? [X4: B] :
( U3
= ( F3 @ X4 ) ) )
= ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_4410_sort__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F7: B > A,Xs3: list @ B] : ( foldr @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F7 ) @ Xs3 @ ( nil @ B ) ) ) ) ) ).
% sort_key_def
thf(fact_4411_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P7: product_prod @ A @ A] :
? [X4: A] :
( P7
= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% Id_def
thf(fact_4412_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F7: A > C,G4: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X4: A,Y5: B] : ( product_Pair @ C @ D @ ( F7 @ X4 ) @ ( G4 @ Y5 ) ) ) ) ) ).
% map_prod_def
thf(fact_4413_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_4414_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
? [Us2: list @ A,Z6: A,Z8: A,Vs3: list @ A] :
( ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ Z6 @ Vs3 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ Z8 ) @ R5 )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ Z8 @ Vs3 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_4415_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X4: list @ A,Y5: list @ A] :
? [A5: A,V4: list @ A] :
( ( Y5
= ( append @ A @ X4 @ ( cons @ A @ A5 @ V4 ) ) )
| ? [U3: list @ A,B4: A,C6: A,W3: list @ A,Z6: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C6 ) @ R5 )
& ( X4
= ( append @ A @ U3 @ ( cons @ A @ B4 @ W3 ) ) )
& ( Y5
= ( append @ A @ U3 @ ( cons @ A @ C6 @ Z6 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_4416_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
@ ^ [X4: A] : X4 )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.eq_fold
thf(fact_4417_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F3: B > A,A4: B] :
( ! [X: B] :
( ( member @ B @ X @ ( set2 @ B @ Xs ) )
=> ( ord_less_eq @ A @ ( F3 @ A4 ) @ ( F3 @ X ) ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A4 @ Xs )
= ( cons @ B @ A4 @ Xs ) ) ) ) ).
% insort_is_Cons
thf(fact_4418_Un__interval,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [B15: A,B23: A,B32: A,F3: A > B] :
( ( ord_less_eq @ A @ B15 @ B23 )
=> ( ( ord_less_eq @ A @ B23 @ B32 )
=> ( ( sup_sup @ ( set @ B )
@ ( collect @ B
@ ^ [Uu3: B] :
? [I4: A] :
( ( Uu3
= ( F3 @ I4 ) )
& ( ord_less_eq @ A @ B15 @ I4 )
& ( ord_less @ A @ I4 @ B23 ) ) )
@ ( collect @ B
@ ^ [Uu3: B] :
? [I4: A] :
( ( Uu3
= ( F3 @ I4 ) )
& ( ord_less_eq @ A @ B23 @ I4 )
& ( ord_less @ A @ I4 @ B32 ) ) ) )
= ( collect @ B
@ ^ [Uu3: B] :
? [I4: A] :
( ( Uu3
= ( F3 @ I4 ) )
& ( ord_less_eq @ A @ B15 @ I4 )
& ( ord_less @ A @ I4 @ B32 ) ) ) ) ) ) ) ).
% Un_interval
thf(fact_4419_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys2: list @ A,X4: A,Y5: A,Xs6: list @ A,Ys6: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y5 @ Ys6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_4420_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ).
% sorted_insort_key
thf(fact_4421_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),N2: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N2 )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 )
& ? [Xys2: list @ A,X4: A,Y5: A,Xs6: list @ A,Ys6: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y5 @ Ys6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_4422_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs3: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I4: nat] :
( ( Uu3
= ( nth @ A @ Xs3 @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_4423_distinct__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs ) ) )
= ( ~ ( member @ A @ ( F3 @ X3 ) @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) )
& ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ) ).
% distinct_insort_key
thf(fact_4424_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,P: B > $o,X3: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( P @ X3 )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs ) )
= ( linorder_insort_key @ B @ A @ F3 @ X3 @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ) ).
% filter_insort
thf(fact_4425_brk__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( brk_rel @ A @ B )
= ( ^ [R4: 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 )] :
? [X4: A,Y5: B] :
( ( Uu3
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $false @ X4 ) @ ( product_Pair @ $o @ B @ $false @ Y5 ) ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R4 ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu3: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X4: A,Y5: B] :
( Uu3
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $true @ X4 ) @ ( product_Pair @ $o @ B @ $false @ Y5 ) ) ) ) ) ) ) ).
% brk_rel_def
thf(fact_4426_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,Xs: list @ A] :
( ( member @ A @ A4 @ ( set2 @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A4
@ ( remove1 @ A @ A4 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_4427_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_4428_set__drop__conv,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( set2 @ A @ ( drop @ A @ N @ L ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I4: nat] :
( ( Uu3
= ( nth @ A @ L @ I4 ) )
& ( ord_less_eq @ nat @ N @ I4 )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) ) ) ) ) ).
% set_drop_conv
thf(fact_4429_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A4: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ A4 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A4
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_4430_pairself__image__eq,axiom,
! [B: $tType,A: $tType,F3: B > A,P: B > B > $o] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F3 ) @ ( collect @ ( product_prod @ B @ B ) @ ( product_case_prod @ B @ B @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [A5: B,B4: B] :
( ( Uu3
= ( product_Pair @ A @ A @ ( F3 @ A5 ) @ ( F3 @ B4 ) ) )
& ( P @ A5 @ B4 ) ) ) ) ).
% pairself_image_eq
thf(fact_4431_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K4: code_integer] :
( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
@ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K4 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_4432_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L4: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K4 @ L4 ) @ ( modulo_modulo @ code_integer @ K4 @ L4 ) ) ) ) ).
% divmod_integer_def
thf(fact_4433_list__collect__set__alt,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F7: B > ( set @ A ),L4: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [I4: nat] :
( ( Uu3
= ( F7 @ ( nth @ B @ L4 @ I4 ) ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ B ) @ L4 ) ) ) ) ) ) ) ).
% list_collect_set_alt
thf(fact_4434_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F3: B > ( set @ A ),A4: B] :
( ( list_collect_set @ B @ A @ F3 @ ( cons @ B @ A4 @ ( nil @ B ) ) )
= ( F3 @ A4 ) ) ).
% list_collect_set_simps(2)
thf(fact_4435_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F3: B > ( set @ A )] :
( ( list_collect_set @ B @ A @ F3 @ ( nil @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_simps(1)
thf(fact_4436_list__collect__set__simps_I3_J,axiom,
! [A: $tType,B: $tType,F3: B > ( set @ A ),A4: B,L: list @ B] :
( ( list_collect_set @ B @ A @ F3 @ ( cons @ B @ A4 @ L ) )
= ( sup_sup @ ( set @ A ) @ ( F3 @ A4 ) @ ( list_collect_set @ B @ A @ F3 @ L ) ) ) ).
% list_collect_set_simps(3)
thf(fact_4437_list__collect__set__simps_I4_J,axiom,
! [A: $tType,B: $tType,F3: B > ( set @ A ),L: list @ B,L2: list @ B] :
( ( list_collect_set @ B @ A @ F3 @ ( append @ B @ L @ L2 ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F3 @ L ) @ ( list_collect_set @ B @ A @ F3 @ L2 ) ) ) ).
% list_collect_set_simps(4)
thf(fact_4438_list__collect__set__map__simps_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > ( set @ A ),X3: C > B,A4: C] :
( ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X3 @ ( cons @ C @ A4 @ ( nil @ C ) ) ) )
= ( F3 @ ( X3 @ A4 ) ) ) ).
% list_collect_set_map_simps(2)
thf(fact_4439_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > ( set @ A ),X3: C > B] :
( ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X3 @ ( nil @ C ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_map_simps(1)
thf(fact_4440_list__collect__set__map__simps_I3_J,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > ( set @ A ),X3: C > B,A4: C,L: list @ C] :
( ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X3 @ ( cons @ C @ A4 @ L ) ) )
= ( sup_sup @ ( set @ A ) @ ( F3 @ ( X3 @ A4 ) ) @ ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X3 @ L ) ) ) ) ).
% list_collect_set_map_simps(3)
thf(fact_4441_list__collect__set__map__simps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > ( set @ A ),X3: C > B,L: list @ C,L2: list @ C] :
( ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X3 @ ( append @ C @ L @ L2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X3 @ L ) ) @ ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X3 @ L2 ) ) ) ) ).
% list_collect_set_map_simps(4)
thf(fact_4442_pairself_Oelims,axiom,
! [B: $tType,A: $tType,X3: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X3 @ Xa )
= Y )
=> ~ ! [A6: A,B5: A] :
( ( Xa
= ( product_Pair @ A @ A @ A6 @ B5 ) )
=> ( Y
!= ( product_Pair @ B @ B @ ( X3 @ A6 ) @ ( X3 @ B5 ) ) ) ) ) ).
% pairself.elims
thf(fact_4443_pairself_Osimps,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: A,B3: A] :
( ( pairself @ A @ B @ F3 @ ( product_Pair @ A @ A @ A4 @ B3 ) )
= ( product_Pair @ B @ B @ ( F3 @ A4 ) @ ( F3 @ B3 ) ) ) ).
% pairself.simps
thf(fact_4444_Pow__Compl,axiom,
! [A: $tType,A3: set @ A] :
( ( pow @ A @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [B7: set @ A] :
( ( Uu3
= ( uminus_uminus @ ( set @ A ) @ B7 ) )
& ( member @ ( set @ A ) @ A3 @ ( pow @ A @ B7 ) ) ) ) ) ).
% Pow_Compl
thf(fact_4445_list__collect__set__def,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F7: B > ( set @ A ),L4: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [A5: B] :
( ( Uu3
= ( F7 @ A5 ) )
& ( member @ B @ A5 @ ( set2 @ B @ L4 ) ) ) ) ) ) ) ).
% list_collect_set_def
thf(fact_4446_list__collect__set__as__map,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F7: B > ( set @ A ),L4: list @ B] : ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( map @ B @ ( set @ A ) @ F7 @ L4 ) ) ) ) ) ).
% list_collect_set_as_map
thf(fact_4447_pairself_Opelims,axiom,
! [B: $tType,A: $tType,X3: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X3 @ Xa ) )
=> ~ ! [A6: A,B5: A] :
( ( Xa
= ( product_Pair @ A @ A @ A6 @ B5 ) )
=> ( ( Y
= ( product_Pair @ B @ B @ ( X3 @ A6 ) @ ( X3 @ B5 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X3 @ ( product_Pair @ A @ A @ A6 @ B5 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_4448_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K4: code_integer] :
( if @ ( product_prod @ code_integer @ $o )
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
@ ^ [R5: code_integer,S7: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ R5 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ S7 ) )
@ ( S7
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_4449_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A7: set @ C,F7: C > A,G4: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A5: C] :
( ( Uu3
= ( product_Pair @ A @ B @ ( F7 @ A5 ) @ ( G4 @ A5 ) ) )
& ( member @ C @ A5 @ A7 ) ) ) ) ) ).
% image2_def
thf(fact_4450_image__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ U )
=> ( ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U )
= ( image2 @ nat @ code_integer @ ( semiring_1_of_nat @ code_integer ) @ ( set_ord_lessThan @ nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_integer
thf(fact_4451_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B3: A,F3: B > A,X3: B,C2: C,G: B > C,A3: set @ B] :
( ( B3
= ( F3 @ X3 ) )
=> ( ( C2
= ( G @ X3 ) )
=> ( ( member @ B @ X3 @ A3 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A3 @ F3 @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_4452_divmod__abs__code_I6_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).
% divmod_abs_code(6)
thf(fact_4453_divmod__abs__code_I5_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).
% divmod_abs_code(5)
thf(fact_4454_nat__of__integer__less__iff,axiom,
! [X3: code_integer,Y: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ X3 )
=> ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ Y )
=> ( ( ord_less @ nat @ ( code_nat_of_integer @ X3 ) @ ( code_nat_of_integer @ Y ) )
= ( ord_less @ code_integer @ X3 @ Y ) ) ) ) ).
% nat_of_integer_less_iff
thf(fact_4455_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K4: code_integer,L4: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L4 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L4 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_4456_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L4: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L4 )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ ( code_divmod_abs @ K4 @ L4 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S7: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S7
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L4 @ S7 ) ) )
@ ( code_divmod_abs @ K4 @ L4 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
@ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K4 @ L4 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S7: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S7
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L4 ) @ S7 ) ) )
@ ( code_divmod_abs @ K4 @ L4 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_4457_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [X4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y5: A,Z6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y5 @ Z6 ) @ R2 )
@ X4 ) ) ) ) ).
% listrel_iff_zip
thf(fact_4458_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,A3: set @ A,B2: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( finite_fold @ A @ B @ F3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A3 ) @ B2 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_4459_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F7: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( compow @ ( A > A ) @ N2 @ ( times_times @ A @ A5 ) @ ( F7 @ ( nth @ B @ Xs3 @ N2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_4460_ball__empty,axiom,
! [A: $tType,P: A > $o,X8: A] :
( ( member @ A @ X8 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X8 ) ) ).
% ball_empty
thf(fact_4461_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B,X3: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_Pair @ A @ C @ X3 @ Y ) )
= ( product_Pair @ A @ B @ X3 @ ( F3 @ Y ) ) ) ).
% apsnd_conv
thf(fact_4462_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P2: A > $o] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( P2 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_4463_funpow__times__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [F3: A > nat,X3: A] :
( ( compow @ ( A > A ) @ ( F3 @ X3 ) @ ( times_times @ A @ X3 ) )
= ( times_times @ A @ ( power_power @ A @ X3 @ ( F3 @ X3 ) ) ) ) ) ).
% funpow_times_power
thf(fact_4464_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) )
= ( ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_4465_Field__not__elem,axiom,
! [A: $tType,V2: A,R: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ V2 @ ( field2 @ A @ R ) )
=> ! [X8: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X8 @ R )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Z6: A] :
( ( Y5 != V2 )
& ( Z6 != V2 ) )
@ X8 ) ) ) ).
% Field_not_elem
thf(fact_4466_rotate__def,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N2: nat] : ( compow @ ( ( list @ A ) > ( list @ A ) ) @ N2 @ ( rotate1 @ A ) ) ) ) ).
% rotate_def
thf(fact_4467_irrefl__distinct,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [A5: A,B4: A] : A5 != B4
@ X4 ) ) ) ) ).
% irrefl_distinct
thf(fact_4468_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N2: nat,P2: A > A > $o,X4: A,Y5: A] :
? [F7: nat > A] :
( ( ( F7 @ ( zero_zero @ nat ) )
= X4 )
& ( ( F7 @ N2 )
= Y5 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( P2 @ ( F7 @ I4 ) @ ( F7 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_4469_sorted__wrt_Oelims_I3_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A] :
( ~ ( sorted_wrt @ A @ X3 @ Xa )
=> ~ ! [X: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ X @ Ys5 ) )
=> ( ! [Xa4: A] :
( ( member @ A @ Xa4 @ ( set2 @ A @ Ys5 ) )
=> ( X3 @ X @ Xa4 ) )
& ( sorted_wrt @ A @ X3 @ Ys5 ) ) ) ) ).
% sorted_wrt.elims(3)
thf(fact_4470_sorted__wrt_Osimps_I2_J,axiom,
! [A: $tType,P: A > A > $o,X3: A,Ys: list @ A] :
( ( sorted_wrt @ A @ P @ ( cons @ A @ X3 @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( P @ X3 @ X4 ) )
& ( sorted_wrt @ A @ P @ Ys ) ) ) ).
% sorted_wrt.simps(2)
thf(fact_4471_relpowp__bot,axiom,
! [A: $tType,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( compow @ ( A > A > $o ) @ N @ ( bot_bot @ ( A > A > $o ) ) )
= ( bot_bot @ ( A > A > $o ) ) ) ) ).
% relpowp_bot
thf(fact_4472_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Y13: A] :
! [Z6: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z6 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y24: A,Aa3: A] :
( ( Y13 = Y24 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Aa3 ) @ R5 ) )
@ Z6 ) )
@ X4 ) ) ) ) ).
% trans_join
thf(fact_4473_takeWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) )
& ( ~ ! [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
=> ( P @ X8 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% takeWhile_append
thf(fact_4474_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Z6: A] :
( ( member @ A @ Y5 @ A7 )
& ( member @ A @ Z6 @ A7 ) )
@ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R5 ) ) ) ) ) ).
% refl_on_def'
thf(fact_4475_sorted__wrt_Oelims_I1_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X3 @ Xa )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ Y )
=> ~ ! [X: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ X @ Ys5 ) )
=> ( Y
= ( ~ ( ! [Y5: A] :
( ( member @ A @ Y5 @ ( set2 @ A @ Ys5 ) )
=> ( X3 @ X @ Y5 ) )
& ( sorted_wrt @ A @ X3 @ Ys5 ) ) ) ) ) ) ) ).
% sorted_wrt.elims(1)
thf(fact_4476_sorted__wrt_Oelims_I2_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A] :
( ( sorted_wrt @ A @ X3 @ Xa )
=> ( ( Xa
!= ( nil @ A ) )
=> ~ ! [X: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ X @ Ys5 ) )
=> ~ ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys5 ) )
=> ( X3 @ X @ Xa3 ) )
& ( sorted_wrt @ A @ X3 @ Ys5 ) ) ) ) ) ).
% sorted_wrt.elims(2)
thf(fact_4477_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_4478_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z4: list @ A] : Y4 = Z4 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,Z4: A] : Y4 = Z4
@ X4 ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_4479_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y5: list @ A,Z6: list @ A] :
( ( size_size @ ( list @ A ) @ Y5 )
= ( size_size @ ( list @ A ) @ Z6 ) )
@ X ) )
=> ( ( ( 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_4480_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 ) )
=> ( ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y5: list @ A,Z6: list @ A] :
( ( size_size @ ( list @ A ) @ Y5 )
= ( size_size @ ( list @ A ) @ Z6 ) )
@ X ) )
=> ( Xs = Ys ) ) ) ) ).
% concat_injective
thf(fact_4481_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( F3 @ X3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A3 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X3 @ Z2 ) @ A3 ) ) ) ) ) ).
% comp_fun_commute_on.fold_fun_left_comm
thf(fact_4482_comp__fun__commute__on_Ofold__insert2,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert3 @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X3 @ Z2 ) @ A3 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert2
thf(fact_4483_comp__fun__commute__on_Ofold__insert,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert3 @ A @ X3 @ A3 ) )
= ( F3 @ X3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A3 ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert
thf(fact_4484_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,A3: set @ A,X3: A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ A3 )
= ( F3 @ X3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_4485_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert3 @ A @ X3 @ A3 ) )
= ( F3 @ X3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_4486_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ X4 ) @ R5 ) ) ) ) ) ) ).
% Under_def
thf(fact_4487_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ( B4 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_4488_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B4 ) @ R5 ) ) ) ) ) ) ).
% Above_def
thf(fact_4489_Gcd__eq__Max,axiom,
! [M5: set @ nat] :
( ( finite_finite2 @ nat @ M5 )
=> ( ( M5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( gcd_Gcd @ nat @ M5 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M: nat] :
( collect @ nat
@ ^ [D6: nat] : ( dvd_dvd @ nat @ D6 @ M ) )
@ M5 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_4490_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Inf_Inf @ A @ A3 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A3 )
& ( ord_less @ A @ Y5 @ X4 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_4491_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_4492_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_4493_cInf__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X3: A] :
( ( complete_Inf_Inf @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% cInf_singleton
thf(fact_4494_Inf__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: A,A3: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( inf_inf @ A @ A4 @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ).
% Inf_insert
thf(fact_4495_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X3 @ Y ) )
= X3 ) ) ) ).
% Inf_atLeastLessThan
thf(fact_4496_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X3 ) )
= Y ) ) ) ).
% cInf_atLeastLessThan
thf(fact_4497_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X3 ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_4498_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X3 @ Y ) )
= X3 ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_4499_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X3 ) )
= Y ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_4500_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X3 @ Y ) )
= X3 ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_4501_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X3 ) )
= Y ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_4502_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F3: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A3 ) )
= F3 ) ) ) ).
% INF_const
thf(fact_4503_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% cINF_const
thf(fact_4504_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A3: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y5: B] :
( ( member @ B @ Y5 @ A3 )
& ( ord_less @ A @ ( F3 @ Y5 ) @ X4 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_4505_INT__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : C2
@ A3 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_4506_INT__insert,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A4: B,A3: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( insert3 @ B @ A4 @ A3 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B2 @ A4 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ A3 ) ) ) ) ).
% INT_insert
thf(fact_4507_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X3 )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X3 ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_4508_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C3: set @ D,A3: set @ C,B2: D > ( set @ C )] :
( ( ( C3
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B2 @ X4 ) )
@ C3 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B2 @ X4 ) )
@ C3 ) )
= ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C3 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_4509_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C3: set @ A,A3: A > ( set @ B ),B2: set @ B] :
( ( ( C3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X4 ) @ B2 )
@ C3 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X4 ) @ B2 )
@ C3 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ C3 ) ) @ B2 ) ) ) ) ).
% INT_simps(1)
thf(fact_4510_INT__simps_I3_J,axiom,
! [E: $tType,F6: $tType,C3: set @ E,A3: E > ( set @ F6 ),B2: set @ F6] :
( ( ( C3
= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F6 )
@ ( image2 @ E @ ( set @ F6 )
@ ^ [X4: E] : ( minus_minus @ ( set @ F6 ) @ ( A3 @ X4 ) @ B2 )
@ C3 ) )
= ( top_top @ ( set @ F6 ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F6 )
@ ( image2 @ E @ ( set @ F6 )
@ ^ [X4: E] : ( minus_minus @ ( set @ F6 ) @ ( A3 @ X4 ) @ B2 )
@ C3 ) )
= ( minus_minus @ ( set @ F6 ) @ ( complete_Inf_Inf @ ( set @ F6 ) @ ( image2 @ E @ ( set @ F6 ) @ A3 @ C3 ) ) @ B2 ) ) ) ) ).
% INT_simps(3)
thf(fact_4511_INT__simps_I4_J,axiom,
! [G2: $tType,H6: $tType,C3: set @ H6,A3: set @ G2,B2: H6 > ( set @ G2 )] :
( ( ( C3
= ( bot_bot @ ( set @ H6 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G2 )
@ ( image2 @ H6 @ ( set @ G2 )
@ ^ [X4: H6] : ( minus_minus @ ( set @ G2 ) @ A3 @ ( B2 @ X4 ) )
@ C3 ) )
= ( top_top @ ( set @ G2 ) ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ H6 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G2 )
@ ( image2 @ H6 @ ( set @ G2 )
@ ^ [X4: H6] : ( minus_minus @ ( set @ G2 ) @ A3 @ ( B2 @ X4 ) )
@ C3 ) )
= ( minus_minus @ ( set @ G2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image2 @ H6 @ ( set @ G2 ) @ B2 @ C3 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_4512_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S: set @ A,A4: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S ) @ A4 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S )
& ( ord_less @ A @ X4 @ A4 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_4513_refl__on__INTER,axiom,
! [B: $tType,A: $tType,S: set @ A,A3: A > ( set @ B ),R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X: A] :
( ( member @ A @ X @ S )
=> ( refl_on @ B @ ( A3 @ X ) @ ( R2 @ X ) ) )
=> ( refl_on @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ S ) ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% refl_on_INTER
thf(fact_4514_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A,X3: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X3 )
= ( ! [Y5: A] :
( ( ord_less @ A @ X3 @ Y5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less @ A @ X4 @ Y5 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_4515_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,Z2: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ X5 )
=> ( ord_less_eq @ A @ Z2 @ X ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ X5 ) ) ) ) ) ).
% cInf_greatest
thf(fact_4516_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A4: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ X5 )
=> ( ord_less_eq @ A @ A4 @ X ) )
=> ( ! [Y2: A] :
( ! [X8: A] :
( ( member @ A @ X8 @ X5 )
=> ( ord_less_eq @ A @ Y2 @ X8 ) )
=> ( ord_less_eq @ A @ Y2 @ A4 ) )
=> ( ( complete_Inf_Inf @ A @ X5 )
= A4 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4517_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A3 )
=> ( ord_less_eq @ A @ V3 @ U ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_4518_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 )
=> ? [X: A] :
( ( member @ A @ X @ X5 )
& ( ord_less @ A @ X @ Z2 ) ) ) ) ) ).
% cInf_lessD
thf(fact_4519_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,X3: A,A4: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( member @ A @ X3 @ X5 )
=> ( ! [X: A] :
( ( member @ A @ X @ X5 )
=> ( ord_less @ A @ A4 @ X ) )
=> ( ord_less @ A @ A4 @ ( complete_Inf_Inf @ A @ X5 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_4520_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I3: set @ B,F3: B > A,X3: A] :
( ( I3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ I3 )
=> ( ( F3 @ I5 )
= X3 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I3 ) )
= X3 ) ) ) ) ).
% INF_eq_const
thf(fact_4521_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_4522_Inf__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A4: A,A3: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( inf_inf @ A @ A4 @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ).
% Inf_finite_insert
thf(fact_4523_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A,F3: B > A,A3: set @ B,I: B] :
( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ Y @ ( F3 @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_4524_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A3: set @ B,A4: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ A4 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ ( F3 @ X4 ) @ A4 ) ) ) ) ) ).
% INF_less_iff
thf(fact_4525_INT__extend__simps_I5_J,axiom,
! [I9: $tType,J6: $tType,A4: I9,B2: J6 > ( set @ I9 ),C3: set @ J6] :
( ( insert3 @ I9 @ A4 @ ( complete_Inf_Inf @ ( set @ I9 ) @ ( image2 @ J6 @ ( set @ I9 ) @ B2 @ C3 ) ) )
= ( complete_Inf_Inf @ ( set @ I9 )
@ ( image2 @ J6 @ ( set @ I9 )
@ ^ [X4: J6] : ( insert3 @ I9 @ A4 @ ( B2 @ X4 ) )
@ C3 ) ) ) ).
% INT_extend_simps(5)
thf(fact_4526_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A3: set @ A,A4: B,B2: A > ( set @ B )] :
( ( member @ A @ U @ A3 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( insert3 @ B @ A4 @ ( B2 @ X4 ) )
@ A3 ) )
= ( insert3 @ B @ A4 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B2 @ A3 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_4527_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_4528_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A3: set @ B,X3: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ X3 )
= ( ! [Y5: A] :
( ( ord_less @ A @ X3 @ Y5 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ ( F3 @ X4 ) @ Y5 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_4529_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,M2: A,F3: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ord_less_eq @ A @ M2 @ ( F3 @ X ) ) )
=> ( ord_less_eq @ A @ M2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_4530_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I3: set @ B,F3: B > A,C2: A] :
( ( I3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ I3 )
=> ( ord_less_eq @ A @ ( F3 @ I5 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I3 ) )
= C2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I3 )
=> ( ( F3 @ X4 )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4531_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X5: set @ A,A4: A] :
( ( finite_finite2 @ A @ X5 )
=> ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A4 @ ( complete_Inf_Inf @ A @ X5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X5 )
=> ( ord_less @ A @ A4 @ X4 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_4532_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Inf_le_Sup
thf(fact_4533_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A,Y2: A] :
( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( member @ A @ ( inf_inf @ A @ X @ Y2 ) @ A3 ) ) )
=> ( member @ A @ ( complete_Inf_Inf @ A @ A3 ) @ A3 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_4534_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ A4 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S ) ) @ A4 ) ) ) ) ).
% cInf_abs_ge
thf(fact_4535_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_4536_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_4537_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_4538_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A3 )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ).
% Min_Inf
thf(fact_4539_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_4540_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y5: B] : C2
@ A3 ) )
= ( top_top @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y5: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% INF_constant
thf(fact_4541_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I3: set @ B,F3: B > A,X3: A] :
( ( I3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ ( F3 @ I4 ) @ X3 )
@ I3 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I3 ) ) @ X3 ) ) ) ) ).
% INF_inf_const2
thf(fact_4542_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I3: set @ B,X3: A,F3: B > A] :
( ( I3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ X3 @ ( F3 @ I4 ) )
@ I3 ) )
= ( inf_inf @ A @ X3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I3 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_4543_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A4: B,A3: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( insert3 @ B @ A4 @ A3 ) ) )
= ( inf_inf @ A @ ( F3 @ A4 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) ) ) ) ).
% INF_insert
thf(fact_4544_INT__empty,axiom,
! [B: $tType,A: $tType,B2: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_4545_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C3: set @ A,A3: A > ( set @ B ),B2: set @ B] :
( ( ( C3
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ C3 ) ) @ B2 )
= B2 ) )
& ( ( C3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ C3 ) ) @ B2 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X4 ) @ B2 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_4546_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C3: set @ D,A3: set @ C,B2: D > ( set @ C )] :
( ( ( C3
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C3 ) ) )
= A3 ) )
& ( ( C3
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B2 @ C3 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B2 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_4547_UN__extend__simps_I7_J,axiom,
! [M8: $tType,N11: $tType,A3: set @ M8,B2: N11 > ( set @ M8 ),C3: set @ N11] :
( ( minus_minus @ ( set @ M8 ) @ A3 @ ( complete_Inf_Inf @ ( set @ M8 ) @ ( image2 @ N11 @ ( set @ M8 ) @ B2 @ C3 ) ) )
= ( complete_Sup_Sup @ ( set @ M8 )
@ ( image2 @ N11 @ ( set @ M8 )
@ ^ [X4: N11] : ( minus_minus @ ( set @ M8 ) @ A3 @ ( B2 @ X4 ) )
@ C3 ) ) ) ).
% UN_extend_simps(7)
thf(fact_4548_Union__maximal__sets,axiom,
! [A: $tType,F11: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F11 )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [T7: set @ A] :
( ( member @ ( set @ A ) @ T7 @ F11 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ F11 )
=> ~ ( ord_less @ ( set @ A ) @ T7 @ X4 ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ F11 ) ) ) ).
% Union_maximal_sets
thf(fact_4549_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F3: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_4550_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S ) @ L ) ) @ E3 ) ) ) ) ).
% cInf_asclose
thf(fact_4551_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X3: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( S
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert3 @ A @ X3 @ S ) )
= X3 ) )
& ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert3 @ A @ X3 @ S ) )
= ( ord_min @ A @ X3 @ ( complete_Inf_Inf @ A @ S ) ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_4552_INT__greaterThan__UNIV,axiom,
( ( complete_Inf_Inf @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% INT_greaterThan_UNIV
thf(fact_4553_INT__extend__simps_I3_J,axiom,
! [F6: $tType,E: $tType,C3: set @ E,A3: E > ( set @ F6 ),B2: set @ F6] :
( ( ( C3
= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F6 ) @ ( complete_Inf_Inf @ ( set @ F6 ) @ ( image2 @ E @ ( set @ F6 ) @ A3 @ C3 ) ) @ B2 )
= ( minus_minus @ ( set @ F6 ) @ ( top_top @ ( set @ F6 ) ) @ B2 ) ) )
& ( ( C3
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F6 ) @ ( complete_Inf_Inf @ ( set @ F6 ) @ ( image2 @ E @ ( set @ F6 ) @ A3 @ C3 ) ) @ B2 )
= ( complete_Inf_Inf @ ( set @ F6 )
@ ( image2 @ E @ ( set @ F6 )
@ ^ [X4: E] : ( minus_minus @ ( set @ F6 ) @ ( A3 @ X4 ) @ B2 )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_4554_INT__extend__simps_I4_J,axiom,
! [G2: $tType,H6: $tType,C3: set @ H6,A3: set @ G2,B2: H6 > ( set @ G2 )] :
( ( ( C3
= ( bot_bot @ ( set @ H6 ) ) )
=> ( ( minus_minus @ ( set @ G2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image2 @ H6 @ ( set @ G2 ) @ B2 @ C3 ) ) )
= A3 ) )
& ( ( C3
!= ( bot_bot @ ( set @ H6 ) ) )
=> ( ( minus_minus @ ( set @ G2 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image2 @ H6 @ ( set @ G2 ) @ B2 @ C3 ) ) )
= ( complete_Inf_Inf @ ( set @ G2 )
@ ( image2 @ H6 @ ( set @ G2 )
@ ^ [X4: H6] : ( minus_minus @ ( set @ G2 ) @ A3 @ ( B2 @ X4 ) )
@ C3 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_4555_sort__mergesort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4 )
= ( mergesort @ A ) ) ) ).
% sort_mergesort
thf(fact_4556_times__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U3 ) @ ( times_times @ nat @ Y5 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V4 ) @ ( times_times @ nat @ Y5 @ U3 ) ) ) )
@ Xa
@ X3 ) ) ) ).
% times_int.abs_eq
thf(fact_4557_min__list_Oelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A,Y: A] :
( ( ( min_list @ A @ X3 )
= Y )
=> ( ! [X: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ Xs2 ) )
=> ( Y
!= ( case_list @ A @ A @ X
@ ^ [A5: A,List2: list @ A] : ( ord_min @ A @ X @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) ) )
=> ~ ( ( X3
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ).
% min_list.elims
thf(fact_4558_filter__rev__alt,axiom,
! [A: $tType] :
( ( filter_rev @ A )
= ( ^ [P2: A > $o,L4: list @ A] : ( filter2 @ A @ P2 @ ( rev @ A @ L4 ) ) ) ) ).
% filter_rev_alt
thf(fact_4559_subset__mset_OcINF__const,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X4: B] : C2
@ A3 ) )
= C2 ) ) ).
% subset_mset.cINF_const
thf(fact_4560_Inf__nat__def1,axiom,
! [K6: set @ nat] :
( ( K6
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( complete_Inf_Inf @ nat @ K6 ) @ K6 ) ) ).
% Inf_nat_def1
thf(fact_4561_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I4: set @ ( product_prod @ A @ B ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ I4 )
@ S ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% INF_Int_eq2
thf(fact_4562_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X: nat,Y2: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X @ Y2 ) ) ) ).
% eq_Abs_Integ
thf(fact_4563_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_4564_trans__INTER,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X: A] :
( ( member @ A @ X @ S )
=> ( trans @ B @ ( R2 @ X ) ) )
=> ( trans @ B @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S ) ) ) ) ).
% trans_INTER
thf(fact_4565_INF__filter__not__bot,axiom,
! [I9: $tType,A: $tType,B2: set @ I9,F: I9 > ( filter @ A )] :
( ! [X11: set @ I9] :
( ( ord_less_eq @ ( set @ I9 ) @ X11 @ B2 )
=> ( ( finite_finite2 @ I9 @ X11 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I9 @ ( filter @ A ) @ F @ X11 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I9 @ ( filter @ A ) @ F @ B2 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_4566_Inf__INT__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( A > $o ) )
= ( ^ [S2: set @ ( A > $o ),X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S2 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_4567_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S2: set @ ( A > B > $o ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S2 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_4568_subset__mset_OcINF__greatest,axiom,
! [A: $tType,B: $tType,A3: set @ B,M2: multiset @ A,F3: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( subseteq_mset @ A @ M2 @ ( F3 @ X ) ) )
=> ( subseteq_mset @ A @ M2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) ) ) ) ).
% subset_mset.cINF_greatest
thf(fact_4569_hd__def,axiom,
! [A: $tType] :
( ( hd @ A )
= ( case_list @ A @ A @ ( undefined @ A )
@ ^ [X213: A,X224: list @ A] : X213 ) ) ).
% hd_def
thf(fact_4570_INF__Int__eq,axiom,
! [A: $tType,S: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I4: set @ A,X4: A] : ( member @ A @ X4 @ I4 )
@ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ S ) ) ) ) ).
% INF_Int_eq
thf(fact_4571_INF__INT__eq,axiom,
! [B: $tType,A: $tType,R2: B > ( set @ A ),S: set @ B] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I4: B,X4: A] : ( member @ A @ X4 @ ( R2 @ I4 ) )
@ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R2 @ S ) ) ) ) ) ).
% INF_INT_eq
thf(fact_4572_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_4573_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N2 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_4574_zipf_Oelims,axiom,
! [B: $tType,A: $tType,C: $tType,X3: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
( ( ( zipf @ A @ B @ C @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
!= ( nil @ C ) ) ) )
=> ( ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ( Y
!= ( cons @ C @ ( X3 @ A6 @ B5 ) @ ( zipf @ A @ B @ C @ X3 @ As @ Bs2 ) ) ) ) )
=> ( ( ? [V3: A,Va: list @ A] :
( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V3: B,Va: list @ B] :
( Xb
= ( cons @ B @ V3 @ Va ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) ) ) ) ) ) ).
% zipf.elims
thf(fact_4575_uminus__int_Oabs__eq,axiom,
! [X3: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X4 )
@ X3 ) ) ) ).
% uminus_int.abs_eq
thf(fact_4576_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_4577_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: A > B,Xa: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X3 @ Xa )
= Y )
=> ( ! [X: A] :
( ( Xa
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( Y != X ) )
=> ( ! [X: A,Y2: A,Zs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ ( cons @ A @ Y2 @ Zs2 ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X3 @ X ) @ ( X3 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) @ X @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_4578_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X3: product_prod @ nat @ nat] :
( ( ring_1_of_int @ A @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J2 ) )
@ X3 ) ) ) ).
% of_int.abs_eq
thf(fact_4579_less__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X3 ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ U3 @ Y5 ) ) )
@ Xa
@ X3 ) ) ).
% less_int.abs_eq
thf(fact_4580_plus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U3 ) @ ( plus_plus @ nat @ Y5 @ V4 ) ) )
@ Xa
@ X3 ) ) ) ).
% plus_int.abs_eq
thf(fact_4581_minus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X3: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X3 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ Y5 @ U3 ) ) )
@ Xa
@ X3 ) ) ) ).
% minus_int.abs_eq
thf(fact_4582_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A,Y: A] :
( ( ( min_list @ A @ X3 )
= Y )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X3 )
=> ( ! [X: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ Xs2 ) )
=> ( ( Y
= ( case_list @ A @ A @ X
@ ^ [A5: A,List2: list @ A] : ( ord_min @ A @ X @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X @ Xs2 ) ) ) )
=> ~ ( ( X3
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_4583_mod__h__bot__normalize,axiom,
! [A: $tType,H2: heap_ext @ product_unit,P: assn] :
( ( syntax7388354845996824322omatch @ A @ ( heap_ext @ product_unit ) @ ( undefined @ A ) @ H2 )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ ( undefined @ ( heap_ext @ product_unit ) ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_normalize
thf(fact_4584_zipf_Opelims,axiom,
! [C: $tType,A: $tType,B: $tType,X3: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
( ( ( zipf @ A @ B @ C @ X3 @ 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 ) ) @ X3 @ ( 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
=> ( ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ( ( Y
= ( cons @ C @ ( X3 @ A6 @ B5 ) @ ( zipf @ A @ B @ C @ X3 @ 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ ( cons @ B @ B5 @ Bs2 ) ) ) ) ) ) )
=> ( ! [V3: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( ( Y
= ( undefined @ ( list @ C ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: B,Va: list @ B] :
( ( Xb
= ( cons @ B @ V3 @ Va ) )
=> ( ( Y
= ( undefined @ ( list @ C ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% zipf.pelims
thf(fact_4585_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L4: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
@ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L4
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K4 )
= ( sgn_sgn @ code_integer @ L4 ) )
@ ( code_divmod_abs @ K4 @ L4 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S7: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S7
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L4 ) @ S7 ) ) )
@ ( code_divmod_abs @ K4 @ L4 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_4586_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C,G: A > B] :
( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F3 ) @ ( map @ A @ B @ G ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F3 @ G ) ) ) ).
% List.map.comp
thf(fact_4587_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F3: A > B,V2: list @ A] :
( ( map @ B @ C @ G @ ( map @ A @ B @ F3 @ V2 ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F3 ) @ V2 ) ) ).
% list.map_comp
thf(fact_4588_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F3: B > C,G: A > B,List: list @ A] :
( ( map @ B @ C @ F3 @ ( map @ A @ B @ G @ List ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F3 @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_4589_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > A,G: C > B,Xs: list @ C] :
( ( map @ B @ A @ F3 @ ( map @ C @ B @ G @ Xs ) )
= ( map @ C @ A @ ( comp @ B @ A @ C @ F3 @ G ) @ Xs ) ) ).
% map_map
thf(fact_4590_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,G: A > C] :
( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F3 ) @ ( map @ A @ C @ G ) )
= ( map @ A @ B @ ( comp @ C @ B @ A @ F3 @ G ) ) ) ).
% map_comp_map
thf(fact_4591_Inf__filter__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( filter @ A ) )
= ( ^ [S2: set @ ( filter @ A )] :
( complete_Sup_Sup @ ( filter @ A )
@ ( collect @ ( filter @ A )
@ ^ [F4: filter @ A] :
! [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ S2 )
=> ( ord_less_eq @ ( filter @ A ) @ F4 @ X4 ) ) ) ) ) ) ).
% Inf_filter_def
thf(fact_4592_Inf__filter__not__bot,axiom,
! [A: $tType,B2: set @ ( filter @ A )] :
( ! [X11: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X11 @ B2 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X11 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X11 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B2 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_4593_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I3: set @ A,F: A > ( filter @ B )] :
( ! [I5: A] :
( ( member @ A @ I5 @ I3 )
=> ! [J3: A] :
( ( member @ A @ J3 @ I3 )
=> ? [X8: A] :
( ( member @ A @ X8 @ I3 )
& ( ord_less_eq @ ( filter @ B ) @ ( F @ X8 ) @ ( inf_inf @ ( filter @ B ) @ ( F @ I5 ) @ ( F @ J3 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F @ I3 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ I3 )
& ( ( F @ X4 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_4594_fun__comp__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,G: A > C,Fg: A > B] :
( ( ( comp @ C @ B @ A @ F3 @ G )
= Fg )
= ( ! [X4: A] :
( ( F3 @ ( G @ X4 ) )
= ( Fg @ X4 ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_4595_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X3: A > B,Y: A > B,F3: B > C] :
( ( X3 = Y )
=> ( ( comp @ B @ C @ A @ F3 @ X3 )
= ( comp @ B @ C @ A @ F3 @ Y ) ) ) ).
% comp_cong_right
thf(fact_4596_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X3: A > B,Y: A > B,F3: C > A] :
( ( X3 = Y )
=> ( ( comp @ A @ B @ C @ X3 @ F3 )
= ( comp @ A @ B @ C @ Y @ F3 ) ) ) ).
% comp_cong_left
thf(fact_4597_foldr_Ofoldr__Cons,axiom,
! [B: $tType,A: $tType,F3: A > B > B,X3: A,Xs: list @ A] :
( ( foldr @ A @ B @ F3 @ ( cons @ A @ X3 @ Xs ) )
= ( comp @ B @ B @ B @ ( F3 @ X3 ) @ ( foldr @ A @ B @ F3 @ Xs ) ) ) ).
% foldr.foldr_Cons
thf(fact_4598_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: B > A > A,F3: C > B,Xs: list @ C,A4: A] :
( ( foldr @ B @ A @ G @ ( map @ C @ B @ F3 @ Xs ) @ A4 )
= ( foldr @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F3 ) @ Xs @ A4 ) ) ).
% foldr_map
thf(fact_4599_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G4 ) @ ( one_one @ A ) ) ) ) ) ).
% prod.eq_fold
thf(fact_4600_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ ( bot_bot @ A ) @ A3 ) ) ) ) ).
% SUP_fold_sup
thf(fact_4601_mult_Osafe__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [X3: A,Y: A,A4: A,B3: A] :
( ( syntax7388354845996824322omatch @ A @ A @ ( times_times @ A @ X3 @ Y ) @ A4 )
=> ( ( times_times @ A @ A4 @ B3 )
= ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% mult.safe_commute
thf(fact_4602_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X4: int,Xa5: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y5: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y5 @ V4 ) @ ( plus_plus @ nat @ U3 @ Z6 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa5 ) ) ) ) ).
% less_int.rep_eq
thf(fact_4603_filter__rev__aux__alt,axiom,
! [A: $tType] :
( ( filter_rev_aux @ A )
= ( ^ [A5: list @ A,P2: A > $o,L4: list @ A] : ( append @ A @ ( filter2 @ A @ P2 @ ( rev @ A @ L4 ) ) @ A5 ) ) ) ).
% filter_rev_aux_alt
thf(fact_4604_length__filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F3 @ Xs ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F3 ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_4605_set__mset__comp__mset,axiom,
! [A: $tType] :
( ( comp @ ( multiset @ A ) @ ( set @ A ) @ ( list @ A ) @ ( set_mset @ A ) @ ( mset @ A ) )
= ( set2 @ A ) ) ).
% set_mset_comp_mset
thf(fact_4606_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3 ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3 )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
thf(fact_4607_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: A > C,G: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) )
@ F3 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image2 @ D @ B @ G )
@ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_4608_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F4: filter @ A,F8: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ F8 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F8 @ F4 ) ) ) ) ).
% less_filter_def
thf(fact_4609_filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: B > A,Xs: list @ B] :
( ( filter2 @ A @ P @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F3 ) @ Xs ) ) ) ).
% filter_map
thf(fact_4610_takeWhile__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: B > A,Xs: list @ B] :
( ( takeWhile @ A @ P @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( takeWhile @ B @ ( comp @ A @ $o @ B @ P @ F3 ) @ Xs ) ) ) ).
% takeWhile_map
thf(fact_4611_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_4612_filter__rev__aux_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,A4: list @ A,Xs: list @ A] :
( ( ( P @ X3 )
=> ( ( filter_rev_aux @ A @ A4 @ P @ ( cons @ A @ X3 @ Xs ) )
= ( filter_rev_aux @ A @ ( cons @ A @ X3 @ A4 ) @ P @ Xs ) ) )
& ( ~ ( P @ X3 )
=> ( ( filter_rev_aux @ A @ A4 @ P @ ( cons @ A @ X3 @ Xs ) )
= ( filter_rev_aux @ A @ A4 @ P @ Xs ) ) ) ) ).
% filter_rev_aux.simps(2)
thf(fact_4613_filter__rev__aux_Osimps_I1_J,axiom,
! [A: $tType,A4: list @ A,P: A > $o] :
( ( filter_rev_aux @ A @ A4 @ P @ ( nil @ A ) )
= A4 ) ).
% filter_rev_aux.simps(1)
thf(fact_4614_folding__insort__key_Oinsort__key__commute,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A,X3: B,Y: B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( member @ B @ X3 @ S )
=> ( ( member @ B @ Y @ S )
=> ( ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ Y ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ X3 ) )
= ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ X3 ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ Y ) ) ) ) ) ) ).
% folding_insort_key.insort_key_commute
thf(fact_4615_set__oo__map__alt,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( comp @ ( list @ B ) @ ( set @ B ) @ ( list @ A ) @ ( set2 @ B ) @ ( map @ A @ B @ F3 ) )
= ( ^ [L4: list @ A] : ( image2 @ A @ B @ F3 @ ( set2 @ A @ L4 ) ) ) ) ).
% set_oo_map_alt
thf(fact_4616_assn__aci_I11_J,axiom,
! [X3: assn,Y: assn,A4: assn,B3: assn] :
( ( syntax7388354845996824322omatch @ assn @ assn @ ( times_times @ assn @ X3 @ Y ) @ A4 )
=> ( ( times_times @ assn @ A4 @ B3 )
= ( times_times @ assn @ B3 @ A4 ) ) ) ).
% assn_aci(11)
thf(fact_4617_mult_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( times_times @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 ) ) ) ).
% mult.right_commute
thf(fact_4618_mult_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( times_times @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) ) ) ) ).
% mult.right_assoc
thf(fact_4619_remove__rev__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X4: A] :
( filter_rev @ A
@ ( comp @ $o @ $o @ A @ (~)
@ ( ^ [Y4: A,Z4: A] : Y4 = Z4
@ X4 ) ) ) ) ) ).
% remove_rev_def
thf(fact_4620_remdup__sort__mergesort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( remdups @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4 ) )
= ( mergesort_remdups @ A ) ) ) ).
% remdup_sort_mergesort_remdups
thf(fact_4621_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C3: set @ ( set @ B ),G: B > A] :
( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C3 )
=> ( finite_finite2 @ B @ X ) )
=> ( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C3 )
=> ! [Xa4: set @ B] :
( ( member @ ( set @ B ) @ Xa4 @ C3 )
=> ( ( X != Xa4 )
=> ( ( inf_inf @ ( set @ B ) @ X @ Xa4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C3 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ C3 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_4622_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C3: set @ ( set @ B ),G: B > A] :
( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C3 )
=> ( finite_finite2 @ B @ X ) )
=> ( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C3 )
=> ! [Xa4: set @ B] :
( ( member @ ( set @ B ) @ Xa4 @ C3 )
=> ( ( X != Xa4 )
=> ( ( inf_inf @ ( set @ B ) @ X @ Xa4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C3 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ C3 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_4623_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_4624_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_4625_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X4: int] :
( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J2 ) )
@ ( rep_Integ @ X4 ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_4626_filter__rev__def,axiom,
! [A: $tType] :
( ( filter_rev @ A )
= ( filter_rev_aux @ A @ ( nil @ A ) ) ) ).
% filter_rev_def
thf(fact_4627_uminus__int__def,axiom,
( ( uminus_uminus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X4 ) ) ) ) ).
% uminus_int_def
thf(fact_4628_SUP__principal,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),I3: set @ B] :
( ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [I4: B] : ( principal @ A @ ( A3 @ I4 ) )
@ I3 ) )
= ( principal @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A3 @ I3 ) ) ) ) ).
% SUP_principal
thf(fact_4629_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I3: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I3 )
& ( ( P4 @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I3 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert3 @ B @ I @ I3 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P4 @ I3 ) ) )
& ( ~ ( member @ B @ I @ I3 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert3 @ B @ I @ I3 ) )
= ( times_times @ A @ ( P4 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P4 @ I3 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_4630_principal__inject,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( principal @ A @ A3 )
= ( principal @ A @ B2 ) )
= ( A3 = B2 ) ) ).
% principal_inject
thf(fact_4631_sup__principal,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( sup_sup @ ( filter @ A ) @ ( principal @ A @ A3 ) @ ( principal @ A @ B2 ) )
= ( principal @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% sup_principal
thf(fact_4632_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_4633_principal__le__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( principal @ A @ A3 ) @ ( principal @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% principal_le_iff
thf(fact_4634_inf__principal,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( inf_inf @ ( filter @ A ) @ ( principal @ A @ A3 ) @ ( principal @ A @ B2 ) )
= ( principal @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% inf_principal
thf(fact_4635_top__eq__principal__UNIV,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( principal @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% top_eq_principal_UNIV
thf(fact_4636_bot__eq__principal__empty,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_eq_principal_empty
thf(fact_4637_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_4638_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I3 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I4: B] : ( times_times @ A @ ( G @ I4 ) @ ( H2 @ I4 ) )
@ I3 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I3 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I3 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_4639_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T4: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( one_one @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_4640_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T4: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ I5 )
= ( one_one @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_4641_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T4 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_4642_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S )
= ( groups1962203154675924110t_prod @ B @ A @ G @ T4 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_4643_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I3 )
& ( ( G @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I3 )
& ( ( H2 @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I4: B] : ( times_times @ A @ ( G @ I4 ) @ ( H2 @ I4 ) )
@ I3 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I3 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I3 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_4644_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X5: set @ A,F3: A > ( set @ B )] :
( ( finite_finite2 @ A @ X5 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [X4: A] : ( principal @ B @ ( F3 @ X4 ) )
@ X5 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ X5 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_4645_times__int__def,axiom,
( ( times_times @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U3 ) @ ( times_times @ nat @ Y5 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V4 ) @ ( times_times @ nat @ Y5 @ U3 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_4646_minus__int__def,axiom,
( ( minus_minus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ Y5 @ U3 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_4647_plus__int__def,axiom,
( ( plus_plus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U3 ) @ ( plus_plus @ nat @ Y5 @ V4 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_4648_at__bot__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K4: A] : ( principal @ A @ ( set_ord_atMost @ A @ K4 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_bot_def
thf(fact_4649_at__top__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K4: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K4 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_top_def
thf(fact_4650_at__bot__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K4: A] : ( principal @ A @ ( set_ord_atMost @ A @ K4 ) )
@ ( set_ord_atMost @ A @ C2 ) ) ) ) ) ).
% at_bot_sub
thf(fact_4651_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_top @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_top_linorder
thf(fact_4652_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_bot @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_bot_linorder
thf(fact_4653_at__top__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K4: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K4 ) )
@ ( set_ord_atLeast @ A @ C2 ) ) ) ) ) ).
% at_top_sub
thf(fact_4654_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F7: A > $o,Xs3: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F7 @ Xs3 ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F7 ) @ Xs3 ) ) ) ) ).
% partition_filter_conv
thf(fact_4655_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_4656_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 )
@ ^ [X4: list @ A] :
( X4
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_4657_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A] :
( ( ord_max @ A @ A4 @ A4 )
= A4 ) ) ).
% max.idem
thf(fact_4658_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_max @ A @ A4 @ ( ord_max @ A @ A4 @ B3 ) )
= ( ord_max @ A @ A4 @ B3 ) ) ) ).
% max.left_idem
thf(fact_4659_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_max @ A @ ( ord_max @ A @ A4 @ B3 ) @ B3 )
= ( ord_max @ A @ A4 @ B3 ) ) ) ).
% max.right_idem
thf(fact_4660_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C2 ) @ A4 )
= ( ( ord_less_eq @ A @ B3 @ A4 )
& ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% max.bounded_iff
thf(fact_4661_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_max @ A @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb2
thf(fact_4662_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_max @ A @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb1
thf(fact_4663_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( ord_less @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 )
= ( ( ord_less @ A @ X3 @ Z2 )
& ( ord_less @ A @ Y @ Z2 ) ) ) ) ).
% max_less_iff_conj
thf(fact_4664_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_max @ A @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb4
thf(fact_4665_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_max @ A @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb3
thf(fact_4666_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X3 )
= X3 ) ) ).
% max_bot
thf(fact_4667_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X3: A] :
( ( ord_max @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% max_bot2
thf(fact_4668_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_max @ A @ X3 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_4669_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X3: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X3 )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_4670_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_max @ A @ X3 @ ( ord_min @ A @ X3 @ Y ) )
= X3 ) ) ).
% max_min_same(1)
thf(fact_4671_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y ) @ X3 )
= X3 ) ) ).
% max_min_same(2)
thf(fact_4672_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X3 @ Y ) @ Y )
= Y ) ) ).
% max_min_same(3)
thf(fact_4673_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ( ord_max @ A @ Y @ ( ord_min @ A @ X3 @ Y ) )
= Y ) ) ).
% max_min_same(4)
thf(fact_4674_of__bool__or__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
| Q ) )
= ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_or_iff
thf(fact_4675_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).
% Max_insert
thf(fact_4676_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A4 @ B3 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_4677_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ A4 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A4 @ B3 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_4678_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( A5
= ( ord_max @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_4679_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less @ A @ ( ord_max @ A @ B3 @ C2 ) @ A4 )
=> ~ ( ( ord_less @ A @ B3 @ A4 )
=> ~ ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% max.strict_boundedE
thf(fact_4680_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less @ A @ Z2 @ ( ord_max @ A @ X3 @ Y ) )
= ( ( ord_less @ A @ Z2 @ X3 )
| ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_4681_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A4 @ B3 ) @ C2 )
= ( ord_max @ A @ A4 @ ( ord_max @ A @ B3 @ C2 ) ) ) ) ).
% max.assoc
thf(fact_4682_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( ord_max @ A @ B4 @ A5 ) ) ) ) ).
% max.commute
thf(fact_4683_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_max @ A @ B3 @ ( ord_max @ A @ A4 @ C2 ) )
= ( ord_max @ A @ A4 @ ( ord_max @ A @ B3 @ C2 ) ) ) ) ).
% max.left_commute
thf(fact_4684_min__max__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_min @ A @ A4 @ ( ord_max @ A @ B3 @ C2 ) )
= ( ord_max @ A @ ( ord_min @ A @ A4 @ B3 ) @ ( ord_min @ A @ A4 @ C2 ) ) ) ) ).
% min_max_distrib2
thf(fact_4685_min__max__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_min @ A @ ( ord_max @ A @ B3 @ C2 ) @ A4 )
= ( ord_max @ A @ ( ord_min @ A @ B3 @ A4 ) @ ( ord_min @ A @ C2 @ A4 ) ) ) ) ).
% min_max_distrib1
thf(fact_4686_max__min__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_max @ A @ A4 @ ( ord_min @ A @ B3 @ C2 ) )
= ( ord_min @ A @ ( ord_max @ A @ A4 @ B3 ) @ ( ord_max @ A @ A4 @ C2 ) ) ) ) ).
% max_min_distrib2
thf(fact_4687_max__min__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_max @ A @ ( ord_min @ A @ B3 @ C2 ) @ A4 )
= ( ord_min @ A @ ( ord_max @ A @ B3 @ A4 ) @ ( ord_max @ A @ C2 @ A4 ) ) ) ) ).
% max_min_distrib1
thf(fact_4688_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 )
= ( ord_max @ A @ ( plus_plus @ A @ X3 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% max_add_distrib_left
thf(fact_4689_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X3 @ ( ord_max @ A @ Y @ Z2 ) )
= ( ord_max @ A @ ( plus_plus @ A @ X3 @ Y ) @ ( plus_plus @ A @ X3 @ Z2 ) ) ) ) ).
% max_add_distrib_right
thf(fact_4690_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X3: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X3 @ Y ) @ Z2 )
= ( ord_max @ A @ ( minus_minus @ A @ X3 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% max_diff_distrib_left
thf(fact_4691_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ B4 @ A5 ) ) ) ) ).
% max_def
thf(fact_4692_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_max @ A @ X3 @ Y )
= X3 ) ) ) ).
% max_absorb1
thf(fact_4693_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_max @ A @ X3 @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_4694_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A4: A,D3: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ( ord_less_eq @ A @ D3 @ B3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D3 ) @ ( ord_max @ A @ A4 @ B3 ) ) ) ) ) ).
% max.mono
thf(fact_4695_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4
= ( ord_max @ A @ A4 @ B3 ) ) ) ) ).
% max.orderE
thf(fact_4696_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] :
( ( A4
= ( ord_max @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% max.orderI
thf(fact_4697_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C2 ) @ A4 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A4 )
=> ~ ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% max.boundedE
thf(fact_4698_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B3 @ C2 ) @ A4 ) ) ) ) ).
% max.boundedI
thf(fact_4699_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( A5
= ( ord_max @ A @ A5 @ B4 ) ) ) ) ) ).
% max.order_iff
thf(fact_4700_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: A,B3: A] : ( ord_less_eq @ A @ A4 @ ( ord_max @ A @ A4 @ B3 ) ) ) ).
% max.cobounded1
thf(fact_4701_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B3: A,A4: A] : ( ord_less_eq @ A @ B3 @ ( ord_max @ A @ A4 @ B3 ) ) ) ).
% max.cobounded2
thf(fact_4702_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X3: A,Y: A] :
( ( ord_less_eq @ A @ Z2 @ ( ord_max @ A @ X3 @ Y ) )
= ( ( ord_less_eq @ A @ Z2 @ X3 )
| ( ord_less_eq @ A @ Z2 @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_4703_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_max @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% max.absorb_iff1
thf(fact_4704_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_max @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% max.absorb_iff2
thf(fact_4705_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A4 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A4 @ B3 ) ) ) ) ).
% max.coboundedI1
thf(fact_4706_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A4 @ B3 ) ) ) ) ).
% max.coboundedI2
thf(fact_4707_sup__max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% sup_max
thf(fact_4708_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X3: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X3 @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_4709_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X3: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X3 @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_4710_partition__rev_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A,X3: A,Xs: list @ A] :
( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ ( cons @ A @ X3 @ Xs ) )
= ( partition_rev @ A @ P @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Yes2 ) @ No2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X3 @ No2 ) ) ) @ Xs ) ) ).
% partition_rev.simps(2)
thf(fact_4711_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_4712_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X4: nat,Y5: nat] : ( ord_less_eq @ nat @ Y5 @ X4 )
@ ^ [X4: nat,Y5: nat] : ( ord_less @ nat @ Y5 @ X4 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_4713_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_4714_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 ) )
=> ( ! [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Yes2 ) )
=> ( P @ X8 ) )
& ! [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ No2 ) )
=> ~ ( P @ X8 ) ) ) ) ).
% partition_P
thf(fact_4715_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X3 @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_4716_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X3 @ Y ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X3 @ P4 ) @ ( times_times @ A @ Y @ P4 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_4717_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X3 @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X3 @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_4718_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X3: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X3 @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X3 @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X3 ) @ ( times_times @ A @ P4 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_4719_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X3: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( S
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert3 @ A @ X3 @ S ) )
= X3 ) )
& ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert3 @ A @ X3 @ S ) )
= ( ord_max @ A @ X3 @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_4720_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N7: set @ A] :
( ! [X: A,Y2: A] :
( ( H2 @ ( ord_max @ A @ X @ Y2 ) )
= ( ord_max @ A @ ( H2 @ X ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N7 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_4721_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B2 ) @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ).
% Max.subset
thf(fact_4722_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A,Y2: A] : ( member @ A @ ( ord_max @ A @ X @ Y2 ) @ ( insert3 @ A @ X @ ( insert3 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Max.closed
thf(fact_4723_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ A3 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_4724_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A3 ) @ ( lattic643756798349783984er_Max @ A @ B2 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_4725_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X3 @ A3 ) ) ) ) ).
% Max.eq_fold
thf(fact_4726_partition_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( partition @ A @ P @ ( cons @ A @ X3 @ 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 @ X3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Yes3 ) @ No3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes3 @ ( cons @ A @ X3 @ No3 ) ) )
@ ( partition @ A @ P @ Xs ) ) ) ).
% partition.simps(2)
thf(fact_4727_partition__rev_Oelims,axiom,
! [A: $tType,X3: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( partition_rev @ A @ X3 @ 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 ) )
=> ! [X: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X @ Xs2 ) )
=> ( Y
!= ( partition_rev @ A @ X3 @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X3 @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X @ No ) ) ) @ Xs2 ) ) ) ) ) ) ).
% partition_rev.elims
thf(fact_4728_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_4729_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert3 @ A @ X3 @ A3 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_4730_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A3 )
= ( ord_max @ A @ X3 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_4731_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_4732_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
@ ^ [X4: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X4 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss2 )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_4733_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_4734_partition__rev_Opelims,axiom,
! [A: $tType,X3: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( partition_rev @ A @ X3 @ 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 ) ) @ X3 @ ( 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 ) ) @ X3 @ ( 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 ) )
=> ! [X: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X @ Xs2 ) )
=> ( ( Y
= ( partition_rev @ A @ X3 @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X3 @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X @ 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 ) ) @ X3 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% partition_rev.pelims
thf(fact_4735_quicksort__by__rel_Oelims,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X @ Xs2 ) )
=> ( Y
!= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X3 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ X3 @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y5: A] : ( X3 @ Y5 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) ) ) ) ) ) ).
% quicksort_by_rel.elims
thf(fact_4736_quicksort__by__rel_Osimps_I2_J,axiom,
! [A: $tType,R: A > A > $o,Sl2: list @ A,X3: A,Xs: list @ A] :
( ( quicksort_by_rel @ A @ R @ Sl2 @ ( cons @ A @ X3 @ 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 @ X3 @ ( quicksort_by_rel @ A @ R @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y5: A] : ( R @ Y5 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ).
% quicksort_by_rel.simps(2)
thf(fact_4737_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_4738_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_4739_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_4740_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_4741_quicksort__by__rel__remove__acc,axiom,
! [A: $tType] :
( ( quicksort_by_rel @ A )
= ( ^ [R4: A > A > $o,Sl3: list @ A,Xs3: list @ A] : ( append @ A @ ( quicksort_by_rel @ A @ R4 @ ( nil @ A ) @ Xs3 ) @ Sl3 ) ) ) ).
% quicksort_by_rel_remove_acc
thf(fact_4742_sorted__wrt__quicksort__by__rel,axiom,
! [X14: $tType,R: X14 > X14 > $o,Xs: list @ X14] :
( ! [X: X14,Y2: X14] :
( ( R @ X @ Y2 )
| ( R @ Y2 @ X ) )
=> ( ! [X: X14,Y2: X14,Z3: X14] :
( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( sorted_wrt @ X14 @ R @ ( quicksort_by_rel @ X14 @ R @ ( nil @ X14 ) @ Xs ) ) ) ) ).
% sorted_wrt_quicksort_by_rel
thf(fact_4743_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_4744_sort__quicksort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4 )
= ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ) ).
% sort_quicksort_by_rel
thf(fact_4745_quicksort__by__rel_Opsimps_I2_J,axiom,
! [A: $tType,R: A > A > $o,Sl2: list @ A,X3: 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 @ X3 @ Xs ) ) ) )
=> ( ( quicksort_by_rel @ A @ R @ Sl2 @ ( cons @ A @ X3 @ 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 @ X3 @ ( quicksort_by_rel @ A @ R @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y5: A] : ( R @ Y5 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ) ).
% quicksort_by_rel.psimps(2)
thf(fact_4746_quicksort__by__rel_Opelims,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X3 @ 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 ) ) @ X3 @ ( 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X @ Xs2 ) )
=> ( ( Y
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X3 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ X3 @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y5: A] : ( X3 @ Y5 @ X )
@ ( 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( cons @ A @ X @ Xs2 ) ) ) ) ) ) ) ) ) ).
% quicksort_by_rel.pelims
thf(fact_4747_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 ) ) )
=> ( ! [R6: 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 ) ) @ R6 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ( P @ R6 @ Sl @ ( nil @ A ) ) )
=> ( ! [R6: A > A > $o,Sl: list @ A,X: 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 ) ) @ R6 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X @ Xs2 ) ) ) )
=> ( ! [Xa3: product_prod @ ( list @ A ) @ ( list @ A ),Xb3: list @ A,Y3: list @ A] :
( ( Xa3
= ( partition_rev @ A
@ ^ [Z6: A] : ( R6 @ Z6 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb3 @ Y3 )
= Xa3 )
=> ( P @ R6 @ Sl @ Y3 ) ) )
=> ( ! [Xa3: product_prod @ ( list @ A ) @ ( list @ A ),Xb3: list @ A,Y3: list @ A] :
( ( Xa3
= ( partition_rev @ A
@ ^ [Z6: A] : ( R6 @ Z6 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb3 @ Y3 )
= Xa3 )
=> ( P @ R6 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R6 @ Sl @ Y3 ) ) @ Xb3 ) ) )
=> ( P @ R6 @ Sl @ ( cons @ A @ X @ Xs2 ) ) ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% quicksort_by_rel.pinduct
thf(fact_4748_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_4749_set__nths,axiom,
! [A: $tType,Xs: list @ A,I3: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I3 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I4: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I4 @ I3 ) ) ) ) ).
% set_nths
thf(fact_4750_sorted__list__of__set__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( linord144544945434240204of_set @ A @ A
@ ^ [X4: A] : X4 ) ) ) ).
% sorted_list_of_set_def
thf(fact_4751_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A] :
( ( finite_finite2 @ A @ ( image2 @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F3 )
=> ( ! [N5: nat] :
( ( ( F3 @ N5 )
= ( F3 @ ( suc @ N5 ) ) )
=> ( ( F3 @ ( suc @ N5 ) )
= ( F3 @ ( suc @ ( suc @ N5 ) ) ) ) )
=> ? [N12: nat] :
( ! [N8: nat] :
( ( ord_less_eq @ nat @ N8 @ N12 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N12 )
=> ( ( ord_less @ nat @ M3 @ N8 )
=> ( ord_less @ A @ ( F3 @ M3 ) @ ( F3 @ N8 ) ) ) ) )
& ! [N8: nat] :
( ( ord_less_eq @ nat @ N12 @ N8 )
=> ( ( F3 @ N12 )
= ( F3 @ N8 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_4752_nths__nil,axiom,
! [A: $tType,A3: set @ nat] :
( ( nths @ A @ ( nil @ A ) @ A3 )
= ( nil @ A ) ) ).
% nths_nil
thf(fact_4753_nths__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_4754_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_4755_nths__singleton,axiom,
! [A: $tType,A3: set @ nat,X3: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ( nths @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ A3 )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ( nths @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ A3 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_4756_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F7: A > B] :
! [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( ord_less_eq @ B @ ( F7 @ X4 ) @ ( F7 @ Y5 ) ) ) ) ) ) ).
% mono_def
thf(fact_4757_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( order_mono @ A @ B @ F3 ) ) ) ).
% monoI
thf(fact_4758_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ).
% monoE
thf(fact_4759_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ).
% monoD
thf(fact_4760_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A4: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A4 ) ) ) ).
% mono_add
thf(fact_4761_distinct__nthsI,axiom,
! [A: $tType,Xs: list @ A,I3: set @ nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( nths @ A @ Xs @ I3 ) ) ) ).
% distinct_nthsI
thf(fact_4762_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,M2: A,N: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_min @ B @ ( F3 @ M2 ) @ ( F3 @ N ) )
= ( F3 @ ( ord_min @ A @ M2 @ N ) ) ) ) ) ).
% min_of_mono
thf(fact_4763_nths__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,I3: set @ nat] :
( ( nths @ A @ ( map @ B @ A @ F3 @ Xs ) @ I3 )
= ( map @ B @ A @ F3 @ ( nths @ B @ Xs @ I3 ) ) ) ).
% nths_map
thf(fact_4764_in__set__nthsD,axiom,
! [A: $tType,X3: A,Xs: list @ A,I3: set @ nat] :
( ( member @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs @ I3 ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_4765_notin__set__nthsI,axiom,
! [A: $tType,X3: A,Xs: list @ A,I3: set @ nat] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ~ ( member @ A @ X3 @ ( set2 @ A @ ( nths @ A @ Xs @ I3 ) ) ) ) ).
% notin_set_nthsI
thf(fact_4766_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_4767_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ord_less @ A @ X3 @ Y ) ) ) ) ).
% mono_strict_invE
thf(fact_4768_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,M2: A,N: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_max @ B @ ( F3 @ M2 ) @ ( F3 @ N ) )
= ( F3 @ ( ord_max @ A @ M2 @ N ) ) ) ) ) ).
% max_of_mono
thf(fact_4769_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ).
% mono_invE
thf(fact_4770_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F3: A > B,A3: A,B2: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( inf_inf @ A @ A3 @ B2 ) ) @ ( inf_inf @ B @ ( F3 @ A3 ) @ ( F3 @ B2 ) ) ) ) ) ).
% mono_inf
thf(fact_4771_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F3: A > B,A3: A,B2: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F3 @ A3 ) @ ( F3 @ B2 ) ) @ ( F3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% mono_sup
thf(fact_4772_set__nths__subset,axiom,
! [A: $tType,Xs: list @ A,I3: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( nths @ A @ Xs @ I3 ) ) @ ( set2 @ A @ Xs ) ) ).
% set_nths_subset
thf(fact_4773_nths__all,axiom,
! [A: $tType,Xs: list @ A,I3: set @ nat] :
( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ nat @ I5 @ I3 ) )
=> ( ( nths @ A @ Xs @ I3 )
= Xs ) ) ).
% nths_all
thf(fact_4774_sorted__nths,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I3: set @ nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs @ I3 ) ) ) ) ).
% sorted_nths
thf(fact_4775_Rings_Omono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A4 ) ) ) ) ).
% Rings.mono_mult
thf(fact_4776_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( order_mono @ nat @ nat @ ( times_times @ nat @ N ) ) ) ).
% mono_times_nat
thf(fact_4777_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F3: A > B,M2: A,N: A,M9: B,N6: B] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ( image2 @ A @ B @ F3 @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
= ( set_or7035219750837199246ssThan @ B @ M9 @ N6 ) )
=> ( ( ord_less @ A @ M2 @ N )
=> ( ( F3 @ M2 )
= M9 ) ) ) ) ) ).
% mono_image_least
thf(fact_4778_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F3: A > A,P4: A,K3: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ P4 ) @ P4 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K3 @ F3 @ ( bot_bot @ A ) ) @ P4 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_4779_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N2: nat,Xs3: list @ A] : ( nths @ A @ Xs3 @ ( collect @ nat @ ( ord_less_eq @ nat @ N2 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_4780_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M2: nat,N: nat,F3: A > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M2 @ F3 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F3 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_4781_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F3 @ ( lattic643756798349783984er_Max @ A @ A3 ) )
= ( lattic643756798349783984er_Max @ B @ ( image2 @ A @ B @ F3 @ A3 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_4782_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F3 @ ( lattic643756798350308766er_Min @ A @ A3 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F3 @ A3 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_4783_nths__append,axiom,
! [A: $tType,L: list @ A,L2: list @ A,A3: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L2 ) @ A3 )
= ( append @ A @ ( nths @ A @ L @ A3 )
@ ( nths @ A @ L2
@ ( collect @ nat
@ ^ [J2: nat] : ( member @ nat @ ( plus_plus @ nat @ J2 @ ( size_size @ ( list @ A ) @ L ) ) @ A3 ) ) ) ) ) ).
% nths_append
thf(fact_4784_nths__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I3: set @ nat] :
( ( nths @ A @ ( drop @ A @ N @ Xs ) @ I3 )
= ( nths @ A @ Xs @ ( image2 @ nat @ nat @ ( plus_plus @ nat @ N ) @ I3 ) ) ) ).
% nths_drop
thf(fact_4785_filter__in__nths,axiom,
! [A: $tType,Xs: list @ A,S3: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ ( nths @ A @ Xs @ S3 ) ) )
@ Xs )
= ( nths @ A @ Xs @ S3 ) ) ) ).
% filter_in_nths
thf(fact_4786_length__nths,axiom,
! [A: $tType,Xs: list @ A,I3: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I3 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I4 @ I3 ) ) ) ) ) ).
% length_nths
thf(fact_4787_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P2: A > $o,Xs3: list @ A] :
( nths @ A @ Xs3
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P2 @ ( nth @ A @ Xs3 @ I4 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_4788_nths__Cons,axiom,
! [A: $tType,X3: A,L: list @ A,A3: set @ nat] :
( ( nths @ A @ ( cons @ A @ X3 @ L ) @ A3 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J2: nat] : ( member @ nat @ ( suc @ J2 ) @ A3 ) ) ) ) ) ).
% nths_Cons
thf(fact_4789_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys )
= ( ? [F7: nat > nat] :
( ( order_mono @ nat @ nat @ F7 )
& ( ( image2 @ nat @ nat @ F7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys @ ( F7 @ I4 ) ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) )
= ( ( F7 @ I4 )
= ( F7 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_4790_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M2: nat,A4: A] :
( ( enumerate @ A @ N @ ( replicate @ A @ M2 @ A4 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q8: nat] : ( product_Pair @ nat @ A @ Q8 @ A4 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M2 ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_4791_finite__subsets__at__top__finite,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite5375528669736107172at_top @ A @ A3 )
= ( principal @ ( set @ A ) @ ( insert3 @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ).
% finite_subsets_at_top_finite
thf(fact_4792_replicate__eq__replicate,axiom,
! [A: $tType,M2: nat,X3: A,N: nat,Y: A] :
( ( ( replicate @ A @ M2 @ X3 )
= ( replicate @ A @ N @ Y ) )
= ( ( M2 = N )
& ( ( M2
!= ( zero_zero @ nat ) )
=> ( X3 = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_4793_length__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X3 ) )
= N ) ).
% length_replicate
thf(fact_4794_concat__replicate__trivial,axiom,
! [A: $tType,I: nat] :
( ( concat @ A @ ( replicate @ ( list @ A ) @ I @ ( nil @ A ) ) )
= ( nil @ A ) ) ).
% concat_replicate_trivial
thf(fact_4795_map__replicate,axiom,
! [A: $tType,B: $tType,F3: B > A,N: nat,X3: B] :
( ( map @ B @ A @ F3 @ ( replicate @ B @ N @ X3 ) )
= ( replicate @ A @ N @ ( F3 @ X3 ) ) ) ).
% map_replicate
thf(fact_4796_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_4797_remdups__adj__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups_adj @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% remdups_adj_set
thf(fact_4798_rev__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( rev @ A @ ( replicate @ A @ N @ X3 ) )
= ( replicate @ A @ N @ X3 ) ) ).
% rev_replicate
thf(fact_4799_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_4800_hd__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% hd_remdups_adj
thf(fact_4801_last__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( last @ A @ ( remdups_adj @ A @ Xs ) )
= ( last @ A @ Xs ) ) ).
% last_remdups_adj
thf(fact_4802_replicate__empty,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( replicate @ A @ N @ X3 )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_4803_empty__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X3 ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_4804_in__set__replicate,axiom,
! [A: $tType,X3: A,N: nat,Y: A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( replicate @ A @ N @ Y ) ) )
= ( ( X3 = Y )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_4805_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A4: A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N @ A4 ) ) )
& ( P @ X4 ) ) )
= ( ( P @ A4 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_4806_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A4: A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N @ A4 ) ) )
=> ( P @ X4 ) ) )
= ( ( P @ A4 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_4807_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X3: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X3 ) @ I )
= X3 ) ) ).
% nth_replicate
thf(fact_4808_hd__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X3 ) )
= X3 ) ) ).
% hd_replicate
thf(fact_4809_drop__replicate,axiom,
! [A: $tType,I: nat,K3: nat,X3: A] :
( ( drop @ A @ I @ ( replicate @ A @ K3 @ X3 ) )
= ( replicate @ A @ ( minus_minus @ nat @ K3 @ I ) @ X3 ) ) ).
% drop_replicate
thf(fact_4810_takeWhile__replicate,axiom,
! [A: $tType,P: A > $o,X3: A,N: nat] :
( ( ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( replicate @ A @ N @ X3 ) )
= ( replicate @ A @ N @ X3 ) ) )
& ( ~ ( P @ X3 )
=> ( ( takeWhile @ A @ P @ ( replicate @ A @ N @ X3 ) )
= ( nil @ A ) ) ) ) ).
% takeWhile_replicate
thf(fact_4811_remdups__adj__Cons__alt,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_4812_last__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X3 ) )
= X3 ) ) ).
% last_replicate
thf(fact_4813_foldr__replicate,axiom,
! [A: $tType,B: $tType,F3: B > A > A,N: nat,X3: B] :
( ( foldr @ B @ A @ F3 @ ( replicate @ B @ N @ X3 ) )
= ( compow @ ( A > A ) @ N @ ( F3 @ X3 ) ) ) ).
% foldr_replicate
thf(fact_4814_take__replicate,axiom,
! [A: $tType,I: nat,K3: nat,X3: A] :
( ( take @ A @ I @ ( replicate @ A @ K3 @ X3 ) )
= ( replicate @ A @ ( ord_min @ nat @ I @ K3 ) @ X3 ) ) ).
% take_replicate
thf(fact_4815_tl__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( tl @ A @ ( replicate @ A @ N @ X3 ) )
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X3 ) ) ).
% tl_replicate
thf(fact_4816_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X3: A,J: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I @ X3 ) @ ( replicate @ B @ J @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X3 @ Y ) ) ) ).
% zip_replicate
thf(fact_4817_set__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X3 ) )
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_4818_finite__subsets__at__top__neq__bot,axiom,
! [A: $tType,A3: set @ A] :
( ( finite5375528669736107172at_top @ A @ A3 )
!= ( bot_bot @ ( filter @ ( set @ A ) ) ) ) ).
% finite_subsets_at_top_neq_bot
thf(fact_4819_remdups__adj__replicate,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X3 ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X3 ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_replicate
thf(fact_4820_append__replicate__commute,axiom,
! [A: $tType,N: nat,X3: A,K3: nat] :
( ( append @ A @ ( replicate @ A @ N @ X3 ) @ ( replicate @ A @ K3 @ X3 ) )
= ( append @ A @ ( replicate @ A @ K3 @ X3 ) @ ( replicate @ A @ N @ X3 ) ) ) ).
% append_replicate_commute
thf(fact_4821_remdups__adj_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups_adj @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups_adj.simps(1)
thf(fact_4822_remdups__adj__singleton,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( remdups_adj @ A @ Xs )
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Xs
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X3 ) ) ) ).
% remdups_adj_singleton
thf(fact_4823_remdups__adj_Osimps_I3_J,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A] :
( ( ( X3 = Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
& ( ( X3 != Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
= ( cons @ A @ X3 @ ( remdups_adj @ A @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_4824_remdups__adj__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups_adj @ A @ Xs )
= Xs ) ) ).
% remdups_adj_distinct
thf(fact_4825_mono__Int,axiom,
! [B: $tType,A: $tType,F3: ( set @ A ) > ( set @ B ),A3: set @ A,B2: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) @ ( inf_inf @ ( set @ B ) @ ( F3 @ A3 ) @ ( F3 @ B2 ) ) ) ) ).
% mono_Int
thf(fact_4826_mono__Un,axiom,
! [B: $tType,A: $tType,F3: ( set @ A ) > ( set @ B ),A3: set @ A,B2: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F3 @ A3 ) @ ( F3 @ B2 ) ) @ ( F3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% mono_Un
thf(fact_4827_replicate__Suc,axiom,
! [A: $tType,N: nat,X3: A] :
( ( replicate @ A @ ( suc @ N ) @ X3 )
= ( cons @ A @ X3 @ ( replicate @ A @ N @ X3 ) ) ) ).
% replicate_Suc
thf(fact_4828_replicate__0,axiom,
! [A: $tType,X3: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X3 )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_4829_replicate__app__Cons__same,axiom,
! [A: $tType,N: nat,X3: A,Xs: list @ A] :
( ( append @ A @ ( replicate @ A @ N @ X3 ) @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( append @ A @ ( replicate @ A @ N @ X3 ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_4830_replicate__length__same,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( X = X3 ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X3 )
= Xs ) ) ).
% replicate_length_same
thf(fact_4831_replicate__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( Y2 = X3 ) )
=> ( Xs
= ( replicate @ A @ N @ X3 ) ) ) ) ).
% replicate_eqI
thf(fact_4832_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: nat,X3: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X3 ) ) ) ).
% sorted_replicate
thf(fact_4833_replicate__add,axiom,
! [A: $tType,N: nat,M2: nat,X3: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N @ M2 ) @ X3 )
= ( append @ A @ ( replicate @ A @ N @ X3 ) @ ( replicate @ A @ M2 @ X3 ) ) ) ).
% replicate_add
thf(fact_4834_filter__replicate,axiom,
! [A: $tType,P: A > $o,X3: A,N: nat] :
( ( ( P @ X3 )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N @ X3 ) )
= ( replicate @ A @ N @ X3 ) ) )
& ( ~ ( P @ X3 )
=> ( ( filter2 @ A @ P @ ( replicate @ A @ N @ X3 ) )
= ( nil @ A ) ) ) ) ).
% filter_replicate
thf(fact_4835_comm__append__are__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [M6: nat,N5: nat,Zs2: list @ A] :
( ( ( concat @ A @ ( replicate @ ( list @ A ) @ M6 @ Zs2 ) )
= Xs )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N5 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_4836_remdups__adj_Oelims,axiom,
! [A: $tType,X3: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X3 )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ( ! [X: A] :
( ( X3
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( Y
!= ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ~ ( ( ( X = Y2 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs2 ) ) ) )
& ( ( X != Y2 )
=> ( Y
= ( cons @ A @ X @ ( remdups_adj @ A @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_4837_remdups__adj_Osimps_I2_J,axiom,
! [A: $tType,X3: A] :
( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% remdups_adj.simps(2)
thf(fact_4838_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_4839_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_4840_map__replicate__const,axiom,
! [B: $tType,A: $tType,K3: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X4: B] : K3
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K3 ) ) ).
% map_replicate_const
thf(fact_4841_replicate__length__filter,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y4: A,Z4: A] : Y4 = Z4
@ X3 )
@ Xs ) )
@ X3 )
= ( filter2 @ A
@ ( ^ [Y4: A,Z4: A] : Y4 = Z4
@ X3 )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_4842_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_4843_replicate__append__same,axiom,
! [A: $tType,I: nat,X3: A] :
( ( append @ A @ ( replicate @ A @ I @ X3 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( cons @ A @ X3 @ ( replicate @ A @ I @ X3 ) ) ) ).
% replicate_append_same
thf(fact_4844_sum__list__replicate,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,C2: A] :
( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N @ C2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ C2 ) ) ) ).
% sum_list_replicate
thf(fact_4845_remdups__adj__append__two,axiom,
! [A: $tType,Xs: list @ A,X3: A,Y: A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X3 = Y ) @ ( nil @ A ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_append_two
thf(fact_4846_map__replicate__trivial,axiom,
! [A: $tType,X3: A,I: nat] :
( ( map @ nat @ A
@ ^ [I4: nat] : X3
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X3 ) ) ).
% map_replicate_trivial
thf(fact_4847_remdups__adj__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) )
= ( case_list @ ( list @ A ) @ A @ ( cons @ A @ X3 @ ( nil @ A ) )
@ ^ [Y5: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( X3 = Y5 ) @ ( cons @ A @ Y5 @ Xs3 ) @ ( cons @ A @ X3 @ ( cons @ A @ Y5 @ Xs3 ) ) )
@ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_4848_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X3: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X3 ) )
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_4849_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X3: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X3 ) )
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_4850_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N: nat,X3: A] :
( ( replicate @ A @ ( suc @ N ) @ X3 )
= ( append @ A @ ( replicate @ A @ N @ X3 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% replicate_Suc_conv_snoc
thf(fact_4851_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X3: A,Ys: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N @ X3 ) @ Ys )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ ( take @ B @ N @ Ys ) ) ) ).
% zip_replicate1
thf(fact_4852_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_4853_ord_Olexordp_Omono,axiom,
! [A: $tType,Less: A > A > $o] :
( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P7: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X24: list @ A] :
( ? [Y5: A,Ys3: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) )
& ( Less @ X4 @ Y5 ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( Less @ X4 @ Y5 )
& ~ ( Less @ Y5 @ X4 )
& ( P7 @ Xs3 @ Ys3 ) ) ) ) ).
% ord.lexordp.mono
thf(fact_4854_remdups__adj__append,axiom,
! [A: $tType,Xs_1: list @ A,X3: A,Xs_2: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ Xs_2 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_4855_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P7: ( set @ A ) > $o,X4: set @ A] :
( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,A5: A] :
( ( X4
= ( insert3 @ A @ A5 @ A7 ) )
& ( P7 @ A7 ) ) ) ) ).
% finite.mono
thf(fact_4856_map__zip2,axiom,
! [A: $tType,B: $tType,K3: A,L: list @ B] :
( ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 ) @ L )
= ( zip @ A @ B @ ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K3 ) @ L ) ) ).
% map_zip2
thf(fact_4857_map__zip1,axiom,
! [A: $tType,B: $tType,K3: B,L: list @ A] :
( ( map @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ K3 )
@ L )
= ( zip @ A @ B @ L @ ( replicate @ B @ ( size_size @ ( list @ A ) @ L ) @ K3 ) ) ) ).
% map_zip1
thf(fact_4858_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_4859_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 )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ Y )
@ ( take @ A @ N @ Xs ) ) ) ).
% zip_replicate2
thf(fact_4860_finite__subsets__at__top__def,axiom,
! [A: $tType] :
( ( finite5375528669736107172at_top @ A )
= ( ^ [A7: set @ A] :
( complete_Inf_Inf @ ( filter @ ( set @ A ) )
@ ( image2 @ ( set @ A ) @ ( filter @ ( set @ A ) )
@ ^ [X9: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y9: set @ A] :
( ( finite_finite2 @ A @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ X9 @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ Y9 @ A7 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X9: set @ A] :
( ( finite_finite2 @ A @ X9 )
& ( ord_less_eq @ ( set @ A ) @ X9 @ A7 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_4861_Cons__replicate__eq,axiom,
! [A: $tType,X3: A,Xs: list @ A,N: nat,Y: A] :
( ( ( cons @ A @ X3 @ Xs )
= ( replicate @ A @ N @ Y ) )
= ( ( X3 = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X3 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_4862_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 ) )
=> ? [N5: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N5 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N5 @ Zs2 ) )
= ( append @ A @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_4863_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P7: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X24: list @ A] :
( ? [Y5: A,Ys3: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ord_less @ A @ X4 @ Y5 ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( ord_less @ A @ X4 @ Y5 )
& ~ ( ord_less @ A @ Y5 @ X4 )
& ( P7 @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp.mono
thf(fact_4864_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_4865_remdups__adj_Opelims,axiom,
! [A: $tType,X3: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X3 )
= Y )
=> ( ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ X3 )
=> ( ( ( X3
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( nil @ A ) ) ) )
=> ( ! [X: A] :
( ( X3
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ( Y
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( ( ( ( X = Y2 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs2 ) ) ) )
& ( ( X != Y2 )
=> ( Y
= ( cons @ A @ X @ ( remdups_adj @ A @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_4866_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ).
% listrel_def
thf(fact_4867_ring__1__class_Oof__int__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ A @ A @ rep_Integ @ ( id @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J2 ) ) ) ) ) ) ).
% ring_1_class.of_int_def
thf(fact_4868_list_Omap__id0,axiom,
! [A: $tType] :
( ( map @ A @ A @ ( id @ A ) )
= ( id @ ( list @ A ) ) ) ).
% list.map_id0
thf(fact_4869_case__prod__Pair,axiom,
! [B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% case_prod_Pair
thf(fact_4870_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_4871_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q: A > B > C,F3: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q )
=> ( order_mono @ A @ ( D > C )
@ ^ [I4: A,X4: D] : ( Q @ I4 @ ( F3 @ X4 ) ) ) ) ) ).
% mono_compose
thf(fact_4872_list_Omap__id,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% list.map_id
thf(fact_4873_List_Omap_Oidentity,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X4: A] : X4 )
= ( id @ ( list @ A ) ) ) ).
% List.map.identity
thf(fact_4874_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X3: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( R2 @ X3 @ Y )
=> ( ( listrelp @ A @ B @ R2 @ Xs @ Ys )
=> ( listrelp @ A @ B @ R2 @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_4875_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] : ( listrelp @ A @ B @ R2 @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_4876_foldr__Nil,axiom,
! [A: $tType,B: $tType,F3: A > B > B] :
( ( foldr @ A @ B @ F3 @ ( nil @ A ) )
= ( id @ B ) ) ).
% foldr_Nil
thf(fact_4877_less__int__def,axiom,
( ( ord_less @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ U3 @ Y5 ) ) ) ) ) ) ).
% less_int_def
thf(fact_4878_foldr__filter,axiom,
! [A: $tType,B: $tType,F3: B > A > A,P: B > $o,Xs: list @ B] :
( ( foldr @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs ) )
= ( foldr @ B @ A
@ ^ [X4: B] : ( if @ ( A > A ) @ ( P @ X4 ) @ ( F3 @ X4 ) @ ( id @ A ) )
@ Xs ) ) ).
% foldr_filter
thf(fact_4879_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R5: A > B > $o,A13: list @ A,A23: list @ B] :
( ( ( A13
= ( nil @ A ) )
& ( A23
= ( nil @ B ) ) )
| ? [X4: A,Y5: B,Xs3: list @ A,Ys3: list @ B] :
( ( A13
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ B @ Y5 @ Ys3 ) )
& ( R5 @ X4 @ Y5 )
& ( listrelp @ A @ B @ R5 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_4880_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A1: list @ A,A22: list @ B] :
( ( listrelp @ A @ B @ R2 @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X: A,Y2: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys5: list @ B] :
( ( A22
= ( cons @ B @ Y2 @ Ys5 ) )
=> ( ( R2 @ X @ Y2 )
=> ~ ( listrelp @ A @ B @ R2 @ Xs2 @ Ys5 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_4881_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R2 ) )
= ( ^ [X4: list @ A,Y5: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X4 @ Y5 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_4882_map__fst__mk__fst,axiom,
! [B: $tType,A: $tType,K3: A,L: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 ) @ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K3 ) ) ).
% map_fst_mk_fst
thf(fact_4883_finite__def,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P7: ( set @ A ) > $o,X4: set @ A] :
( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,A5: A] :
( ( X4
= ( insert3 @ A @ A5 @ A7 ) )
& ( P7 @ A7 ) ) ) ) ) ).
% finite_def
thf(fact_4884_comp__fun__idem__on_Ofold__insert__idem2,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert3 @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X3 @ Z2 ) @ A3 ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem2
thf(fact_4885_img__fst,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,S: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ S )
=> ( member @ A @ A4 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) ) ) ).
% img_fst
thf(fact_4886_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_4887_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_4888_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_4889_sorted__wrt__map__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X4: product_prod @ A @ B,Y5: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ X4 ) @ ( product_fst @ A @ B @ Y5 ) )
@ 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_4890_map__fst__mk__snd,axiom,
! [B: $tType,A: $tType,K3: B,L: list @ A] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ K3 )
@ L ) )
= L ) ).
% map_fst_mk_snd
thf(fact_4891_sorted__wrt__map__rev__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X4: product_prod @ A @ B,Y5: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ Y5 ) @ ( product_fst @ A @ B @ X4 ) )
@ 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_4892_fst__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_4893_fn__fst__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > C] :
( ( ^ [X4: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ X4 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [A5: A,Uu3: B] : ( F3 @ A5 ) ) ) ).
% fn_fst_conv
thf(fact_4894_fstE,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,A4: A,B3: B,P: A > $o] :
( ( X3
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( ( P @ ( product_fst @ A @ B @ X3 ) )
=> ( P @ A4 ) ) ) ).
% fstE
thf(fact_4895_fst__eqD,axiom,
! [B: $tType,A: $tType,X3: A,Y: B,A4: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y ) )
= A4 )
=> ( X3 = A4 ) ) ).
% fst_eqD
thf(fact_4896_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_4897_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X3: A,S: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X3 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) )
=> ~ ! [Y2: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ S ) ) ).
% in_fst_imageE
thf(fact_4898_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_4899_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_4900_fst__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% fst_diag_id
thf(fact_4901_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_4902_fst__image__mp,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B ),B2: set @ A,X3: A,Y: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) @ B2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ A3 )
=> ( member @ A @ X3 @ B2 ) ) ) ).
% fst_image_mp
thf(fact_4903_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_4904_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_4905_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > C > A,G: A > B > C > B,A4: A,B3: B,Xs: list @ C] :
( ( product_fst @ A @ B
@ ( foldl @ ( product_prod @ A @ B ) @ C
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ B ) )
@ ^ [A5: A,B4: B,X4: C] : ( product_Pair @ A @ B @ ( F3 @ A5 @ X4 ) @ ( G @ A5 @ B4 @ X4 ) ) )
@ ( product_Pair @ A @ B @ A4 @ B3 )
@ Xs ) )
= ( foldl @ A @ C @ F3 @ A4 @ Xs ) ) ).
% fst_foldl
thf(fact_4906_distinct__map__fstD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),X3: A,Y: B,Z2: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z2 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( Y = Z2 ) ) ) ) ).
% distinct_map_fstD
thf(fact_4907_lfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,K3: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K3 ) @ F3 @ ( bot_bot @ A ) )
= ( compow @ ( A > A ) @ K3 @ F3 @ ( bot_bot @ A ) ) )
=> ( ( complete_lattice_lfp @ A @ F3 )
= ( compow @ ( A > A ) @ K3 @ F3 @ ( bot_bot @ A ) ) ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_4908_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_4909_comp__fun__idem__on_Ofold__insert__idem,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert3 @ A @ X3 @ A3 ) )
= ( F3 @ X3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A3 ) ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem
thf(fact_4910_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),K3: A,V1: B,V22: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_4911_bezw_Oelims,axiom,
! [X3: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X3 @ Xa )
= Y )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_4912_bezw_Osimps,axiom,
( bezw
= ( ^ [X4: nat,Y5: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y5
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y5 @ ( modulo_modulo @ nat @ X4 @ Y5 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y5 @ ( modulo_modulo @ nat @ X4 @ Y5 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y5 @ ( modulo_modulo @ nat @ X4 @ Y5 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X4 @ Y5 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_4913_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_4914_img__snd,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,S: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ S )
=> ( member @ B @ B3 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S ) ) ) ).
% img_snd
thf(fact_4915_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_4916_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_4917_range__snd,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_snd
thf(fact_4918_map__snd__mk__fst,axiom,
! [B: $tType,A: $tType,K3: B,L: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K3 ) @ L ) )
= L ) ).
% map_snd_mk_fst
thf(fact_4919_map__snd__mk__snd,axiom,
! [B: $tType,A: $tType,K3: A,L: list @ B] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( map @ B @ ( product_prod @ B @ A )
@ ^ [X4: B] : ( product_Pair @ B @ A @ X4 @ K3 )
@ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K3 ) ) ).
% map_snd_mk_snd
thf(fact_4920_Ex__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ? [A5: A,X9: B] : ( P @ A5 @ X9 ) )
= ( ? [Z6: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z6 ) @ ( product_snd @ A @ B @ Z6 ) ) ) ) ).
% Ex_prod_contract
thf(fact_4921_All__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ! [A5: A,X9: B] : ( P @ A5 @ X9 ) )
= ( ! [Z6: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z6 ) @ ( product_snd @ A @ B @ Z6 ) ) ) ) ).
% All_prod_contract
thf(fact_4922_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_4923_snd__eqD,axiom,
! [B: $tType,A: $tType,X3: B,Y: A,A4: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) )
= A4 )
=> ( Y = A4 ) ) ).
% snd_eqD
thf(fact_4924_sndE,axiom,
! [A: $tType,B: $tType,X3: product_prod @ A @ B,A4: A,B3: B,P: B > $o] :
( ( X3
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( ( P @ ( product_snd @ A @ B @ X3 ) )
=> ( P @ B3 ) ) ) ).
% sndE
thf(fact_4925_fn__snd__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: B > C] :
( ( ^ [X4: product_prod @ A @ B] : ( F3 @ ( product_snd @ A @ B @ X4 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [Uu3: A] : F3 ) ) ).
% fn_snd_conv
thf(fact_4926_snd__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X4: B] : ( product_Pair @ B @ B @ X4 @ X4 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_4927_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X3: A,Y: B,A4: product_prod @ A @ B] :
( ( P @ X3 @ Y )
=> ( ( A4
= ( product_Pair @ A @ B @ X3 @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A4 ) @ ( product_snd @ A @ B @ A4 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_4928_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_4929_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_4930_in__snd__imageE,axiom,
! [A: $tType,B: $tType,Y: A,S: set @ ( product_prod @ B @ A )] :
( ( member @ A @ Y @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S ) )
=> ~ ! [X: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ S ) ) ).
% in_snd_imageE
thf(fact_4931_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_4932_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_4933_snd__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_4934_fst__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X4: B] : ( product_Pair @ B @ B @ X4 @ X4 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_4935_snd__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% snd_diag_id
thf(fact_4936_Id__fstsnd__eq,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [X4: product_prod @ A @ A] :
( ( product_fst @ A @ A @ X4 )
= ( product_snd @ A @ A @ X4 ) ) ) ) ).
% Id_fstsnd_eq
thf(fact_4937_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_4938_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_4939_snd__image__mp,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ B @ A ),B2: set @ A,X3: B,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A3 ) @ B2 )
=> ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y ) @ A3 )
=> ( member @ A @ Y @ B2 ) ) ) ).
% snd_image_mp
thf(fact_4940_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_4941_fs__contract,axiom,
! [B: $tType,C: $tType,A: $tType,F3: 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] :
? [P7: product_prod @ A @ B] :
( ( Uu3 = P7 )
& ( member @ C @ ( F3 @ ( product_fst @ A @ B @ P7 ) @ ( product_snd @ A @ B @ P7 ) ) @ S ) ) ) )
= ( collect @ A
@ ^ [A5: A] :
? [B4: B] : ( member @ C @ ( F3 @ A5 @ B4 ) @ S ) ) ) ).
% fs_contract
thf(fact_4942_lfp__induct2,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,F3: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F3 ) )
=> ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F3 )
=> ( ! [A6: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( F3 @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) ) ) )
=> ( P @ A6 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% lfp_induct2
thf(fact_4943_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_4944_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_4945_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_4946_in__set__zip,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( ? [N2: nat] :
( ( ( nth @ A @ Xs @ N2 )
= ( product_fst @ A @ B @ P4 ) )
& ( ( nth @ B @ Ys @ N2 )
= ( product_snd @ A @ B @ P4 ) )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_4947_finite__range__prod,axiom,
! [A: $tType,C: $tType,B: $tType,F3: B > ( product_prod @ A @ C )] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ ( comp @ ( product_prod @ A @ C ) @ A @ B @ ( product_fst @ A @ C ) @ F3 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ( finite_finite2 @ C @ ( image2 @ B @ C @ ( comp @ ( product_prod @ A @ C ) @ C @ B @ ( product_snd @ A @ C ) @ F3 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ C ) @ ( image2 @ B @ ( product_prod @ A @ C ) @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% finite_range_prod
thf(fact_4948_bezw__non__0,axiom,
! [Y: nat,X3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X3 @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X3 @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_4949_fst__snd__flip,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X4: A,Y5: B] : ( product_Pair @ B @ A @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_4950_snd__fst__flip,axiom,
! [A: $tType,B: $tType] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X4: B,Y5: A] : ( product_Pair @ A @ B @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_4951_bezw_Opelims,axiom,
! [X3: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_4952_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_4953_mod__pure,axiom,
! [B3: $o,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( pure_assn @ B3 ) @ H2 )
= ( ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 )
= ( bot_bot @ ( set @ nat ) ) )
& B3 ) ) ).
% mod_pure
thf(fact_4954_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 )
@ ^ [P7: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P7 ) ) )
@ ( zip @ A @ nat @ Xs @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P7: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P7 ) )
@ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_4955_mod__emp,axiom,
! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( one_one @ assn ) @ H2 )
= ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 )
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_emp
thf(fact_4956_fi__match__entails,axiom,
! [M2: list @ ( product_prod @ assn @ assn )] :
( ! [X: product_prod @ assn @ assn] :
( ( member @ ( product_prod @ assn @ assn ) @ X @ ( set2 @ ( product_prod @ assn @ assn ) @ M2 ) )
=> ( product_case_prod @ assn @ assn @ $o @ entails @ X ) )
=> ( 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_4957_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_4958_nths__shift__lemma,axiom,
! [A: $tType,A3: set @ nat,Xs: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P7: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P7 ) @ A3 )
@ ( 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 )
@ ^ [P7: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P7 ) @ I ) @ A3 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_4959_mod__star__trueE_H,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H2 )
=> ~ ! [H5: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( ( product_fst @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 )
= ( product_fst @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 ) )
=> ( ( ord_less_eq @ ( set @ nat ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 ) )
=> ~ ( rep_assn @ P @ H5 ) ) ) ) ).
% mod_star_trueE'
thf(fact_4960_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 )
@ ^ [P7: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P7 ) @ A7 )
@ ( zip @ A @ nat @ Xs3 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ) ).
% nths_def
thf(fact_4961_FI__RESULT__def,axiom,
( fI_RESULT
= ( ^ [M10: list @ ( product_prod @ assn @ assn ),UP: assn,UQ: assn,F4: assn] :
( ! [X4: product_prod @ assn @ assn] :
( ( member @ ( product_prod @ assn @ assn ) @ X4 @ ( set2 @ ( product_prod @ assn @ assn ) @ M10 ) )
=> ( product_case_prod @ assn @ assn @ $o @ entails @ X4 ) )
=> ( entails @ ( times_times @ assn @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_fst @ assn @ assn ) @ M10 ) @ ( 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 ) @ M10 ) @ ( one_one @ assn ) ) @ UQ ) @ F4 ) ) ) ) ) ).
% FI_RESULT_def
thf(fact_4962_fstI,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,Y: A,Z2: B] :
( ( X3
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_fst @ A @ B @ X3 )
= Y ) ) ).
% fstI
thf(fact_4963_sndI,axiom,
! [A: $tType,B: $tType,X3: product_prod @ A @ B,Y: A,Z2: B] :
( ( X3
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_snd @ A @ B @ X3 )
= Z2 ) ) ).
% sndI
thf(fact_4964_in__set__enumerate__eq,axiom,
! [A: $tType,P4: product_prod @ nat @ A,N: nat,Xs: list @ A] :
( ( member @ ( 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_4965_prod__decode__aux_Opelims,axiom,
! [X3: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa @ X3 )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X3 @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X3 )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus @ nat @ Xa @ ( suc @ X3 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_4966_mergesort__by__rel__split__length,axiom,
! [A: $tType,Xs12: 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 ) @ Xs12 @ Xs23 ) @ Xs ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs12 ) @ ( 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 ) @ Xs12 @ 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_4967_mset__mergesort__by__rel__split,axiom,
! [A: $tType,Xs12: 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 ) @ Xs12 @ Xs23 ) @ Xs ) ) ) @ ( mset @ A @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ Xs ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Xs12 ) ) @ ( mset @ A @ Xs23 ) ) ) ).
% mset_mergesort_by_rel_split
thf(fact_4968_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs12: list @ A,Xs23: list @ A,X1: A,X22: A,Xs: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( cons @ A @ X1 @ ( cons @ A @ X22 @ Xs ) ) )
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs12 ) @ ( cons @ A @ X22 @ Xs23 ) ) @ Xs ) ) ).
% mergesort_by_rel_split.simps(3)
thf(fact_4969_mergesort__by__rel__split_Osimps_I1_J,axiom,
! [A: $tType,Xs12: list @ A,Xs23: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) ) ).
% mergesort_by_rel_split.simps(1)
thf(fact_4970_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs12: list @ A,Xs23: list @ A,X3: A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs12 ) @ Xs23 ) ) ).
% mergesort_by_rel_split.simps(2)
thf(fact_4971_mergesort__by__rel__split_Oelims,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X3 @ Xa )
= Y )
=> ( ! [Xs1: list @ A,Xs22: list @ A] :
( ( X3
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) ) ) )
=> ( ! [Xs1: list @ A,Xs22: list @ A] :
( ( X3
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X: A] :
( ( Xa
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs1 ) @ Xs22 ) ) ) )
=> ~ ! [Xs1: list @ A,Xs22: list @ A] :
( ( X3
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X12: A,X23: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs2 ) ) )
=> ( Y
!= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ Xs1 ) @ ( cons @ A @ X23 @ Xs22 ) ) @ Xs2 ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
thf(fact_4972_FI__def,axiom,
( fi
= ( ^ [M: list @ ( product_prod @ assn @ assn ),P7: assn,Q8: assn,Up: assn,Uq: assn,F7: assn] :
( ! [X4: product_prod @ assn @ assn] :
( ( member @ ( product_prod @ assn @ assn ) @ X4 @ ( set2 @ ( product_prod @ assn @ assn ) @ M ) )
=> ( product_case_prod @ assn @ assn @ $o @ entails @ X4 ) )
=> ( entails @ ( times_times @ assn @ ( times_times @ assn @ ( foldr @ assn @ assn @ ( times_times @ assn ) @ ( map @ ( product_prod @ assn @ assn ) @ assn @ ( product_fst @ assn @ assn ) @ M ) @ ( one_one @ assn ) ) @ P7 ) @ Up ) @ ( 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 ) @ M ) @ ( one_one @ assn ) ) @ Q8 ) @ Uq ) @ F7 ) ) ) ) ) ).
% FI_def
thf(fact_4973_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P4: A,Q: B > $o,Q4: B] :
( ( P @ P4 )
=> ( ( Q @ Q4 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q4 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_4974_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X3: B] :
( ( P @ Y @ X3 )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_4975_FI__p__nomatch,axiom,
! [M2: list @ ( product_prod @ assn @ assn ),Ps3: assn,Qs: assn,Q4: assn,P4: assn,Up2: assn,Uq2: assn,F3: assn] :
( ( fi @ M2 @ Ps3 @ ( times_times @ assn @ Qs @ Q4 ) @ ( times_times @ assn @ P4 @ Up2 ) @ Uq2 @ F3 )
=> ( fi @ M2 @ ( times_times @ assn @ Ps3 @ P4 ) @ ( times_times @ assn @ Qs @ Q4 ) @ Up2 @ Uq2 @ F3 ) ) ).
% FI_p_nomatch
thf(fact_4976_FI__finalize,axiom,
! [M2: list @ ( product_prod @ assn @ assn ),P4: assn,Up2: assn,Q4: assn,Uq2: assn,F3: assn] :
( ( fI_RESULT @ M2 @ ( times_times @ assn @ P4 @ Up2 ) @ ( times_times @ assn @ Q4 @ Uq2 ) @ F3 )
=> ( fi @ M2 @ P4 @ Q4 @ Up2 @ Uq2 @ F3 ) ) ).
% FI_finalize
thf(fact_4977_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_4978_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 ) )
=> ( ! [R6: 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 ) @ R6 @ Xs2 ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R6 @ ( 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 @ R6 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
=> ( P @ R6 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% mergesort_by_rel.pinduct
thf(fact_4979_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),X3: product_prod @ C @ A,X5: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert3 @ ( product_prod @ C @ A ) @ X3 @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S ) @ X5 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A16: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X3 )
= W3 )
@ ( insert3 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X3 ) @ Z6 ) @ A16 )
@ A16 ) )
@ X5
@ S ) ) ) ).
% insert_relcomp_union_fold
thf(fact_4980_relcomp__empty1,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) @ R )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty1
thf(fact_4981_relcomp__empty2,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ C )] :
( ( relcomp @ A @ C @ B @ R @ ( bot_bot @ ( set @ ( product_prod @ C @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty2
thf(fact_4982_relcomp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S: set @ ( product_prod @ A @ C ),T4: set @ ( product_prod @ A @ C ),R: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ C ) ) @ S @ T4 ) @ R )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ S @ R ) @ ( relcomp @ A @ C @ B @ T4 @ R ) ) ) ).
% relcomp_distrib2
thf(fact_4983_relcomp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B ),T4: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ R @ ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ S @ T4 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ R @ S ) @ ( relcomp @ A @ C @ B @ R @ T4 ) ) ) ).
% relcomp_distrib
thf(fact_4984_Id__O__R,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( relcomp @ A @ A @ B @ ( id2 @ A ) @ R )
= R ) ).
% Id_O_R
thf(fact_4985_R__O__Id,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( relcomp @ A @ B @ B @ R @ ( id2 @ B ) )
= R ) ).
% R_O_Id
thf(fact_4986_relcomp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R7: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ B @ C ),S3: set @ ( product_prod @ B @ C )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R7 @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S5 @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R7 @ S5 ) @ ( relcomp @ A @ B @ C @ R2 @ S3 ) ) ) ) ).
% relcomp_mono
thf(fact_4987_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A4: A,C2: B,R2: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ C2 ) @ ( relcomp @ A @ C @ B @ R2 @ S3 ) )
=> ~ ! [B5: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A4 @ B5 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B5 @ C2 ) @ S3 ) ) ) ).
% relcompEpair
thf(fact_4988_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R2: set @ ( product_prod @ A @ C ),S3: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R2 @ S3 ) )
=> ~ ! [X: A,Y2: C,Z3: B] :
( ( Xz
= ( product_Pair @ A @ B @ X @ Z3 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ Y2 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y2 @ Z3 ) @ S3 ) ) ) ) ).
% relcompE
thf(fact_4989_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B ),C2: C,S3: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ C2 ) @ S3 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A4 @ C2 ) @ ( relcomp @ A @ B @ C @ R2 @ S3 ) ) ) ) ).
% relcomp.relcompI
thf(fact_4990_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R2 @ S3 ) )
= ( ? [A5: A,B4: B,C6: C] :
( ( A1 = A5 )
& ( A22 = C6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R2 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ C6 ) @ S3 ) ) ) ) ).
% relcomp.simps
thf(fact_4991_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R2 @ S3 ) )
=> ~ ! [B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B5 ) @ R2 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ A22 ) @ S3 ) ) ) ).
% relcomp.cases
thf(fact_4992_O__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ A @ D ),S: set @ ( product_prod @ D @ C ),T4: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( relcomp @ A @ D @ C @ R @ S ) @ T4 )
= ( relcomp @ A @ D @ B @ R @ ( relcomp @ D @ C @ B @ S @ T4 ) ) ) ).
% O_assoc
thf(fact_4993_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 ) @ ( insert3 @ ( 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_4994_trans__O__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ R2 ) @ R2 ) ) ).
% trans_O_subset
thf(fact_4995_relcomp__UNION__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R2: D > ( set @ ( product_prod @ A @ C ) ),I3: set @ D,S3: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ C ) ) @ ( image2 @ D @ ( set @ ( product_prod @ A @ C ) ) @ R2 @ I3 ) ) @ S3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I4: D] : ( relcomp @ A @ C @ B @ ( R2 @ I4 ) @ S3 )
@ I3 ) ) ) ).
% relcomp_UNION_distrib2
thf(fact_4996_relcomp__UNION__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S3: set @ ( product_prod @ A @ C ),R2: D > ( set @ ( product_prod @ C @ B ) ),I3: set @ D] :
( ( relcomp @ A @ C @ B @ S3 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ C @ B ) ) @ ( image2 @ D @ ( set @ ( product_prod @ C @ B ) ) @ R2 @ I3 ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I4: D] : ( relcomp @ A @ C @ B @ S3 @ ( R2 @ I4 ) )
@ I3 ) ) ) ).
% relcomp_UNION_distrib
thf(fact_4997_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R5: set @ ( product_prod @ A @ C ),S7: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Z6: B] :
? [Y5: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y5 ) @ R5 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y5 @ Z6 ) @ S7 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_4998_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S )
=> ( ( relcomp @ A @ B @ C @ R @ S )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [X4: A,Y5: B,A7: set @ ( product_prod @ A @ C )] :
( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [W3: B,Z6: C,A16: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y5 = W3 ) @ ( insert3 @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Z6 ) @ A16 ) @ A16 ) )
@ A7
@ S ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R ) ) ) ) ).
% relcomp_fold
thf(fact_4999_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),X3: product_prod @ C @ A,R: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( ( relcomp @ C @ A @ B @ ( insert3 @ ( product_prod @ C @ A ) @ X3 @ R ) @ S )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A16: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X3 )
= W3 )
@ ( insert3 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X3 ) @ Z6 ) @ A16 )
@ A16 ) )
@ ( relcomp @ C @ A @ B @ R @ S )
@ S ) ) ) ).
% insert_relcomp_fold
thf(fact_5000_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: A > B,Xa: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ Xa ) )
=> ( ! [X: A] :
( ( Xa
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ( Y = X )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) )
=> ( ! [X: A,Y2: A,Zs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ ( cons @ A @ Y2 @ Zs2 ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X3 @ X ) @ ( X3 @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) @ X @ ( arg_min_list @ A @ B @ X3 @ ( cons @ A @ Y2 @ Zs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_5001_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) ) ) )
=> ~ ! [X: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ X @ Ys5 ) )
=> ( ( Y
= ( ! [Y5: A] :
( ( member @ A @ Y5 @ ( set2 @ A @ Ys5 ) )
=> ( X3 @ X @ Y5 ) )
& ( sorted_wrt @ A @ X3 @ Ys5 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( cons @ A @ X @ Ys5 ) ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
thf(fact_5002_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A] :
( ( sorted_wrt @ A @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [X: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ X @ Ys5 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( cons @ A @ X @ Ys5 ) ) )
=> ~ ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys5 ) )
=> ( X3 @ X @ Xa3 ) )
& ( sorted_wrt @ A @ X3 @ Ys5 ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
thf(fact_5003_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A] :
( ~ ( sorted_wrt @ A @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ Xa ) )
=> ~ ! [X: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ X @ Ys5 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( cons @ A @ X @ Ys5 ) ) )
=> ( ! [Xa4: A] :
( ( member @ A @ Xa4 @ ( set2 @ A @ Ys5 ) )
=> ( X3 @ X @ Xa4 ) )
& ( sorted_wrt @ A @ X3 @ Ys5 ) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
thf(fact_5004_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 ) @ ( insert3 @ ( 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_5005_reduction__pairI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S ) @ R )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) ) ) ) ).
% reduction_pairI
thf(fact_5006_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_5007_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_5008_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_5009_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_5010_sorted__wrt__mergesort__by__rel__merge,axiom,
! [A: $tType,R: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ! [X: A,Y2: A] :
( ( R @ X @ Y2 )
| ( R @ Y2 @ X ) )
=> ( ! [X: A,Y2: A,Z3: A] :
( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( ( 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_5011_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_5012_mergesort__by__rel__simps_I2_J,axiom,
! [A: $tType,R: A > A > $o,X3: A] :
( ( mergesort_by_rel @ A @ R @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% mergesort_by_rel_simps(2)
thf(fact_5013_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_5014_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_5015_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R: A > A > $o,X1: A,X22: A,Xs: list @ A] :
( ( mergesort_by_rel @ A @ R @ ( cons @ A @ X1 @ ( cons @ A @ X22 @ Xs ) ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [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 @ X1 @ ( nil @ A ) ) @ ( cons @ A @ X22 @ ( nil @ A ) ) ) @ Xs ) ) ) ).
% mergesort_by_rel_simps(3)
thf(fact_5016_mergesort__by__rel__merge__simps_I1_J,axiom,
! [A: $tType,R: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ( R @ X3 @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ R @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ) )
& ( ~ ( R @ X3 @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ Y @ ( merges9089515139780605204_merge @ A @ R @ ( cons @ A @ X3 @ Xs ) @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_simps(1)
thf(fact_5017_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_5018_sorted__wrt__mergesort__by__rel,axiom,
! [X14: $tType,R: X14 > X14 > $o,Xs: list @ X14] :
( ! [X: X14,Y2: X14] :
( ( R @ X @ Y2 )
| ( R @ Y2 @ X ) )
=> ( ! [X: X14,Y2: X14,Z3: X14] :
( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( sorted_wrt @ X14 @ R @ ( mergesort_by_rel @ X14 @ R @ Xs ) ) ) ) ).
% sorted_wrt_mergesort_by_rel
thf(fact_5019_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_5020_mergesort__by__rel__merge_Osimps_I3_J,axiom,
! [A: $tType,R: A > A > $o,V2: A,Va2: list @ A] :
( ( merges9089515139780605204_merge @ A @ R @ ( nil @ A ) @ ( cons @ A @ V2 @ Va2 ) )
= ( cons @ A @ V2 @ Va2 ) ) ).
% mergesort_by_rel_merge.simps(3)
thf(fact_5021_mergesort__by__rel__merge_Oelims,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X3 @ Xa @ Xb )
= Y )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y2: A,Ys5: list @ A] :
( ( Xb
= ( cons @ A @ Y2 @ Ys5 ) )
=> ~ ( ( ( X3 @ X @ Y2 )
=> ( Y
= ( cons @ A @ X @ ( merges9089515139780605204_merge @ A @ X3 @ Xs2 @ ( cons @ A @ Y2 @ Ys5 ) ) ) ) )
& ( ~ ( X3 @ X @ Y2 )
=> ( Y
= ( cons @ A @ Y2 @ ( merges9089515139780605204_merge @ A @ X3 @ ( cons @ A @ X @ Xs2 ) @ Ys5 ) ) ) ) ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V3 @ Va ) )
=> ( Y
!= ( cons @ A @ V3 @ Va ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.elims
thf(fact_5022_sort__mergesort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4 )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% sort_mergesort_by_rel
thf(fact_5023_mergesort__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( mergesort @ A )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% mergesort_def
thf(fact_5024_mergesort__by__rel_Oelims,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X3 @ 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 @ X3 @ ( mergesort_by_rel @ A @ X3 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X3 @ ( 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_5025_mergesort__by__rel_Osimps,axiom,
! [A: $tType] :
( ( mergesort_by_rel @ A )
= ( ^ [R4: 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 @ R4 @ ( mergesort_by_rel @ A @ R4 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) @ ( mergesort_by_rel @ A @ R4 @ ( 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_5026_mergesort__by__rel_Opelims,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ 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 @ X3 @ ( mergesort_by_rel @ A @ X3 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X3 @ ( 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 ) @ X3 @ Xa ) ) ) ) ) ).
% mergesort_by_rel.pelims
thf(fact_5027_mergesort__by__rel__merge_Opelims,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X3 @ 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y2: A,Ys5: list @ A] :
( ( Xb
= ( cons @ A @ Y2 @ Ys5 ) )
=> ( ( ( ( X3 @ X @ Y2 )
=> ( Y
= ( cons @ A @ X @ ( merges9089515139780605204_merge @ A @ X3 @ Xs2 @ ( cons @ A @ Y2 @ Ys5 ) ) ) ) )
& ( ~ ( X3 @ X @ Y2 )
=> ( Y
= ( cons @ A @ Y2 @ ( merges9089515139780605204_merge @ A @ X3 @ ( cons @ A @ X @ Xs2 ) @ Ys5 ) ) ) ) )
=> ~ ( 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys5 ) ) ) ) ) ) )
=> ( ( ( 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V3 @ Va ) )
=> ( ( Y
= ( cons @ A @ V3 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
thf(fact_5028_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [X4: C,Y5: A,A7: set @ ( product_prod @ C @ B )] :
( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A16: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y5 = W3 ) @ ( insert3 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X4 @ Z6 ) @ A16 ) @ A16 ) )
@ A7
@ S ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_5029_FI__init,axiom,
! [P: assn,Q: assn,F: assn] :
( ( fi @ ( nil @ ( product_prod @ assn @ assn ) ) @ ( times_times @ assn @ sln @ P ) @ ( times_times @ assn @ sln @ Q ) @ sln @ sln @ F )
=> ( fI_QUERY @ P @ Q @ F ) ) ).
% FI_init
thf(fact_5030_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P: A > $o] :
( finite6289374366891150609ommute @ A @ ( set @ A )
@ ^ [X4: A,A16: set @ A] : ( if @ ( set @ A ) @ ( P @ X4 ) @ ( insert3 @ A @ X4 @ A16 ) @ A16 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_5031_SLN__right,axiom,
! [P: assn] :
( ( times_times @ assn @ P @ sln )
= P ) ).
% SLN_right
thf(fact_5032_SLN__left,axiom,
! [P: assn] :
( ( times_times @ assn @ sln @ P )
= P ) ).
% SLN_left
thf(fact_5033_comp__fun__commute_Ofoldl__f__commute,axiom,
! [B: $tType,A: $tType,F3: A > B > B,A4: A,B3: B,Xs: list @ A] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( ( F3 @ A4
@ ( foldl @ B @ A
@ ^ [A5: B,B4: A] : ( F3 @ B4 @ A5 )
@ B3
@ Xs ) )
= ( foldl @ B @ A
@ ^ [A5: B,B4: A] : ( F3 @ B4 @ A5 )
@ ( F3 @ A4 @ B3 )
@ Xs ) ) ) ).
% comp_fun_commute.foldl_f_commute
thf(fact_5034_SLN__def,axiom,
( sln
= ( one_one @ assn ) ) ).
% SLN_def
thf(fact_5035_comp__fun__commute_Ofoldr__conv__foldl,axiom,
! [B: $tType,A: $tType,F3: A > B > B,Xs: list @ A,A4: B] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( ( foldr @ A @ B @ F3 @ Xs @ A4 )
= ( foldl @ B @ A
@ ^ [A5: B,B4: A] : ( F3 @ B4 @ A5 )
@ A4
@ Xs ) ) ) ).
% comp_fun_commute.foldr_conv_foldl
thf(fact_5036_comp__fun__commute__Image__fold,axiom,
! [B: $tType,A: $tType,S: set @ A] :
( finite6289374366891150609ommute @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X4: A,Y5: B,A7: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X4 @ S ) @ ( insert3 @ B @ Y5 @ A7 ) @ A7 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_5037_comp__fun__commute__fold__insert,axiom,
! [B: $tType,A: $tType,F3: A > B > B,A3: set @ A,X3: A,Z2: B] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert3 @ A @ X3 @ A3 ) )
= ( F3 @ X3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A3 ) ) ) ) ) ) ).
% comp_fun_commute_fold_insert
thf(fact_5038_FI__q__nomatch,axiom,
! [M2: list @ ( product_prod @ assn @ assn ),Up2: assn,Qs: assn,Q4: assn,Uq2: assn,F3: assn] :
( ( fi @ M2 @ ( times_times @ assn @ sln @ Up2 ) @ Qs @ sln @ ( times_times @ assn @ Q4 @ Uq2 ) @ F3 )
=> ( fi @ M2 @ sln @ ( times_times @ assn @ Qs @ Q4 ) @ Up2 @ Uq2 @ F3 ) ) ).
% FI_q_nomatch
thf(fact_5039_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B2: set @ A] :
( ( finite_finite2 @ A @ B2 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X4: B,Z6: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y5: A] : ( insert3 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y5 ) )
@ Z6
@ B2 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_5040_FI__match,axiom,
! [P4: assn,Q4: assn,M2: list @ ( product_prod @ assn @ assn ),Ps3: assn,Up2: assn,Qs: assn,Uq2: assn,F3: assn] :
( ( entails @ P4 @ Q4 )
=> ( ( fi @ ( cons @ ( product_prod @ assn @ assn ) @ ( product_Pair @ assn @ assn @ P4 @ Q4 ) @ M2 ) @ ( times_times @ assn @ Ps3 @ Up2 ) @ ( times_times @ assn @ Qs @ Uq2 ) @ sln @ sln @ F3 )
=> ( fi @ M2 @ ( times_times @ assn @ Ps3 @ P4 ) @ ( times_times @ assn @ Qs @ Q4 ) @ Up2 @ Uq2 @ F3 ) ) ) ).
% FI_match
thf(fact_5041_range__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F3: C > ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( product_prod @ A @ B ) @ F3 @ ( top_top @ ( set @ C ) ) )
@ ( product_Sigma @ A @ B @ ( image2 @ C @ A @ ( comp @ ( product_prod @ A @ B ) @ A @ C @ ( product_fst @ A @ B ) @ F3 ) @ ( top_top @ ( set @ C ) ) )
@ ^ [Uu3: A] : ( image2 @ C @ B @ ( comp @ ( product_prod @ A @ B ) @ B @ C @ ( product_snd @ A @ B ) @ F3 ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% range_prod
thf(fact_5042_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G: D > A,X3: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G @ X3 ) )
= ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X3 ) ) @ ( F3 @ ( product_snd @ D @ C @ X3 ) ) ) ) ).
% apsnd_apfst
thf(fact_5043_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F3: C > A,G: D > B,X3: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_apsnd @ D @ B @ C @ G @ X3 ) )
= ( product_Pair @ A @ B @ ( F3 @ ( product_fst @ C @ D @ X3 ) ) @ ( G @ ( product_snd @ C @ D @ X3 ) ) ) ) ).
% apfst_apsnd
thf(fact_5044_SigmaI,axiom,
! [B: $tType,A: $tType,A4: A,A3: set @ A,B3: B,B2: A > ( set @ B )] :
( ( member @ A @ A4 @ A3 )
=> ( ( member @ B @ B3 @ ( B2 @ A4 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% SigmaI
thf(fact_5045_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
= ( ( member @ A @ A4 @ A3 )
& ( member @ B @ B3 @ ( B2 @ A4 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_5046_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F3: C > A,X3: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_Pair @ C @ B @ X3 @ Y ) )
= ( product_Pair @ A @ B @ ( F3 @ X3 ) @ Y ) ) ).
% apfst_conv
thf(fact_5047_Field__square,axiom,
! [A: $tType,X3: set @ A] :
( ( field2 @ A
@ ( product_Sigma @ A @ A @ X3
@ ^ [Uu3: A] : X3 ) )
= X3 ) ).
% Field_square
thf(fact_5048_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_5049_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_5050_Times__empty,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_5051_fst__image__times,axiom,
! [B: $tType,A: $tType,B2: set @ B,A3: set @ A] :
( ( ( B2
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) )
= A3 ) ) ) ).
% fst_image_times
thf(fact_5052_snd__image__times,axiom,
! [B: $tType,A: $tType,A3: set @ B,B2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu3: B] : B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu3: B] : B2 ) )
= B2 ) ) ) ).
% snd_image_times
thf(fact_5053_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,A3: set @ A,X5: set @ B] :
( ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : X5 )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_UNIV_cancel
thf(fact_5054_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_5055_pairself__image__cart,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: set @ B,B2: set @ B] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F3 )
@ ( product_Sigma @ B @ B @ A3
@ ^ [Uu3: B] : B2 ) )
= ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F3 @ A3 )
@ ^ [Uu3: A] : ( image2 @ B @ A @ F3 @ B2 ) ) ) ).
% pairself_image_cart
thf(fact_5056_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A4: A,A3: set @ A,B3: B,B2: set @ B] :
( ( product_Sigma @ A @ B @ ( insert3 @ A @ A4 @ A3 )
@ ^ [Uu3: A] : ( insert3 @ B @ B3 @ B2 ) )
= ( insert3 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : ( insert3 @ B @ B3 @ B2 ) )
@ ( product_Sigma @ A @ B @ ( insert3 @ A @ A4 @ A3 )
@ ^ [Uu3: A] : B2 ) ) ) ) ).
% insert_Times_insert
thf(fact_5057_times__eq__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B,C3: set @ A,D2: set @ B] :
( ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 )
= ( product_Sigma @ A @ B @ C3
@ ^ [Uu3: A] : D2 ) )
= ( ( ( A3 = C3 )
& ( B2 = D2 ) )
| ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C3
= ( bot_bot @ ( set @ A ) ) )
| ( D2
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_5058_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod @ A @ B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ~ ! [X: A] :
( ( member @ A @ X @ A3 )
=> ! [Y2: B] :
( ( member @ B @ Y2 @ ( B2 @ X ) )
=> ( C2
!= ( product_Pair @ A @ B @ X @ Y2 ) ) ) ) ) ).
% SigmaE
thf(fact_5059_SigmaD1,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ( member @ A @ A4 @ A3 ) ) ).
% SigmaD1
thf(fact_5060_SigmaD2,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ( member @ B @ B3 @ ( B2 @ A4 ) ) ) ).
% SigmaD2
thf(fact_5061_SigmaE2,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,A3: set @ A,B2: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
=> ~ ( ( member @ A @ A4 @ A3 )
=> ~ ( member @ B @ B3 @ ( B2 @ A4 ) ) ) ) ).
% SigmaE2
thf(fact_5062_mult__fun__commute,axiom,
! [A: $tType,F3: A > assn] :
( finite6289374366891150609ommute @ A @ assn
@ ^ [I4: A,A5: assn] : ( times_times @ assn @ A5 @ ( F3 @ I4 ) ) ) ).
% mult_fun_commute
thf(fact_5063_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I3: set @ A,J4: set @ A,C3: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I3 @ J4 ) @ C3 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I3 @ C3 ) @ ( product_Sigma @ A @ B @ J4 @ C3 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_5064_comp__fun__commute__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite6289374366891150609ommute @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4 ) ) ) ).
% comp_fun_commute_insort
thf(fact_5065_in__prod__fst__sndI,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,A3: set @ A,B2: set @ B] :
( ( member @ A @ ( product_fst @ A @ B @ X3 ) @ A3 )
=> ( ( member @ B @ ( product_snd @ A @ B @ X3 ) @ B2 )
=> ( member @ ( product_prod @ A @ B ) @ X3
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) ) ) ) ).
% in_prod_fst_sndI
thf(fact_5066_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_5067_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I3: set @ A,X5: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I3 @ X5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ I3 )
=> ( ( X5 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_5068_listrel__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R2 )
@ ( product_Sigma @ ( list @ A ) @ ( list @ A ) @ ( lists @ A @ A3 )
@ ^ [Uu3: list @ A] : ( lists @ A @ A3 ) ) ) ) ).
% listrel_subset
thf(fact_5069_relcomp__subset__Sigma,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),A3: set @ A,B2: set @ B,S3: set @ ( product_prod @ B @ C ),C3: set @ C] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S3
@ ( product_Sigma @ B @ C @ B2
@ ^ [Uu3: B] : C3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R2 @ S3 )
@ ( product_Sigma @ A @ C @ A3
@ ^ [Uu3: A] : C3 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_5070_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ A,C3: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A3 @ B2 )
@ ^ [Uu3: A] : C3 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : C3 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu3: A] : C3 ) ) ) ).
% Times_Diff_distrib1
thf(fact_5071_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I3: set @ A,A3: A > ( set @ B ),B2: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I3
@ ^ [I4: A] : ( minus_minus @ ( set @ B ) @ ( A3 @ I4 ) @ ( B2 @ I4 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I3 @ A3 ) @ ( product_Sigma @ A @ B @ I3 @ B2 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_5072_Id__on__subset__Times,axiom,
! [A: $tType,A3: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( id_on @ A @ A3 )
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) ) ).
% Id_on_subset_Times
thf(fact_5073_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_5074_times__subset__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,C3: set @ B,B2: set @ A,D2: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : C3 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu3: A] : D2 ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( C3
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
& ( ord_less_eq @ ( set @ B ) @ C3 @ D2 ) ) ) ) ).
% times_subset_iff
thf(fact_5075_wfI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : B2 ) )
=> ( ! [X: A,P6: A > $o] :
( ! [Xa3: A] :
( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa3 ) @ R2 )
=> ( P6 @ Y2 ) )
=> ( P6 @ Xa3 ) )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ X @ B2 )
=> ( P6 @ X ) ) ) )
=> ( wf @ A @ R2 ) ) ) ).
% wfI
thf(fact_5076_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ( B2 @ X4 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( finite_finite2 @ B @ ( B2 @ A6 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) ) ) ) ).
% finite_SigmaI2
thf(fact_5077_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_cartesian_productD1
thf(fact_5078_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B2 ) ) ) ).
% finite_cartesian_productD2
thf(fact_5079_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( B2
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A3 )
& ( finite_finite2 @ B @ B2 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_5080_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B2 ) )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ( B2 @ X4 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_5081_Restr__trancl__mono,axiom,
! [A: $tType,V2: A,W2: A,E5: set @ ( product_prod @ A @ A ),U2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E5
@ ( product_Sigma @ A @ A @ U2
@ ^ [Uu3: A] : U2 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ E5 ) ) ) ).
% Restr_trancl_mono
thf(fact_5082_refl__onI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
=> ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R2 ) )
=> ( refl_on @ A @ A3 @ R2 ) ) ) ).
% refl_onI
thf(fact_5083_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ A7
@ ^ [Uu3: A] : A7 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R5 ) ) ) ) ) ).
% refl_on_def
thf(fact_5084_set__zip__cart,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ B,L: list @ A,L2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ L @ L2 ) ) )
=> ( member @ ( product_prod @ A @ B ) @ X3
@ ( product_Sigma @ A @ B @ ( set2 @ A @ L )
@ ^ [Uu3: A] : ( set2 @ B @ L2 ) ) ) ) ).
% set_zip_cart
thf(fact_5085_swap__product,axiom,
! [B: $tType,A: $tType,A3: set @ B,B2: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I4: B,J2: A] : ( product_Pair @ A @ B @ J2 @ I4 ) )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu3: B] : B2 ) )
= ( product_Sigma @ A @ B @ B2
@ ^ [Uu3: A] : A3 ) ) ).
% swap_product
thf(fact_5086_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F3: C > A,G: D > B,A3: set @ C,B2: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y5: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G @ Y5 ) ) )
@ ( product_Sigma @ C @ D @ A3
@ ^ [Uu3: C] : B2 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F3 @ A3 )
@ ^ [Uu3: A] : ( image2 @ D @ B @ G @ B2 ) ) ) ).
% image_paired_Times
thf(fact_5087_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X3: A,A3: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu3: A] : A3 ) )
= ( finite_card @ B @ A3 ) ) ).
% card_cartesian_product_singleton
thf(fact_5088_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A7: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A7 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert3 @ A @ X4 ) @ A7 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_5089_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A7: set @ A,B7: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y5: B] : ( insert3 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B7 @ X4 ) ) )
@ A7 ) ) ) ) ).
% Sigma_def
thf(fact_5090_product__fold,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: A,Z6: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y5: B] : ( insert3 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) )
@ Z6
@ B2 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A3 ) ) ) ) ).
% product_fold
thf(fact_5091_lists__length__Suc__eq,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A3 )
& ( ( 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,N2: A] : ( cons @ A @ N2 @ Xs3 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) )
@ ^ [Uu3: list @ A] : A3 ) ) ) ).
% lists_length_Suc_eq
thf(fact_5092_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q4: product_prod @ A @ B,F3: C > A,P4: product_prod @ C @ B] :
( ( Q4
= ( product_apfst @ C @ A @ B @ F3 @ P4 ) )
=> ~ ! [X: C,Y2: B] :
( ( P4
= ( product_Pair @ C @ B @ X @ Y2 ) )
=> ( Q4
!= ( product_Pair @ A @ B @ ( F3 @ X ) @ Y2 ) ) ) ) ).
% apfst_convE
thf(fact_5093_rtrancl__last__visit__node,axiom,
! [A: $tType,S3: A,S5: A,R: set @ ( product_prod @ A @ A ),Sh: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S3 @ S5 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( ( S3 != Sh )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S3 @ S5 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : ( uminus_uminus @ ( set @ A ) @ ( insert3 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) )
| ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S3 @ Sh ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Sh @ S5 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : ( uminus_uminus @ ( set @ A ) @ ( insert3 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit_node
thf(fact_5094_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B3: A,P4: product_prod @ B @ A] :
( ( B3
= ( product_snd @ B @ A @ P4 ) )
= ( ? [A5: B] :
( P4
= ( product_Pair @ B @ A @ A5 @ B3 ) ) ) ) ).
% eq_snd_iff
thf(fact_5095_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ).
% listrel_rtrancl_refl
thf(fact_5096_wf__insert,axiom,
! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ R2 ) )
= ( ( wf @ A @ R2 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% wf_insert
thf(fact_5097_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( U != V2 )
=> ~ ! [Vh: A] :
( ( U != Vh )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% converse_rtranclE'
thf(fact_5098_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_5099_converse__rtrancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ B3 )
=> ( ! [Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ Z3 )
=> ( P @ Y2 ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_5100_converse__rtranclE,axiom,
! [A: $tType,X3: A,Z2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( X3 != Z2 )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_5101_rtrancl__induct,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P @ A4 )
=> ( ! [Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R2 )
=> ( ( P @ Y2 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% rtrancl_induct
thf(fact_5102_rtrancl__trans,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_5103_rtranclE,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( A4 != B3 )
=> ~ ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ Y2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ B3 ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_5104_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_5105_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A4: A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( transitive_rtrancl @ A @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_5106_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( ? [A5: A] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: A,B4: A,C6: A] :
( ( A1 = A5 )
& ( A22 = C6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C6 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_5107_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B5 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_5108_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) )
= ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel_rtrancl_eq_rtrancl_listrel1
thf(fact_5109_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ ( transitive_rtrancl @ A @ R2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1_rtrancl_subset_rtrancl_listrel1
thf(fact_5110_rtrancl__mono__rightI,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ S4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ S4 ) ) ) ).
% rtrancl_mono_rightI
thf(fact_5111_rtrancl__mono__mp,axiom,
! [A: $tType,U2: set @ ( product_prod @ A @ A ),V: set @ ( product_prod @ A @ A ),X3: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U2 @ V )
=> ( ( member @ ( product_prod @ A @ A ) @ X3 @ ( transitive_rtrancl @ A @ U2 ) )
=> ( member @ ( product_prod @ A @ A ) @ X3 @ ( transitive_rtrancl @ A @ V ) ) ) ) ).
% rtrancl_mono_mp
thf(fact_5112_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_5113_in__rtrancl__insert,axiom,
! [A: $tType,X3: product_prod @ A @ A,R: set @ ( product_prod @ A @ A ),R2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X3 @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ X3 @ ( transitive_rtrancl @ A @ ( insert3 @ ( product_prod @ A @ A ) @ R2 @ R ) ) ) ) ).
% in_rtrancl_insert
thf(fact_5114_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_5115_tranclD,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% tranclD
thf(fact_5116_rtranclD,axiom,
! [A: $tType,A4: A,B3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( A4 = B3 )
| ( ( A4 != B3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% rtranclD
thf(fact_5117_tranclD2,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ R ) ) ) ).
% tranclD2
thf(fact_5118_trancl__into__rtrancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_5119_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X3: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R ) )
= ( ( X3 = Y )
| ( ( X3 != Y )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_5120_rtrancl__into__trancl1,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_5121_rtrancl__into__trancl2,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_5122_rtrancl__trancl__trancl,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_5123_trancl__rtrancl__trancl,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_5124_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A4: A,B3: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B3 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ Q )
=> ( Y2 = X ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_5125_rtrancl__Un__separatorE,axiom,
! [A: $tType,A4: A,B3: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ Q )
=> ( X = Y2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_5126_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B5 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( P @ A6 @ B5 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_5127_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa: A,Xb: B,Za2: A,Zb: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ~ ! [A6: A,B5: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A6 @ B5 ) ) @ R2 )
=> ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_5128_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( P @ Bx @ By )
=> ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( P @ Aa2 @ Ba )
=> ( P @ A6 @ B5 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_5129_rtrancl__sub__insert__rtrancl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X3: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ ( insert3 @ ( product_prod @ A @ A ) @ X3 @ R ) ) ) ).
% rtrancl_sub_insert_rtrancl
thf(fact_5130_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Zs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_5131_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P: B > $o,K3: B,M2: B > A] :
( ( wf @ A @ R2 )
=> ( ! [X: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ ( transitive_rtrancl @ A @ R2 ) ) ) )
=> ( ( P @ K3 )
=> ? [X: B] :
( ( P @ X )
& ! [Y3: B] :
( ( P @ Y3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M2 @ X ) @ ( M2 @ Y3 ) ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_5132_trancl__union__outside,axiom,
! [A: $tType,V2: A,W2: A,E5: set @ ( product_prod @ A @ A ),U2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E5 @ U2 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ E5 ) )
=> ? [X: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ X ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E5 @ U2 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ U2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ W2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E5 @ U2 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_5133_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X3: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ X3 @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_5134_trancl__over__edgeE,axiom,
! [A: $tType,U: A,W2: A,V1: A,V22: A,E5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W2 ) @ ( transitive_trancl @ A @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V1 @ V22 ) @ E5 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W2 ) @ ( transitive_trancl @ A @ E5 ) )
=> ~ ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V1 ) @ ( transitive_rtrancl @ A @ E5 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V22 @ W2 ) @ ( transitive_rtrancl @ A @ E5 ) ) ) ) ) ).
% trancl_over_edgeE
thf(fact_5135_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( ( size_size @ ( list @ A ) @ X3 )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_5136_trans__rtrancl__eq__reflcl,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ A3 )
=> ( ( transitive_rtrancl @ A @ A3 )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ ( id2 @ A ) ) ) ) ).
% trans_rtrancl_eq_reflcl
thf(fact_5137_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_5138_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% rtrancl_listrel1_if_listrel
thf(fact_5139_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
=> ( ( A4 = B3 )
| ( member @ A @ A4 @ A3 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_5140_Restr__rtrancl__mono,axiom,
! [A: $tType,V2: A,W2: A,E5: set @ ( product_prod @ A @ A ),U2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E5
@ ( product_Sigma @ A @ A @ U2
@ ^ [Uu3: A] : U2 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_rtrancl @ A @ E5 ) ) ) ).
% Restr_rtrancl_mono
thf(fact_5141_pred__nat__trancl__eq__le,axiom,
! [M2: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_5142_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A4: A,B3: A,E5: set @ ( product_prod @ A @ A ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_rtrancl @ A @ E5 ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ A4 ) @ ( F3 @ B3 ) ) @ ( transitive_rtrancl @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( pairself @ A @ B @ F3 ) @ E5 ) ) ) ) ).
% rtrancl_mapI
thf(fact_5143_listrel__subset__rtrancl__listrel1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel_subset_rtrancl_listrel1
thf(fact_5144_rtrancl__last__touch,axiom,
! [A: $tType,Q4: A,Q7: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q7 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ A @ Q4 @ S )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Qt ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q7 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_touch
thf(fact_5145_rtrancl__last__visit_H,axiom,
! [A: $tType,Q4: A,Q7: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q7 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q7 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : S ) ) ) )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Qt ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q7 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit'
thf(fact_5146_rtrancl__insert,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A4 ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ Y5 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_5147_rtrancl__last__visit,axiom,
! [A: $tType,Q4: A,Q7: A,R: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q7 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q7 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : S ) ) ) )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Qt ) @ ( transitive_trancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q7 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit
thf(fact_5148_trancl__insert,axiom,
! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B4 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_5149_trancl__multi__insert2,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),M2: A,X5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( insert3 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu3: A] : X5 ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ! [X: A] :
( ( member @ A @ X @ X5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ M2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_5150_trancl__multi__insert,axiom,
! [A: $tType,A4: A,B3: A,R2: set @ ( product_prod @ A @ A ),X5: set @ A,M2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ X5
@ ^ [Uu3: A] : ( insert3 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ! [X: A] :
( ( member @ A @ X @ X5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ B3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_5151_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A4: A,P4: product_prod @ A @ B] :
( ( A4
= ( product_fst @ A @ B @ P4 ) )
= ( ? [B4: B] :
( P4
= ( product_Pair @ A @ B @ A4 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_5152_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V2: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ E5
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : R ) ) ) )
=> ( ~ ( member @ A @ U @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E5 @ R ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_5153_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F3: C > A,G: C > B,A3: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ A3 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F3 @ A3 )
@ ^ [X4: A] : ( image2 @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F3 @ ( insert3 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_5154_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P7: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X24: list @ A] :
( ? [Y5: A,Ys3: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ord_less @ A @ X4 @ Y5 ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( ord_less @ A @ X4 @ Y5 )
& ~ ( ord_less @ A @ Y5 @ X4 )
& ( P7 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_5155_vimage__eq,axiom,
! [A: $tType,B: $tType,A4: A,F3: A > B,B2: set @ B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F3 @ B2 ) )
= ( member @ B @ ( F3 @ A4 ) @ B2 ) ) ).
% vimage_eq
thf(fact_5156_vimageI,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: B,B3: A,B2: set @ A] :
( ( ( F3 @ A4 )
= B3 )
=> ( ( member @ A @ B3 @ B2 )
=> ( member @ B @ A4 @ ( vimage @ B @ A @ F3 @ B2 ) ) ) ) ).
% vimageI
thf(fact_5157_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,P: B > $o] :
( ( vimage @ A @ B @ F3 @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y5: A] : ( P @ ( F3 @ Y5 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_5158_vimage__ident,axiom,
! [A: $tType,Y6: set @ A] :
( ( vimage @ A @ A
@ ^ [X4: A] : X4
@ Y6 )
= Y6 ) ).
% vimage_ident
thf(fact_5159_vimage__empty,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( vimage @ A @ B @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_5160_vimage__UNIV,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( vimage @ A @ B @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% vimage_UNIV
thf(fact_5161_vimage__Int,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ B,B2: set @ B] :
( ( vimage @ A @ B @ F3 @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A3 ) @ ( vimage @ A @ B @ F3 @ B2 ) ) ) ).
% vimage_Int
thf(fact_5162_vimage__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ B,B2: set @ B] :
( ( vimage @ A @ B @ F3 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A3 ) @ ( vimage @ A @ B @ F3 @ B2 ) ) ) ).
% vimage_Un
thf(fact_5163_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_5164_lexordp__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ ( nil @ A ) ) ) ).
% lexordp_simps(2)
thf(fact_5165_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_5166_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( ord_less @ A @ X3 @ Y )
| ( ~ ( ord_less @ A @ Y @ X3 )
& ( ord_lexordp @ A @ Xs @ Ys ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_5167_vimage__const,axiom,
! [B: $tType,A: $tType,C2: B,A3: set @ B] :
( ( ( member @ B @ C2 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : C2
@ A3 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C2 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : C2
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_5168_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ A] :
( ( image2 @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_5169_vimage__if,axiom,
! [B: $tType,A: $tType,C2: B,A3: set @ B,D3: B,B2: set @ A] :
( ( ( member @ B @ C2 @ A3 )
=> ( ( ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B2 ) @ C2 @ D3 )
@ A3 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B2 ) @ C2 @ D3 )
@ A3 )
= B2 ) ) ) )
& ( ~ ( member @ B @ C2 @ A3 )
=> ( ( ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B2 ) @ C2 @ D3 )
@ A3 )
= ( uminus_uminus @ ( set @ A ) @ B2 ) ) )
& ( ~ ( member @ B @ D3 @ A3 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B2 ) @ C2 @ D3 )
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_5170_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A3 ) @ B2 )
= ( ord_less_eq @ ( set @ B ) @ A3 @ ( vimage @ B @ A @ F3 @ B2 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_5171_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A3 ) ) @ A3 ) ).
% image_vimage_subset
thf(fact_5172_vimage__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ A,F3: B > A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ A3 ) @ ( vimage @ B @ A @ F3 @ B2 ) ) ) ).
% vimage_mono
thf(fact_5173_subset__vimage__iff,axiom,
! [B: $tType,A: $tType,A3: set @ A,F3: A > B,B2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( vimage @ A @ B @ F3 @ B2 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( member @ B @ ( F3 @ X4 ) @ B2 ) ) ) ) ).
% subset_vimage_iff
thf(fact_5174_rel__restrict__sub,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A3 ) @ R ) ).
% rel_restrict_sub
thf(fact_5175_rel__restrict__mono,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A3 @ R ) @ ( rel_restrict @ A @ B2 @ R ) ) ) ).
% rel_restrict_mono
thf(fact_5176_rel__restrict__union,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: set @ A,B2: set @ A] :
( ( rel_restrict @ A @ R @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( rel_restrict @ A @ ( rel_restrict @ A @ R @ A3 ) @ B2 ) ) ).
% rel_restrict_union
thf(fact_5177_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B,G: A > B,Y: set @ B] :
( ! [W: A] :
( ( member @ A @ W @ S )
=> ( ( F3 @ W )
= ( G @ W ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ Y ) @ S )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_5178_finite__rel__restrict,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( rel_restrict @ A @ R @ A3 ) ) ) ).
% finite_rel_restrict
thf(fact_5179_vimage__Diff,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ B,B2: set @ B] :
( ( vimage @ A @ B @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A3 ) @ ( vimage @ A @ B @ F3 @ B2 ) ) ) ).
% vimage_Diff
thf(fact_5180_lexordp__irreflexive_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ Xs ) ) ).
% lexordp_irreflexive'
thf(fact_5181_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_5182_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_5183_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_5184_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F3: A > B,Q: A > $o] :
( ! [X: A] :
( ( P @ ( F3 @ X ) )
= ( Q @ X ) )
=> ( ( vimage @ A @ B @ F3 @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_5185_vimageI2,axiom,
! [B: $tType,A: $tType,F3: B > A,A4: B,A3: set @ A] :
( ( member @ A @ ( F3 @ A4 ) @ A3 )
=> ( member @ B @ A4 @ ( vimage @ B @ A @ F3 @ A3 ) ) ) ).
% vimageI2
thf(fact_5186_vimageE,axiom,
! [A: $tType,B: $tType,A4: A,F3: A > B,B2: set @ B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F3 @ B2 ) )
=> ( member @ B @ ( F3 @ A4 ) @ B2 ) ) ).
% vimageE
thf(fact_5187_vimageD,axiom,
! [A: $tType,B: $tType,A4: A,F3: A > B,A3: set @ B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F3 @ A3 ) )
=> ( member @ B @ ( F3 @ A4 ) @ A3 ) ) ).
% vimageD
thf(fact_5188_lexordp__append__leftI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Us: list @ A,Vs: list @ A,Xs: list @ A] :
( ( ord_lexordp @ A @ Us @ Vs )
=> ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) ) ) ) ).
% lexordp_append_leftI
thf(fact_5189_vimage__Compl,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ B] :
( ( vimage @ A @ B @ F3 @ ( uminus_uminus @ ( set @ B ) @ A3 ) )
= ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A3 ) ) ) ).
% vimage_Compl
thf(fact_5190_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ! [X: A] :
~ ( ord_less @ A @ X @ X )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_5191_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F7: A > B,B7: set @ B] :
( collect @ A
@ ^ [X4: A] : ( member @ B @ ( F7 @ X4 ) @ B7 ) ) ) ) ).
% vimage_def
thf(fact_5192_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A4: A,F3: A > B,B3: B] :
( ( member @ A @ A4 @ ( vimage @ A @ B @ F3 @ ( insert3 @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F3 @ A4 )
= B3 ) ) ).
% vimage_singleton_eq
thf(fact_5193_rel__restrict__lift,axiom,
! [A: $tType,X3: A,Y: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( rel_restrict @ A @ E5 @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ E5 ) ) ).
% rel_restrict_lift
thf(fact_5194_rel__restrictI,axiom,
! [A: $tType,X3: A,R: set @ A,Y: A,E5: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X3 @ R )
=> ( ~ ( member @ A @ Y @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ E5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( rel_restrict @ A @ E5 @ R ) ) ) ) ) ).
% rel_restrictI
thf(fact_5195_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X3: A,Y: A,A3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( rel_restrict @ A @ A3 @ R ) )
=> ~ ( member @ A @ X3 @ R ) ) ).
% rel_restrict_notR(1)
thf(fact_5196_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X3: A,Y: A,A3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( rel_restrict @ A @ A3 @ R ) )
=> ~ ( member @ A @ Y @ R ) ) ).
% rel_restrict_notR(2)
thf(fact_5197_vimage__Suc__insert__Suc,axiom,
! [N: nat,A3: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert3 @ nat @ ( suc @ N ) @ A3 ) )
= ( insert3 @ nat @ N @ ( vimage @ nat @ nat @ suc @ A3 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_5198_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_5199_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% lexordp.Cons
thf(fact_5200_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_5201_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Us: list @ A,Vs: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
=> ( ! [A6: A] :
~ ( ord_less @ A @ A6 @ A6 )
=> ( ord_lexordp @ A @ Us @ Vs ) ) ) ) ).
% lexordp_append_leftD
thf(fact_5202_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_5203_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V2: A,W2: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E5 @ R ) ) )
=> ~ ( member @ A @ W2 @ R ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_5204_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V2: A,W2: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E5 @ R ) ) )
=> ~ ( member @ A @ V2 @ R ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_5205_rel__restrict__trancl__mem,axiom,
! [A: $tType,A4: A,B3: A,A3: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A3 @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A3 ) @ R ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_5206_rel__restrict__mono2,axiom,
! [A: $tType,R: set @ A,S: set @ A,A3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ R @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A3 @ S ) @ ( rel_restrict @ A @ A3 @ R ) ) ) ).
% rel_restrict_mono2
thf(fact_5207_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X3: B,A3: set @ B,F3: B > ( set @ A )] :
( ( ( member @ B @ X3 @ A3 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 ) @ ( product_Sigma @ B @ A @ A3 @ F3 ) )
= ( F3 @ X3 ) ) )
& ( ~ ( member @ B @ X3 @ A3 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 ) @ ( product_Sigma @ B @ A @ A3 @ F3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_5208_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: set @ A] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F3 @ A3 )
= ( bot_bot @ ( set @ B ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_5209_rel__restrict__trancl__sub,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),R: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A3 @ R ) ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A3 ) @ R ) ) ).
% rel_restrict_trancl_sub
thf(fact_5210_vimage__insert,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: B,B2: set @ B] :
( ( vimage @ A @ B @ F3 @ ( insert3 @ B @ A4 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F3 @ B2 ) ) ) ).
% vimage_insert
thf(fact_5211_vimage__Suc__insert__0,axiom,
! [A3: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert3 @ nat @ ( zero_zero @ nat ) @ A3 ) )
= ( vimage @ nat @ nat @ suc @ A3 ) ) ).
% vimage_Suc_insert_0
thf(fact_5212_rel__restrict__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [V4: A,W3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V4 @ W3 ) @ R4 )
& ~ ( member @ A @ V4 @ A7 )
& ~ ( member @ A @ W3 @ A7 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_5213_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,Ys5: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys5 ) )
=> ( ! [X: A,Xs2: list @ A,Y2: A,Ys5: list @ A] :
( ( ord_less @ A @ X @ Y2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys5 ) ) )
=> ( ! [X: A,Xs2: list @ A,Ys5: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys5 )
=> ( ( P @ Xs2 @ Ys5 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ X @ Ys5 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_5214_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 ) ) )
=> ( ! [X: A] :
( ? [Xs5: list @ A] :
( Xs
= ( cons @ A @ X @ Xs5 ) )
=> ! [Y2: A] :
( ? [Ys4: list @ A] :
( Ys
= ( cons @ A @ Y2 @ Ys4 ) )
=> ~ ( ord_less @ A @ X @ Y2 ) ) )
=> ~ ! [X: A,Xs5: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs5 ) )
=> ! [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
=> ~ ( ord_lexordp @ A @ Xs5 @ Ys4 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_5215_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A13: list @ A,A23: list @ A] :
( ? [Y5: A,Ys3: list @ A] :
( ( A13
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A13
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ord_less @ A @ X4 @ Y5 ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A13
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( ord_less @ A @ X4 @ Y5 )
& ~ ( ord_less @ A @ Y5 @ X4 )
& ( ord_lexordp @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_5216_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp @ A @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y2: A,Ys5: list @ A] :
( A22
!= ( cons @ A @ Y2 @ Ys5 ) ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys5: list @ A] :
( A22
= ( cons @ A @ Y2 @ Ys5 ) )
=> ~ ( ord_less @ A @ X @ Y2 ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys5: list @ A] :
( ( A22
= ( cons @ A @ Y2 @ Ys5 ) )
=> ( ~ ( ord_less @ A @ X @ Y2 )
=> ( ~ ( ord_less @ A @ Y2 @ X )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys5 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_5217_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp @ A @ ( append @ A @ Us @ ( cons @ A @ X3 @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_5218_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ? [X4: A,Vs3: list @ A] :
( Ys3
= ( append @ A @ Xs3 @ ( cons @ A @ X4 @ Vs3 ) ) )
| ? [Us2: list @ A,A5: A,B4: A,Vs3: list @ A,Ws3: list @ A] :
( ( ord_less @ A @ A5 @ B4 )
& ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ A5 @ Vs3 ) ) )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ B4 @ Ws3 ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_5219_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ? [X: A] :
( ( member @ A @ X @ ( image2 @ B @ A @ F3 @ A3 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F3 @ ( insert3 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_5220_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ~ ! [Y2: A] :
( ( member @ A @ Y2 @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F3 @ ( insert3 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_5221_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F: set @ A,H2: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ F )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ F )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ ( insert3 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ F ) @ A3 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_5222_rel__restrict__Int__empty,axiom,
! [A: $tType,A3: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ ( field2 @ A @ R ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( rel_restrict @ A @ R @ A3 )
= R ) ) ).
% rel_restrict_Int_empty
thf(fact_5223_Field__rel__restrict,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ ( rel_restrict @ A @ R @ A3 ) ) @ ( minus_minus @ ( set @ A ) @ ( field2 @ A @ R ) @ A3 ) ) ).
% Field_rel_restrict
thf(fact_5224_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F7: A > B,B7: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : ( vimage @ A @ B @ F7 @ ( insert3 @ B @ Y5 @ ( bot_bot @ ( set @ B ) ) ) )
@ B7 ) ) ) ) ).
% vimage_eq_UN
thf(fact_5225_rel__restrict__compl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A3 ) @ ( rel_restrict @ A @ R @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% rel_restrict_compl
thf(fact_5226_homo__rel__restrict__mono,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B2: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R @ A3 )
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ B2 @ A3 )
@ ^ [Uu3: A] : ( minus_minus @ ( set @ A ) @ B2 @ A3 ) ) ) ) ).
% homo_rel_restrict_mono
thf(fact_5227_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ? [X: A] :
( ( member @ A @ X @ ( image2 @ B @ A @ F3 @ A3 ) )
& ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert3 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_5228_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ~ ( finite_finite2 @ B @ A3 )
=> ~ ! [Y2: A] :
( ( member @ A @ Y2 @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert3 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_5229_rel__restrict__alt__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A7: set @ A] :
( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( product_Sigma @ A @ A @ ( uminus_uminus @ ( set @ A ) @ A7 )
@ ^ [Uu3: A] : ( uminus_uminus @ ( set @ A ) @ A7 ) ) ) ) ) ).
% rel_restrict_alt_def
thf(fact_5230_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] : ( member @ ( 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_5231_rel__restrict__Sigma__sub,axiom,
! [A: $tType,A3: set @ A,R: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) )
@ ( rel_restrict @ A
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
@ R )
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ A3 @ R )
@ ^ [Uu3: A] : ( minus_minus @ ( set @ A ) @ A3 @ R ) ) ) ) ).
% rel_restrict_Sigma_sub
thf(fact_5232_inv__image__partition,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys ) )
=> ~ ( P @ Y2 ) )
=> ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( insert3 @ ( 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_5233_ord_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o] :
( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P7: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X24: list @ A] :
( ? [Y5: A,Ys3: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) )
& ( Less2 @ X4 @ Y5 ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( Less2 @ X4 @ Y5 )
& ~ ( Less2 @ Y5 @ X4 )
& ( P7 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord.lexordp_def
thf(fact_5234_sorted__key__list__of__set__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord144544945434240204of_set @ B @ A )
= ( ^ [F7: B > A] : ( finite_folding_F @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F7 ) @ ( nil @ B ) ) ) ) ) ).
% sorted_key_list_of_set_def
thf(fact_5235_size__list__conv__sum__list,axiom,
! [B: $tType] :
( ( size_list @ B )
= ( ^ [F7: B > nat,Xs3: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F7 @ Xs3 ) ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_5236_ord_Olexordp__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp2 @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X3 @ Y )
| ( ~ ( Less @ Y @ X3 )
& ( lexordp2 @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_5237_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_5238_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_5239_size__list__map,axiom,
! [A: $tType,B: $tType,F3: A > nat,G: B > A,Xs: list @ B] :
( ( size_list @ A @ F3 @ ( map @ B @ A @ G @ Xs ) )
= ( size_list @ B @ ( comp @ A @ nat @ B @ F3 @ G ) @ Xs ) ) ).
% size_list_map
thf(fact_5240_size__list__append,axiom,
! [A: $tType,F3: A > nat,Xs: list @ A,Ys: list @ A] :
( ( size_list @ A @ F3 @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_list @ A @ F3 @ Xs ) @ ( size_list @ A @ F3 @ Ys ) ) ) ).
% size_list_append
thf(fact_5241_ord_Olexordp_OCons,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X3 @ Y )
=> ( lexordp2 @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_5242_ord_Olexordp_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X3 @ Y )
=> ( ~ ( Less @ Y @ X3 )
=> ( ( lexordp2 @ A @ Less @ Xs @ Ys )
=> ( lexordp2 @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_5243_ord_Olexordp_Ocong,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( lexordp2 @ A ) ) ).
% ord.lexordp.cong
thf(fact_5244_ord_Olexordp__irreflexive,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ! [X: A] :
~ ( Less @ X @ X )
=> ~ ( lexordp2 @ A @ Less @ Xs @ Xs ) ) ).
% ord.lexordp_irreflexive
thf(fact_5245_ord_Olexordp__append__leftI,axiom,
! [A: $tType,Less: A > A > $o,Us: list @ A,Vs: list @ A,Xs: list @ A] :
( ( lexordp2 @ A @ Less @ Us @ Vs )
=> ( lexordp2 @ A @ Less @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) ) ) ).
% ord.lexordp_append_leftI
thf(fact_5246_ord_Olexordp__append__leftD,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A,Us: list @ A,Vs: list @ A] :
( ( lexordp2 @ A @ Less @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
=> ( ! [A6: A] :
~ ( Less @ A6 @ A6 )
=> ( lexordp2 @ A @ Less @ Us @ Vs ) ) ) ).
% ord.lexordp_append_leftD
thf(fact_5247_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F3: B > nat,G: A > B] :
( ( comp @ ( list @ B ) @ nat @ ( list @ A ) @ ( size_list @ B @ F3 ) @ ( map @ A @ B @ G ) )
= ( size_list @ A @ ( comp @ B @ nat @ A @ F3 @ G ) ) ) ).
% list.size_gen_o_map
thf(fact_5248_ord_Olexordp_Osimps,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o,A13: list @ A,A23: list @ A] :
( ? [Y5: A,Ys3: list @ A] :
( ( A13
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A13
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ( Less2 @ X4 @ Y5 ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A13
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( Less2 @ X4 @ Y5 )
& ~ ( Less2 @ Y5 @ X4 )
& ( lexordp2 @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_5249_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,Ys5: list @ A] :
( A22
!= ( cons @ A @ Y2 @ Ys5 ) ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys5: list @ A] :
( A22
= ( cons @ A @ Y2 @ Ys5 ) )
=> ~ ( Less @ X @ Y2 ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys5: list @ A] :
( ( A22
= ( cons @ A @ Y2 @ Ys5 ) )
=> ( ~ ( Less @ X @ Y2 )
=> ( ~ ( Less @ Y2 @ X )
=> ~ ( lexordp2 @ A @ Less @ Xs2 @ Ys5 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_5250_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_5251_ord_Olexordp__append__left__rightI,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X3 @ Y )
=> ( lexordp2 @ A @ Less @ ( append @ A @ Us @ ( cons @ A @ X3 @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% ord.lexordp_append_left_rightI
thf(fact_5252_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_5253_list_Osize__gen_I1_J,axiom,
! [A: $tType,X3: A > nat] :
( ( size_list @ A @ X3 @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_5254_size__list__estimation,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y @ ( F3 @ X3 ) )
=> ( ord_less @ nat @ Y @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_5255_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F3: A > nat,G: A > nat] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F3 @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F3 @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_5256_size__list__estimation_H,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F3 @ X3 ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_5257_list_Osize__gen_I2_J,axiom,
! [A: $tType,X3: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X3 @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X3 @ X21 ) @ ( size_list @ A @ X3 @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_5258_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,A3: set @ A,X3: A,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z2 @ A3 )
= ( F3 @ X3 @ ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% folding_on.remove
thf(fact_5259_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( insert3 @ A @ X3 @ A3 ) )
= ( F3 @ X3 @ ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% folding_on.insert_remove
thf(fact_5260_folding__idem__on_Oinsert__idem,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite1890593828518410140dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( insert3 @ A @ X3 @ A3 ) )
= ( F3 @ X3 @ ( finite_folding_F @ A @ B @ F3 @ Z2 @ A3 ) ) ) ) ) ) ).
% folding_idem_on.insert_idem
thf(fact_5261_folding__on_Oempty,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( bot_bot @ ( set @ A ) ) )
= Z2 ) ) ).
% folding_on.empty
thf(fact_5262_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
@ ^ [X4: A] : X4 ) ) ) ).
% sorted_list_of_set.fold_insort_key.folding_on_axioms
thf(fact_5263_folding__on_Oinsert,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X3: A,A3: set @ A,Z2: B] :
( ( finite_folding_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ S )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z2 @ ( insert3 @ A @ X3 @ A3 ) )
= ( F3 @ X3 @ ( finite_folding_F @ A @ B @ F3 @ Z2 @ A3 ) ) ) ) ) ) ) ).
% folding_on.insert
thf(fact_5264_mult1__def,axiom,
! [A: $tType] :
( ( mult1 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) )
@ ( product_case_prod @ ( multiset @ A ) @ ( multiset @ A ) @ $o
@ ^ [N9: multiset @ A,M10: multiset @ A] :
? [A5: A,M03: multiset @ A,K8: multiset @ A] :
( ( M10
= ( add_mset @ A @ A5 @ M03 ) )
& ( N9
= ( plus_plus @ ( multiset @ A ) @ M03 @ K8 ) )
& ! [B4: A] :
( ( member @ A @ B4 @ ( set_mset @ A @ K8 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R5 ) ) ) ) ) ) ) ).
% mult1_def
thf(fact_5265_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu3: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X9: set @ A,Y9: set @ A] :
( ( Uu3
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X9 @ Y9 ) )
& ( X9
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ Y9 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ X9 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_5266_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,A3: set @ A,Z2: B,Y: B,A4: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ S )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A3 @ Y )
=> ( ( member @ A @ A4 @ A3 )
=> ? [Y10: B] :
( ( Y
= ( F3 @ A4 @ Y10 ) )
& ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ Y10 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
thf(fact_5267_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X8: A] :
( ( member @ A @ X8 @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X8 ) ) ).
% bex_empty
thf(fact_5268_ball__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) )
=> ( P @ X4 ) ) )
= ( ! [X9: A] : ( P @ X9 ) ) ) ).
% ball_UNIV
thf(fact_5269_bex__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) )
& ( P @ X4 ) ) )
= ( ? [X9: A] : ( P @ X9 ) ) ) ).
% bex_UNIV
thf(fact_5270_Bex__def,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A7: set @ A,P2: A > $o] :
? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( P2 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_5271_empty__fold__graphE,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z2: B,X3: B] :
( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( bot_bot @ ( set @ A ) ) @ X3 )
=> ( X3 = Z2 ) ) ).
% empty_fold_graphE
thf(fact_5272_fold__graph_OemptyI,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z2: B] : ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( bot_bot @ ( set @ A ) ) @ Z2 ) ).
% fold_graph.emptyI
thf(fact_5273_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ A @ B )
= ( ^ [F7: A > B,A7: set @ A] :
( collect @ B
@ ^ [Y5: B] :
? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( Y5
= ( F7 @ X4 ) ) ) ) ) ) ).
% image_def
thf(fact_5274_fold__graph_OinsertI,axiom,
! [A: $tType,B: $tType,X3: A,A3: set @ A,F3: A > B > B,Z2: B,Y: B] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A3 @ Y )
=> ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( insert3 @ A @ X3 @ A3 ) @ ( F3 @ X3 @ Y ) ) ) ) ).
% fold_graph.insertI
thf(fact_5275_Bex__set__list__ex,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) )
= ( list_ex @ A @ P @ Xs ) ) ).
% Bex_set_list_ex
thf(fact_5276_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ A] :
( ( vimage @ A @ B @ F3 @ ( image2 @ A @ B @ F3 @ A3 ) )
= ( collect @ A
@ ^ [Y5: A] :
? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ( F3 @ X4 )
= ( F3 @ Y5 ) ) ) ) ) ).
% vimage_image_eq
thf(fact_5277_fold__graph_Ocases,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z2: B,A1: set @ A,A22: B] :
( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A1 @ A22 )
=> ( ( ( A1
= ( bot_bot @ ( set @ A ) ) )
=> ( A22 != Z2 ) )
=> ~ ! [X: A,A14: set @ A] :
( ( A1
= ( insert3 @ A @ X @ A14 ) )
=> ! [Y2: B] :
( ( A22
= ( F3 @ X @ Y2 ) )
=> ( ~ ( member @ A @ X @ A14 )
=> ~ ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A14 @ Y2 ) ) ) ) ) ) ).
% fold_graph.cases
thf(fact_5278_fold__graph_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold_graph @ A @ B )
= ( ^ [F7: A > B > B,Z6: B,A13: set @ A,A23: B] :
( ( ( A13
= ( bot_bot @ ( set @ A ) ) )
& ( A23 = Z6 ) )
| ? [X4: A,A7: set @ A,Y5: B] :
( ( A13
= ( insert3 @ A @ X4 @ A7 ) )
& ( A23
= ( F7 @ X4 @ Y5 ) )
& ~ ( member @ A @ X4 @ A7 )
& ( finite_fold_graph @ A @ B @ F7 @ Z6 @ A7 @ Y5 ) ) ) ) ) ).
% fold_graph.simps
thf(fact_5279_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X9: set @ A,Y9: set @ A] :
( ( finite_finite2 @ A @ X9 )
& ( finite_finite2 @ A @ Y9 )
& ( Y9
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ X9 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ Y9 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R4 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_5280_nths__nths,axiom,
! [A: $tType,Xs: list @ A,A3: set @ nat,B2: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs @ A3 ) @ B2 )
= ( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I4: nat] :
( ( member @ nat @ I4 @ A3 )
& ( member @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I11: nat] :
( ( member @ nat @ I11 @ A3 )
& ( ord_less @ nat @ I11 @ I4 ) ) ) )
@ B2 ) ) ) ) ) ).
% nths_nths
thf(fact_5281_max__extp_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: set @ A,A22: set @ A] :
( ( max_extp @ A @ R @ A1 @ A22 )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X8: A] :
( ( member @ A @ X8 @ A1 )
=> ? [Xa4: A] :
( ( member @ A @ Xa4 @ A22 )
& ( R @ X8 @ Xa4 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_5282_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R4: A > A > $o,A13: set @ A,A23: set @ A] :
( ( finite_finite2 @ A @ A13 )
& ( finite_finite2 @ A @ A23 )
& ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A13 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A23 )
& ( R4 @ X4 @ Y5 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_5283_max__extp_Omax__extI,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ X5 )
=> ( ( finite_finite2 @ A @ Y6 )
=> ( ( Y6
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ X5 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ Y6 )
& ( R @ X @ Xa3 ) ) )
=> ( max_extp @ A @ R @ X5 @ Y6 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_5284_comp__fun__commute__on_Ofold__graph__insertE,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,X3: A,A3: set @ A,Z2: B,V2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ A3 ) @ S )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( insert3 @ A @ X3 @ A3 ) @ V2 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ~ ! [Y2: B] :
( ( V2
= ( F3 @ X3 @ Y2 ) )
=> ~ ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A3 @ Y2 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE
thf(fact_5285_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_5286_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat,S3: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ S @ ( set_ord_lessThan @ A @ S3 ) ) ) )
=> ( ( infini527867602293511546merate @ A @ ( inf_inf @ ( set @ A ) @ S @ ( set_ord_lessThan @ A @ S3 ) ) @ N )
= ( infini527867602293511546merate @ A @ S @ N ) ) ) ) ) ).
% finite_enumerate_initial_segment
thf(fact_5287_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F3 ) @ Xs @ ( top_top @ A ) ) ) ) ).
% INF_set_fold
thf(fact_5288_fold__append,axiom,
! [A: $tType,B: $tType,F3: B > A > A,Xs: list @ B,Ys: list @ B] :
( ( fold @ B @ A @ F3 @ ( append @ B @ Xs @ Ys ) )
= ( comp @ A @ A @ A @ ( fold @ B @ A @ F3 @ Ys ) @ ( fold @ B @ A @ F3 @ Xs ) ) ) ).
% fold_append
thf(fact_5289_prod__list_OCons,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X3: A,Xs: list @ A] :
( ( groups5270119922927024881d_list @ A @ ( cons @ A @ X3 @ Xs ) )
= ( times_times @ A @ X3 @ ( groups5270119922927024881d_list @ A @ Xs ) ) ) ) ).
% prod_list.Cons
thf(fact_5290_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_5291_fold__replicate,axiom,
! [A: $tType,B: $tType,F3: B > A > A,N: nat,X3: B] :
( ( fold @ B @ A @ F3 @ ( replicate @ B @ N @ X3 ) )
= ( compow @ ( A > A ) @ N @ ( F3 @ X3 ) ) ) ).
% fold_replicate
thf(fact_5292_enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,M2: nat,N: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M2 ) @ ( infini527867602293511546merate @ A @ S @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% enumerate_mono_iff
thf(fact_5293_finite__enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,M2: nat,N: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ M2 @ ( finite_card @ A @ S ) )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S ) )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M2 ) @ ( infini527867602293511546merate @ A @ S @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_5294_fold__simps_I1_J,axiom,
! [B: $tType,A: $tType,F3: B > A > A,S3: A] :
( ( fold @ B @ A @ F3 @ ( nil @ B ) @ S3 )
= S3 ) ).
% fold_simps(1)
thf(fact_5295_foldl__conv__fold,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F7: A > B > A,S7: A,Xs3: list @ B] :
( fold @ B @ A
@ ^ [X4: B,T3: A] : ( F7 @ T3 @ X4 )
@ Xs3
@ S7 ) ) ) ).
% foldl_conv_fold
thf(fact_5296_fold__invariant,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Q: A > $o,P: B > $o,S3: B,F3: A > B > B] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( Q @ X ) )
=> ( ( P @ S3 )
=> ( ! [X: A,S8: B] :
( ( Q @ X )
=> ( ( P @ S8 )
=> ( P @ ( F3 @ X @ S8 ) ) ) )
=> ( P @ ( fold @ A @ B @ F3 @ Xs @ S3 ) ) ) ) ) ).
% fold_invariant
thf(fact_5297_List_Ofold__cong,axiom,
! [B: $tType,A: $tType,A4: A,B3: A,Xs: list @ B,Ys: list @ B,F3: B > A > A,G: B > A > A] :
( ( A4 = B3 )
=> ( ( Xs = Ys )
=> ( ! [X: B] :
( ( member @ B @ X @ ( set2 @ B @ Xs ) )
=> ( ( F3 @ X )
= ( G @ X ) ) )
=> ( ( fold @ B @ A @ F3 @ Xs @ A4 )
= ( fold @ B @ A @ G @ Ys @ B3 ) ) ) ) ) ).
% List.fold_cong
thf(fact_5298_fold__simps_I2_J,axiom,
! [B: $tType,A: $tType,F3: B > A > A,X3: B,Xs: list @ B,S3: A] :
( ( fold @ B @ A @ F3 @ ( cons @ B @ X3 @ Xs ) @ S3 )
= ( fold @ B @ A @ F3 @ Xs @ ( F3 @ X3 @ S3 ) ) ) ).
% fold_simps(2)
thf(fact_5299_fold__Cons,axiom,
! [B: $tType,A: $tType,F3: A > B > B,X3: A,Xs: list @ A] :
( ( fold @ A @ B @ F3 @ ( cons @ A @ X3 @ Xs ) )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F3 @ Xs ) @ ( F3 @ X3 ) ) ) ).
% fold_Cons
thf(fact_5300_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H2: B > C,G: A > B > B,F3: A > C > C,S3: B] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H2 @ ( G @ X ) )
= ( comp @ C @ C @ B @ ( F3 @ X ) @ H2 ) ) )
=> ( ( H2 @ ( fold @ A @ B @ G @ Xs @ S3 ) )
= ( fold @ A @ C @ F3 @ Xs @ ( H2 @ S3 ) ) ) ) ).
% fold_commute_apply
thf(fact_5301_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H2: B > C,G: A > B > B,F3: A > C > C] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H2 @ ( G @ X ) )
= ( comp @ C @ C @ B @ ( F3 @ X ) @ H2 ) ) )
=> ( ( comp @ B @ C @ B @ H2 @ ( fold @ A @ B @ G @ Xs ) )
= ( comp @ C @ C @ B @ ( fold @ A @ C @ F3 @ Xs ) @ H2 ) ) ) ).
% fold_commute
thf(fact_5302_fold__Nil,axiom,
! [A: $tType,B: $tType,F3: A > B > B] :
( ( fold @ A @ B @ F3 @ ( nil @ A ) )
= ( id @ B ) ) ).
% fold_Nil
thf(fact_5303_fold__id,axiom,
! [A: $tType,B: $tType,Xs: list @ A,F3: A > B > B] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( F3 @ X )
= ( id @ B ) ) )
=> ( ( fold @ A @ B @ F3 @ Xs )
= ( id @ B ) ) ) ).
% fold_id
thf(fact_5304_fold__permuted__eq,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ A,P: B > $o,Z2: B,F3: A > B > B] :
( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) )
=> ( ( P @ Z2 )
=> ( ! [X: A,Z3: B] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( P @ Z3 )
=> ( P @ ( F3 @ X @ Z3 ) ) ) )
=> ( ! [X: A,Y2: A,Z3: B] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( P @ Z3 )
=> ( ( F3 @ X @ ( F3 @ Y2 @ Z3 ) )
= ( F3 @ Y2 @ ( F3 @ X @ Z3 ) ) ) ) ) )
=> ( ( fold @ A @ B @ F3 @ Xs @ Z2 )
= ( fold @ A @ B @ F3 @ Ys @ Z2 ) ) ) ) ) ) ).
% fold_permuted_eq
thf(fact_5305_fold__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: B > A > A,F3: C > B,Xs: list @ C] :
( ( fold @ B @ A @ G @ ( map @ C @ B @ F3 @ Xs ) )
= ( fold @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F3 ) @ Xs ) ) ).
% fold_map
thf(fact_5306_foldr__conv__fold,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F7: B > A > A,Xs3: list @ B] : ( fold @ B @ A @ F7 @ ( rev @ B @ Xs3 ) ) ) ) ).
% foldr_conv_fold
thf(fact_5307_fold__filter,axiom,
! [A: $tType,B: $tType,F3: B > A > A,P: B > $o,Xs: list @ B] :
( ( fold @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs ) )
= ( fold @ B @ A
@ ^ [X4: B] : ( if @ ( A > A ) @ ( P @ X4 ) @ ( F3 @ X4 ) @ ( id @ A ) )
@ Xs ) ) ).
% fold_filter
thf(fact_5308_prod__list__zero__iff,axiom,
! [A: $tType] :
( ( ( semiring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [Xs: list @ A] :
( ( ( groups5270119922927024881d_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ ( set2 @ A @ Xs ) ) ) ) ).
% prod_list_zero_iff
thf(fact_5309_union__set__fold,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
= ( fold @ A @ ( set @ A ) @ ( insert3 @ A ) @ Xs @ A3 ) ) ).
% union_set_fold
thf(fact_5310_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_5311_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_5312_fold__multiset__equiv,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B > B,Ys: list @ A] :
( ! [X: A,Y2: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) )
= ( comp @ B @ B @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) ) ) )
=> ( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) )
=> ( ( fold @ A @ B @ F3 @ Xs )
= ( fold @ A @ B @ F3 @ Ys ) ) ) ) ).
% fold_multiset_equiv
thf(fact_5313_fold__rev,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B > B] :
( ! [X: A,Y2: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ) )
=> ( ( fold @ A @ B @ F3 @ ( rev @ A @ Xs ) )
= ( fold @ A @ B @ F3 @ Xs ) ) ) ).
% fold_rev
thf(fact_5314_List_Ounion__def,axiom,
! [A: $tType] :
( ( union @ A )
= ( fold @ A @ ( list @ A ) @ ( insert @ A ) ) ) ).
% List.union_def
thf(fact_5315_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B > B] :
( ! [X: A,Y2: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ) )
=> ( ( foldr @ A @ B @ F3 @ Xs )
= ( fold @ A @ B @ F3 @ Xs ) ) ) ).
% foldr_fold
thf(fact_5316_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B > B,X3: A] :
( ! [X: A,Y2: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) )
= ( comp @ B @ B @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) ) ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( fold @ A @ B @ F3 @ Xs )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F3 @ ( remove1 @ A @ X3 @ Xs ) ) @ ( F3 @ X3 ) ) ) ) ) ).
% fold_remove1_split
thf(fact_5317_minus__set__fold,axiom,
! [A: $tType,A3: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( set2 @ A @ Xs ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs @ A3 ) ) ).
% minus_set_fold
thf(fact_5318_enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ ( infini527867602293511546merate @ A @ S @ ( suc @ N ) ) ) ) ) ).
% enumerate_step
thf(fact_5319_enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M2: nat,N: nat,S: set @ A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ~ ( finite_finite2 @ A @ S )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M2 ) @ ( infini527867602293511546merate @ A @ S @ N ) ) ) ) ) ).
% enumerate_mono
thf(fact_5320_sort__conv__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs )
= ( fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4 )
@ Xs
@ ( nil @ A ) ) ) ) ).
% sort_conv_fold
thf(fact_5321_finite__enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S ) )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ S ) ) ) ) ).
% finite_enumerate_in_set
thf(fact_5322_finite__enumerate__Ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,S3: A] :
( ( finite_finite2 @ A @ S )
=> ( ( member @ A @ S3 @ S )
=> ? [N5: nat] :
( ( ord_less @ nat @ N5 @ ( finite_card @ A @ S ) )
& ( ( infini527867602293511546merate @ A @ S @ N5 )
= S3 ) ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_5323_finite__enum__ext,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X5: set @ A,Y6: set @ A] :
( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( finite_card @ A @ X5 ) )
=> ( ( infini527867602293511546merate @ A @ X5 @ I5 )
= ( infini527867602293511546merate @ A @ Y6 @ I5 ) ) )
=> ( ( finite_finite2 @ A @ X5 )
=> ( ( finite_finite2 @ A @ Y6 )
=> ( ( ( finite_card @ A @ X5 )
= ( finite_card @ A @ Y6 ) )
=> ( X5 = Y6 ) ) ) ) ) ) ).
% finite_enum_ext
thf(fact_5324_inter__coset__fold,axiom,
! [A: $tType,A3: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( coset @ A @ Xs ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs @ A3 ) ) ).
% inter_coset_fold
thf(fact_5325_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_5326_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_5327_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_5328_Max_Oset__eq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs: list @ A] :
( ( lattic643756798349783984er_Max @ A @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) )
= ( fold @ A @ A @ ( ord_max @ A ) @ Xs @ X3 ) ) ) ).
% Max.set_eq_fold
thf(fact_5329_Min_Oset__eq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs: list @ A] :
( ( lattic643756798350308766er_Min @ A @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) )
= ( fold @ A @ A @ ( ord_min @ A ) @ Xs @ X3 ) ) ) ).
% Min.set_eq_fold
thf(fact_5330_comp__fun__idem__on_Ofold__set__fold,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,Xs: list @ A,Y: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ S )
=> ( ( finite_fold @ A @ B @ F3 @ Y @ ( set2 @ A @ Xs ) )
= ( fold @ A @ B @ F3 @ Xs @ Y ) ) ) ) ).
% comp_fun_idem_on.fold_set_fold
thf(fact_5331_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_5332_finite__enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M2: nat,N: nat,S: set @ A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M2 ) @ ( infini527867602293511546merate @ A @ S @ N ) ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_5333_finite__le__enumerate,axiom,
! [S: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ S )
=> ( ( ord_less @ nat @ N @ ( finite_card @ nat @ S ) )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_5334_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,Xs: list @ A,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ S )
=> ( ( finite_fold @ A @ B @ F3 @ Y @ ( set2 @ A @ Xs ) )
= ( fold @ A @ B @ F3 @ ( remdups @ A @ Xs ) @ Y ) ) ) ) ).
% comp_fun_commute_on.fold_set_fold_remdups
thf(fact_5335_finite__enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ ( infini527867602293511546merate @ A @ S @ ( suc @ N ) ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_5336_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 @ ( insert3 @ A @ ( infini527867602293511546merate @ A @ S @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N ) ) ) ).
% enumerate_Suc'
thf(fact_5337_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_5338_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X5: set @ A,Y6: set @ A] :
( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( finite_card @ A @ X5 ) )
=> ( ( infini527867602293511546merate @ A @ X5 @ I5 )
= ( infini527867602293511546merate @ A @ Y6 @ I5 ) ) )
=> ( ( finite_finite2 @ A @ X5 )
=> ( ( finite_finite2 @ A @ Y6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X5 ) @ ( finite_card @ A @ Y6 ) )
=> ( ord_less_eq @ ( set @ A ) @ X5 @ Y6 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_5339_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ Xs @ ( bot_bot @ A ) ) ) ) ).
% SUP_set_fold
thf(fact_5340_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S ) )
=> ( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S7: A] :
( ( member @ A @ S7 @ S )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ S7 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_5341_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S
@ ( insert3 @ A
@ ( ord_Least @ A
@ ^ [N2: A] : ( member @ A @ N2 @ S ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N ) ) ) ).
% enumerate_Suc
thf(fact_5342_subset__mset_OcINF__union,axiom,
! [A: $tType,B: $tType,A3: set @ B,F3: B > ( multiset @ A ),B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ B2 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) )
= ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ B2 ) ) ) ) ) ) ) ) ).
% subset_mset.cINF_union
thf(fact_5343_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K3: A] :
( ( P @ K3 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ K3 ) ) ) ).
% Least_le
thf(fact_5344_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K3: A,P: A > $o] :
( ( ord_less @ A @ K3 @ ( ord_Least @ A @ P ) )
=> ~ ( P @ K3 ) ) ) ).
% not_less_Least
thf(fact_5345_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K3: A] :
( ( P @ K3 )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_5346_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A4: A,Q: A > $o] :
( ( P @ A4 )
=> ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_5347_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_5348_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_5349_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X8: A] :
( ( P @ X8 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X8 @ Y2 ) )
& ! [Y2: A] :
( ( ( P @ Y2 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y2 @ Ya2 ) ) )
=> ( Y2 = X8 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_5350_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z2: A] :
( ? [X8: A] :
( ( P @ X8 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X8 @ Y2 ) )
& ! [Y2: A] :
( ( ( P @ Y2 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y2 @ Ya2 ) ) )
=> ( Y2 = X8 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z2 ) ) ) ) ).
% Least1_le
thf(fact_5351_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) )
=> ( ! [X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( Q @ X ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_5352_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A] :
( ( P @ X3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) )
=> ( ( ord_Least @ A @ P )
= X3 ) ) ) ) ).
% Least_equality
thf(fact_5353_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A4: A,Q: A > $o] :
( ( P @ A4 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B10: A] :
( ( P @ B10 )
=> ( ord_less_eq @ A @ A6 @ B10 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_5354_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B10: A] :
( ( P @ B10 )
=> ( ord_less_eq @ A @ A6 @ B10 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_5355_abort__Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( abort_Bleast @ A )
= ( ^ [S2: set @ A,P2: A > $o] :
( ord_Least @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S2 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_5356_Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( bleast @ A )
= ( ^ [S2: set @ A,P2: A > $o] :
( ord_Least @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S2 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% Bleast_def
thf(fact_5357_subset__mset_Ole__cINF__iff,axiom,
! [A: $tType,B: $tType,A3: set @ B,F3: B > ( multiset @ A ),U: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) )
=> ( ( subseteq_mset @ A @ U @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( subseteq_mset @ A @ U @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% subset_mset.le_cINF_iff
thf(fact_5358_subset__mset_OcINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType,B2: set @ B,F3: C > ( multiset @ A ),A3: set @ C,G: B > ( multiset @ A )] :
( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F3 @ A3 ) )
=> ( ! [M6: B] :
( ( member @ B @ M6 @ B2 )
=> ? [X8: C] :
( ( member @ C @ X8 @ A3 )
& ( subseteq_mset @ A @ ( F3 @ X8 ) @ ( G @ M6 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B2 ) ) ) ) ) ) ).
% subset_mset.cINF_mono
thf(fact_5359_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,S: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ? [X8: A] :
( ( member @ A @ X8 @ S )
& ! [Xa4: A] :
( ( member @ A @ Xa4 @ S )
=> ( ord_less_eq @ A @ X8 @ Xa4 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y5: B] : ( member @ B @ Y5 @ ( image2 @ A @ B @ F3 @ S ) ) )
= ( F3
@ ( ord_Least @ A
@ ^ [X4: A] : ( member @ A @ X4 @ S ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_5360_subset__mset_OcINF__superset__mono,axiom,
! [A: $tType,B: $tType,A3: set @ B,G: B > ( multiset @ A ),B2: set @ B,F3: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B2 )
=> ( ! [X: B] :
( ( member @ B @ X @ B2 )
=> ( subseteq_mset @ A @ ( G @ X ) @ ( F3 @ X ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B2 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) ) ) ) ) ) ).
% subset_mset.cINF_superset_mono
thf(fact_5361_subset__mset_OcINF__inf__distrib,axiom,
! [A: $tType,B: $tType,A3: set @ B,F3: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A3 ) )
=> ( ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A3 ) ) )
= ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A5: B] : ( inter_mset @ A @ ( F3 @ A5 ) @ ( G @ A5 ) )
@ A3 ) ) ) ) ) ) ).
% subset_mset.cINF_inf_distrib
thf(fact_5362_enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( ( infini527867602293511546merate @ A @ S @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S7: A] :
( ( member @ A @ S7 @ S )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N ) @ S7 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_5363_subset__mset_OcINF__insert,axiom,
! [A: $tType,B: $tType,A3: set @ B,F3: B > ( multiset @ A ),A4: B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ ( insert3 @ B @ A4 @ A3 ) ) )
= ( inter_mset @ A @ ( F3 @ A4 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) ) ) ) ) ).
% subset_mset.cINF_insert
thf(fact_5364_subset__mset_Omono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F3: ( multiset @ A ) > B,A3: C > ( multiset @ A ),I3: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A3 @ I3 ) )
=> ( ( I3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A3 @ I3 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A3 @ X4 ) )
@ I3 ) ) ) ) ) ) ) ).
% subset_mset.mono_cINF
thf(fact_5365_list_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R4: A > B > $o,A5: list @ A,B4: list @ B] :
? [Z6: list @ ( product_prod @ A @ B )] :
( ( member @ ( list @ ( product_prod @ A @ B ) ) @ Z6
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) ) ) )
& ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z6 )
= A5 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z6 )
= B4 ) ) ) ) ).
% list.in_rel
thf(fact_5366_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 ) )
@ ^ [X4: A,Y5: B,A7: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X4 @ S ) @ ( insert3 @ B @ Y5 @ A7 ) @ A7 ) )
@ ( bot_bot @ ( set @ B ) )
@ R ) ) ) ).
% Image_fold
thf(fact_5367_ImageI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B ),A3: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ B @ B3 @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% ImageI
thf(fact_5368_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_5369_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_5370_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_5371_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_5372_Image__Id,axiom,
! [A: $tType,A3: set @ A] :
( ( image @ A @ A @ ( id2 @ A ) @ A3 )
= A3 ) ).
% Image_Id
thf(fact_5373_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_5374_Image__Id__on,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( image @ A @ A @ ( id_on @ A @ A3 ) @ B2 )
= ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ).
% Image_Id_on
thf(fact_5375_Image__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o,A3: set @ B] :
( ( image @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) @ A3 )
= ( collect @ A
@ ^ [Y5: A] :
? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( P @ X4 @ Y5 ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_5376_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A ),A4: B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R2 @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A4 @ B3 ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_5377_listrel__Nil,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert3 @ ( list @ B ) @ ( nil @ B ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( insert3 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listrel_Nil
thf(fact_5378_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 @ ( insert3 @ B @ U @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% pair_vimage_is_Image
thf(fact_5379_Image__Int__subset,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A3: set @ B,B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R @ A3 ) @ ( image @ B @ A @ R @ B2 ) ) ) ).
% Image_Int_subset
thf(fact_5380_rtrancl__image__advance,axiom,
! [A: $tType,Q4: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X3: A] :
( ( member @ A @ Q4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ X3 ) @ R )
=> ( member @ A @ X3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_5381_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q4: A,R: set @ ( product_prod @ A @ A ),Q0: set @ A,X3: A] :
( ( member @ A @ Q4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ X3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ A @ X3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_5382_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 ) )
= ( ? [Us2: list @ A,Vs3: list @ A] :
( ( Xs
= ( append @ A @ Us2 @ Vs3 ) )
& ( ( size_size @ ( list @ A ) @ Us2 )
= ( size_size @ ( list @ B ) @ Ys ) )
& ( ( size_size @ ( list @ A ) @ Vs3 )
= ( size_size @ ( list @ B ) @ Zs ) )
& ( list_all2 @ A @ B @ P @ Us2 @ Ys )
& ( list_all2 @ A @ B @ P @ Vs3 @ Zs ) ) ) ) ).
% list_all2_append2
thf(fact_5383_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 )
= ( ? [Us2: list @ B,Vs3: list @ B] :
( ( Zs
= ( append @ B @ Us2 @ Vs3 ) )
& ( ( size_size @ ( list @ B ) @ Us2 )
= ( size_size @ ( list @ A ) @ Xs ) )
& ( ( size_size @ ( list @ B ) @ Vs3 )
= ( size_size @ ( list @ A ) @ Ys ) )
& ( list_all2 @ A @ B @ P @ Xs @ Us2 )
& ( list_all2 @ A @ B @ P @ Ys @ Vs3 ) ) ) ) ).
% list_all2_append1
thf(fact_5384_list__all2__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: A > B > $o,Us: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( list_all2 @ A @ B @ P @ ( append @ A @ Xs @ Us ) @ ( append @ B @ Ys @ Vs ) )
= ( ( list_all2 @ A @ B @ P @ Xs @ Ys )
& ( list_all2 @ A @ B @ P @ Us @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_5385_list_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X3: list @ A,G: B > C,Y: list @ B] :
( ( list_all2 @ A @ C @ Sa @ X3 @ ( map @ B @ C @ G @ Y ) )
= ( list_all2 @ A @ B
@ ^ [X4: A,Y5: B] : ( Sa @ X4 @ ( G @ Y5 ) )
@ X3
@ Y ) ) ).
% list.rel_map(2)
thf(fact_5386_list_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X3: list @ A,Y: list @ B] :
( ( list_all2 @ C @ B @ Sb @ ( map @ A @ C @ I @ X3 ) @ Y )
= ( list_all2 @ A @ B
@ ^ [X4: A] : ( Sb @ ( I @ X4 ) )
@ X3
@ Y ) ) ).
% list.rel_map(1)
thf(fact_5387_list__all2__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F3: C > A,As2: list @ C,Bs: list @ B] :
( ( list_all2 @ A @ B @ P @ ( map @ C @ A @ F3 @ As2 ) @ Bs )
= ( list_all2 @ C @ B
@ ^ [X4: C] : ( P @ ( F3 @ X4 ) )
@ As2
@ Bs ) ) ).
% list_all2_map1
thf(fact_5388_list__all2__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As2: list @ A,F3: C > B,Bs: list @ C] :
( ( list_all2 @ A @ B @ P @ As2 @ ( map @ C @ B @ F3 @ Bs ) )
= ( list_all2 @ A @ C
@ ^ [X4: A,Y5: C] : ( P @ X4 @ ( F3 @ Y5 ) )
@ As2
@ Bs ) ) ).
% list_all2_map2
thf(fact_5389_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 ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys5: list @ B] :
( ( P @ X @ Y2 )
=> ( ( list_all2 @ A @ B @ P @ Xs2 @ Ys5 )
=> ( ( R @ Xs2 @ Ys5 )
=> ( R @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys5 ) ) ) ) )
=> ( R @ Xs @ Ys ) ) ) ) ).
% List.list_all2_induct
thf(fact_5390_list_Orel__induct,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X3: list @ A,Y: list @ B,Q: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ R @ X3 @ 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 @ X3 @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_5391_list_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A4: list @ A,B3: list @ B] :
( ( list_all2 @ A @ B @ R @ A4 @ B3 )
=> ( ( ( A4
= ( nil @ A ) )
=> ( B3
!= ( nil @ B ) ) )
=> ~ ! [X12: A,X23: list @ A] :
( ( A4
= ( cons @ A @ X12 @ X23 ) )
=> ! [Y12: B,Y23: list @ B] :
( ( B3
= ( cons @ B @ Y12 @ Y23 ) )
=> ( ( R @ X12 @ Y12 )
=> ~ ( list_all2 @ A @ B @ R @ X23 @ Y23 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_5392_list_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: B,Y222: list @ B] :
~ ( list_all2 @ A @ B @ R @ ( nil @ A ) @ ( cons @ B @ Y21 @ Y222 ) ) ).
% list.rel_distinct(1)
thf(fact_5393_list_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: A,Y222: list @ A] :
~ ( list_all2 @ A @ B @ R @ ( cons @ A @ Y21 @ Y222 ) @ ( nil @ B ) ) ).
% list.rel_distinct(2)
thf(fact_5394_Misc_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: A > B > $o,L: list @ A,L2: list @ B,Q: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ P @ L @ L2 )
=> ( ( Q @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,X2: B,Ls: list @ A,Ls4: list @ B] :
( ( P @ X @ X2 )
=> ( ( list_all2 @ A @ B @ P @ Ls @ Ls4 )
=> ( ( Q @ Ls @ Ls4 )
=> ( Q @ ( cons @ A @ X @ Ls ) @ ( cons @ B @ X2 @ Ls4 ) ) ) ) )
=> ( Q @ L @ L2 ) ) ) ) ).
% Misc.list_all2_induct
thf(fact_5395_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_5396_relcomp__Image,axiom,
! [A: $tType,C: $tType,B: $tType,X5: set @ ( product_prod @ B @ C ),Y6: set @ ( product_prod @ C @ A ),Z7: set @ B] :
( ( image @ B @ A @ ( relcomp @ B @ C @ A @ X5 @ Y6 ) @ Z7 )
= ( image @ C @ A @ Y6 @ ( image @ B @ C @ X5 @ Z7 ) ) ) ).
% relcomp_Image
thf(fact_5397_list_Orel__mono,axiom,
! [B: $tType,A: $tType,R: A > B > $o,Ra2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra2 )
=> ( ord_less_eq @ ( ( list @ A ) > ( list @ B ) > $o ) @ ( list_all2 @ A @ B @ R ) @ ( list_all2 @ A @ B @ Ra2 ) ) ) ).
% list.rel_mono
thf(fact_5398_Image__Un,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A3: set @ B,B2: set @ B] :
( ( image @ B @ A @ R @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R @ A3 ) @ ( image @ B @ A @ R @ B2 ) ) ) ).
% Image_Un
thf(fact_5399_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_5400_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_5401_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_5402_list__all2__antisym,axiom,
! [A: $tType,P: A > A > $o,Q: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ! [X: A,Y2: A] :
( ( P @ X @ Y2 )
=> ( ( Q @ Y2 @ X )
=> ( X = Y2 ) ) )
=> ( ( list_all2 @ A @ A @ P @ Xs @ Ys )
=> ( ( list_all2 @ A @ A @ Q @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% list_all2_antisym
thf(fact_5403_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,Cs: list @ C] :
( ! [A6: A,B5: B,C5: C] :
( ( P1 @ A6 @ B5 )
=> ( ( P22 @ B5 @ C5 )
=> ( P32 @ A6 @ C5 ) ) )
=> ( ( list_all2 @ A @ B @ P1 @ As2 @ Bs )
=> ( ( list_all2 @ B @ C @ P22 @ Bs @ Cs )
=> ( list_all2 @ A @ C @ P32 @ As2 @ Cs ) ) ) ) ).
% list_all2_trans
thf(fact_5404_list__all2__refl,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ! [X: A] : ( P @ X @ X )
=> ( list_all2 @ A @ A @ P @ Xs @ Xs ) ) ).
% list_all2_refl
thf(fact_5405_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,Ys5: B] :
( ( P @ Xs2 @ Ys5 )
=> ( Q @ Xs2 @ Ys5 ) )
=> ( list_all2 @ A @ B @ Q @ Xs @ Ys ) ) ) ).
% list_all2_mono
thf(fact_5406_list__all2__eq,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z4: list @ A] : Y4 = Z4 )
= ( list_all2 @ A @ A
@ ^ [Y4: A,Z4: A] : Y4 = Z4 ) ) ).
% list_all2_eq
thf(fact_5407_list_Orel__refl,axiom,
! [B: $tType,Ra2: B > B > $o,X3: list @ B] :
( ! [X: B] : ( Ra2 @ X @ X )
=> ( list_all2 @ B @ B @ Ra2 @ X3 @ X3 ) ) ).
% list.rel_refl
thf(fact_5408_list_Orel__eq,axiom,
! [A: $tType] :
( ( list_all2 @ A @ A
@ ^ [Y4: A,Z4: A] : Y4 = Z4 )
= ( ^ [Y4: list @ A,Z4: list @ A] : Y4 = Z4 ) ) ).
% list.rel_eq
thf(fact_5409_list__all2__update__cong,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,X3: A,Y: B,I: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( P @ X3 @ Y )
=> ( list_all2 @ A @ B @ P @ ( list_update @ A @ Xs @ I @ X3 ) @ ( list_update @ B @ Ys @ I @ Y ) ) ) ) ).
% list_all2_update_cong
thf(fact_5410_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( mono @ A @ B )
= ( mono @ A @ B ) ) ) ).
% order.mono.cong
thf(fact_5411_list__all2__appendI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,A4: list @ A,B3: list @ B,C2: list @ A,D3: list @ B] :
( ( list_all2 @ A @ B @ P @ A4 @ B3 )
=> ( ( list_all2 @ A @ B @ P @ C2 @ D3 )
=> ( list_all2 @ A @ B @ P @ ( append @ A @ A4 @ C2 ) @ ( append @ B @ B3 @ D3 ) ) ) ) ).
% list_all2_appendI
thf(fact_5412_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_5413_list_Orel__cong,axiom,
! [A: $tType,B: $tType,X3: list @ A,Ya: list @ A,Y: list @ B,Xa: list @ B,R: A > B > $o,Ra2: A > B > $o] :
( ( X3 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( set2 @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( set2 @ B @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra2 @ Z3 @ Yb ) ) ) )
=> ( ( list_all2 @ A @ B @ R @ X3 @ Y )
= ( list_all2 @ A @ B @ Ra2 @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_5414_list_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X3: list @ A,Y: list @ B,Ra2: A > B > $o] :
( ( list_all2 @ A @ B @ R @ X3 @ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) )
=> ( ( member @ B @ Yb @ ( set2 @ B @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra2 @ Z3 @ Yb ) ) ) )
=> ( list_all2 @ A @ B @ Ra2 @ X3 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_5415_list_Orel__refl__strong,axiom,
! [A: $tType,X3: list @ A,Ra2: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) )
=> ( Ra2 @ Z3 @ Z3 ) )
=> ( list_all2 @ A @ A @ Ra2 @ X3 @ X3 ) ) ).
% list.rel_refl_strong
thf(fact_5416_list__all2__same,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( list_all2 @ A @ A @ P @ Xs @ Xs )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 @ X4 ) ) ) ) ).
% list_all2_same
thf(fact_5417_rev__ImageI,axiom,
! [B: $tType,A: $tType,A4: A,A3: set @ A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ A3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( member @ B @ B3 @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_5418_Image__iff,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A ),A3: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R2 @ A3 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ B3 ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_5419_ImageE,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A ),A3: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ R2 @ A3 ) )
=> ~ ! [X: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ B3 ) @ R2 )
=> ~ ( member @ B @ X @ A3 ) ) ) ).
% ImageE
thf(fact_5420_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 ) )
= ( ? [Z6: A,Zs3: list @ A] :
( ( Xs
= ( cons @ A @ Z6 @ Zs3 ) )
& ( P @ Z6 @ Y )
& ( list_all2 @ A @ B @ P @ Zs3 @ Ys ) ) ) ) ).
% list_all2_Cons2
thf(fact_5421_list__all2__Cons1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X3: A,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( cons @ A @ X3 @ Xs ) @ Ys )
= ( ? [Z6: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z6 @ Zs3 ) )
& ( P @ X3 @ Z6 )
& ( list_all2 @ A @ B @ P @ Xs @ Zs3 ) ) ) ) ).
% list_all2_Cons1
thf(fact_5422_list__all2__Cons,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X3: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( ( P @ X3 @ Y )
& ( list_all2 @ A @ B @ P @ Xs @ Ys ) ) ) ).
% list_all2_Cons
thf(fact_5423_list_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X222: list @ A,Y222: list @ B] :
( ( R @ X21 @ Y21 )
=> ( ( list_all2 @ A @ B @ R @ X222 @ Y222 )
=> ( list_all2 @ A @ B @ R @ ( cons @ A @ X21 @ X222 ) @ ( cons @ B @ Y21 @ Y222 ) ) ) ) ).
% list.rel_intros(2)
thf(fact_5424_list_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,X222: list @ A,Y21: B,Y222: list @ B] :
( ( list_all2 @ A @ B @ R @ ( cons @ A @ X21 @ X222 ) @ ( cons @ B @ Y21 @ Y222 ) )
= ( ( R @ X21 @ Y21 )
& ( list_all2 @ A @ B @ R @ X222 @ Y222 ) ) ) ).
% list.rel_inject(2)
thf(fact_5425_finite__Image__subset,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ B @ A ),B2: set @ B,C3: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ ( image @ B @ A @ A3 @ B2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ C3 @ A3 )
=> ( finite_finite2 @ A @ ( image @ B @ A @ C3 @ B2 ) ) ) ) ).
% finite_Image_subset
thf(fact_5426_rtrancl__reachable__induct,axiom,
! [A: $tType,I3: set @ A,INV: set @ A,E5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ I3 @ INV )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E5 @ INV ) @ INV )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ I3 ) @ INV ) ) ) ).
% rtrancl_reachable_induct
thf(fact_5427_rtrancl__image__unfold__right,axiom,
! [A: $tType,E5: set @ ( product_prod @ A @ A ),V: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E5 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ V ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ V ) ) ).
% rtrancl_image_unfold_right
thf(fact_5428_Image__mono,axiom,
! [B: $tType,A: $tType,R7: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),A8: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R7 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R7 @ A8 ) @ ( image @ A @ B @ R2 @ A3 ) ) ) ) ).
% Image_mono
thf(fact_5429_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_5430_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_5431_finite__Image,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),A3: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( finite_finite2 @ B @ ( image @ A @ B @ R @ A3 ) ) ) ).
% finite_Image
thf(fact_5432_Un__Image,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A ),A3: set @ B] :
( ( image @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R @ S ) @ A3 )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R @ A3 ) @ ( image @ B @ A @ S @ A3 ) ) ) ).
% Un_Image
thf(fact_5433_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B ),S7: set @ A] :
( collect @ B
@ ^ [Y5: B] :
? [X4: A] :
( ( member @ A @ X4 @ S7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% Image_def
thf(fact_5434_Image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),B2: C > ( set @ B ),A3: set @ C] :
( ( image @ B @ A @ R2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B2 @ A3 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R2 @ ( B2 @ X4 ) )
@ A3 ) ) ) ).
% Image_UN
thf(fact_5435_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_5436_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_5437_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A4: list @ A,B3: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ A4 ) )
=> ( P @ ( nth @ A @ A4 @ N5 ) @ ( nth @ B @ B3 @ N5 ) ) )
=> ( list_all2 @ A @ B @ P @ A4 @ B3 ) ) ) ).
% list_all2_all_nthI
thf(fact_5438_list__all2__conv__all__nth,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P2: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P2 @ ( nth @ A @ Xs3 @ I4 ) @ ( nth @ B @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_5439_Image__empty__rtrancl__Image__id,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),V2: A] :
( ( ( image @ A @ A @ R @ ( insert3 @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ ( insert3 @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert3 @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_5440_wfE__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( image @ A @ A @ R @ A3 ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% wfE_pf
thf(fact_5441_wfI__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A14: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A14 @ ( image @ A @ A @ R @ A14 ) )
=> ( A14
= ( bot_bot @ ( set @ A ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_pf
thf(fact_5442_Image__empty__trancl__Image__empty,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),V2: A] :
( ( ( image @ A @ A @ R @ ( insert3 @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_trancl @ A @ R ) @ ( insert3 @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_5443_reachable__mono,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),R8: set @ ( product_prod @ A @ A ),X5: set @ A,X15: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ R8 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ X15 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ X5 ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R8 ) @ X15 ) ) ) ) ).
% reachable_mono
thf(fact_5444_list_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R4: A > B > $o,A5: list @ A,B4: list @ B] :
( ( ( A5
= ( nil @ A ) )
= ( B4
= ( nil @ B ) ) )
& ( ( A5
!= ( nil @ A ) )
=> ( ( B4
!= ( nil @ B ) )
=> ( ( R4 @ ( hd @ A @ A5 ) @ ( hd @ B @ B4 ) )
& ( list_all2 @ A @ B @ R4 @ ( tl @ A @ A5 ) @ ( tl @ B @ B4 ) ) ) ) ) ) ) ) ).
% list.rel_sel
thf(fact_5445_Image__subset__snd__image,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ B @ A ),B2: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ A3 @ B2 ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A3 ) ) ).
% Image_subset_snd_image
thf(fact_5446_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_5447_Image__singleton,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A4: B] :
( ( image @ B @ A @ R2 @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A4 @ B4 ) @ R2 ) ) ) ).
% Image_singleton
thf(fact_5448_Image__subset,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),A3: set @ A,B2: set @ B,C3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R2 @ C3 ) @ B2 ) ) ).
% Image_subset
thf(fact_5449_Image__INT__subset,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),B2: C > ( set @ B ),A3: set @ C] :
( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B2 @ A3 ) ) )
@ ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R2 @ ( B2 @ X4 ) )
@ A3 ) ) ) ).
% Image_INT_subset
thf(fact_5450_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 )
@ ^ [X4: A,Ys3: list @ A] : ( member @ A @ X4 @ ( set2 @ A @ Ys3 ) )
@ Xs3
@ Xss2 ) ) ) ).
% product_lists_set
thf(fact_5451_listrel__Cons,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),X3: B,Xs: list @ B] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert3 @ ( list @ B ) @ ( cons @ B @ X3 @ Xs ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( set_Cons @ A @ ( image @ B @ A @ R2 @ ( insert3 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert3 @ ( list @ B ) @ Xs @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) ) ) ) ).
% listrel_Cons
thf(fact_5452_rtrancl__apply__insert,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X3: A,S: set @ A] :
( ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ ( insert3 @ A @ X3 @ S ) )
= ( insert3 @ A @ X3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ ( sup_sup @ ( set @ A ) @ S @ ( image @ A @ A @ R @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_5453_rtrancl__Image__in__Field,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),V: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ V ) @ ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R ) @ V ) ) ).
% rtrancl_Image_in_Field
thf(fact_5454_list__all2I,axiom,
! [A: $tType,B: $tType,A4: list @ A,B3: list @ B,P: A > B > $o] :
( ! [X: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ A4 @ B3 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P @ X ) )
=> ( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( list_all2 @ A @ B @ P @ A4 @ B3 ) ) ) ).
% list_all2I
thf(fact_5455_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V2: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_rtrancl @ A @ E5 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E5 @ R ) @ R )
=> ( ~ ( member @ A @ V2 @ R )
=> ~ ( ~ ( member @ A @ U @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E5 @ R ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_5456_rel__restrict__tranclI,axiom,
! [A: $tType,X3: A,Y: A,E5: set @ ( product_prod @ A @ A ),R: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ E5 ) )
=> ( ~ ( member @ A @ X3 @ R )
=> ( ~ ( member @ A @ Y @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E5 @ R ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E5 @ R ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_5457_Image__absorb__rtrancl,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B2: set @ A,C3: set @ A] :
( ( trans @ A @ A3 )
=> ( ( refl_on @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ B2 )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ A3 ) @ C3 )
= ( image @ A @ A @ A3 @ C3 ) ) ) ) ) ).
% Image_absorb_rtrancl
thf(fact_5458_refl__on__reflcl__Image,axiom,
! [A: $tType,B2: set @ A,A3: set @ ( product_prod @ A @ A ),C3: set @ A] :
( ( refl_on @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ B2 )
=> ( ( image @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ ( id2 @ A ) ) @ C3 )
= ( image @ A @ A @ A3 @ C3 ) ) ) ) ).
% refl_on_reflcl_Image
thf(fact_5459_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A ),B7: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : ( image @ B @ A @ R5 @ ( insert3 @ B @ Y5 @ ( bot_bot @ ( set @ B ) ) ) )
@ B7 ) ) ) ) ).
% Image_eq_UN
thf(fact_5460_Sigma__Image,axiom,
! [A: $tType,B: $tType,A3: set @ B,B2: B > ( set @ A ),X5: set @ B] :
( ( image @ B @ A @ ( product_Sigma @ B @ A @ A3 @ B2 ) @ X5 )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B2 @ ( inf_inf @ ( set @ B ) @ X5 @ A3 ) ) ) ) ).
% Sigma_Image
thf(fact_5461_UN__Image,axiom,
! [A: $tType,B: $tType,C: $tType,X5: C > ( set @ ( product_prod @ B @ A ) ),I3: set @ C,S: set @ B] :
( ( image @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ X5 @ I3 ) ) @ S )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [I4: C] : ( image @ B @ A @ ( X5 @ I4 ) @ S )
@ I3 ) ) ) ).
% UN_Image
thf(fact_5462_finite__reachable__advance,axiom,
! [A: $tType,E5: set @ ( product_prod @ A @ A ),V0: A,V2: A] :
( ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ ( insert3 @ A @ V0 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V0 @ V2 ) @ ( transitive_rtrancl @ A @ E5 ) )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ ( insert3 @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_5463_rtrancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V2: A,E5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ E5 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ ( insert3 @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ ( insert3 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_5464_trancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V2: A,E5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ E5 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E5 ) @ ( insert3 @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E5 ) @ ( insert3 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_5465_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V2: A,E5: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_trancl @ A @ E5 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E5 @ S ) @ S )
=> ( ( member @ A @ U @ S )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E5
@ ( product_Sigma @ A @ A @ S
@ ^ [Uu3: A] : S ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_5466_list__all2__iff,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P2: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [X4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P2 @ X4 ) ) ) ) ) ).
% list_all2_iff
thf(fact_5467_subset__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_5468_subset__mset_Omono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F3: ( multiset @ A ) > B,A3: C > ( multiset @ A ),I3: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A3 @ I3 ) )
=> ( ( I3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A3 @ X4 ) )
@ I3 ) )
@ ( F3 @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A3 @ I3 ) ) ) ) ) ) ) ) ).
% subset_mset.mono_cSUP
thf(fact_5469_trans__wf__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( wf @ A @ R2 )
= ( ! [A5: A] :
( wf @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( image @ A @ A @ ( converse @ A @ A @ R2 ) @ ( insert3 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) )
@ ^ [Uu3: A] : ( image @ A @ A @ ( converse @ A @ A @ R2 ) @ ( insert3 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% trans_wf_iff
thf(fact_5470_converse__inject,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S3: set @ ( product_prod @ B @ A )] :
( ( ( converse @ B @ A @ R2 )
= ( converse @ B @ A @ S3 ) )
= ( R2 = S3 ) ) ).
% converse_inject
thf(fact_5471_converse__converse,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( converse @ B @ A @ ( converse @ A @ B @ R2 ) )
= R2 ) ).
% converse_converse
thf(fact_5472_converse__iff,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( converse @ B @ A @ R2 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A4 ) @ R2 ) ) ).
% converse_iff
thf(fact_5473_Field__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( converse @ A @ A @ R2 ) )
= ( field2 @ A @ R2 ) ) ).
% Field_converse
thf(fact_5474_converse__mono,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S3: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S3 ) ) ).
% converse_mono
thf(fact_5475_converse__empty,axiom,
! [B: $tType,A: $tType] :
( ( converse @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% converse_empty
thf(fact_5476_trans__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ ( converse @ A @ A @ R2 ) )
= ( trans @ A @ R2 ) ) ).
% trans_converse
thf(fact_5477_converse__Id,axiom,
! [A: $tType] :
( ( converse @ A @ A @ ( id2 @ A ) )
= ( id2 @ A ) ) ).
% converse_Id
thf(fact_5478_refl__on__converse,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A3 @ ( converse @ A @ A @ R2 ) )
= ( refl_on @ A @ A3 @ R2 ) ) ).
% refl_on_converse
thf(fact_5479_finite__converse,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( converse @ B @ A @ R2 ) )
= ( finite_finite2 @ ( product_prod @ B @ A ) @ R2 ) ) ).
% finite_converse
thf(fact_5480_total__on__converse,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A3 @ ( converse @ A @ A @ R2 ) )
= ( total_on @ A @ A3 @ R2 ) ) ).
% total_on_converse
thf(fact_5481_converse__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( converse @ B @ A @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% converse_UNIV
thf(fact_5482_converse__Id__on,axiom,
! [A: $tType,A3: set @ A] :
( ( converse @ A @ A @ ( id_on @ A @ A3 ) )
= ( id_on @ A @ A3 ) ) ).
% converse_Id_on
thf(fact_5483_card__inverse,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A )] :
( ( finite_card @ ( product_prod @ A @ B ) @ ( converse @ B @ A @ R ) )
= ( finite_card @ ( product_prod @ B @ A ) @ R ) ) ).
% card_inverse
thf(fact_5484_pair__set__inverse,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( converse @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [B4: A,A5: B] : ( P @ A5 @ B4 ) ) ) ) ).
% pair_set_inverse
thf(fact_5485_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_5486_in__listrel1__converse,axiom,
! [A: $tType,X3: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( listrel1 @ A @ ( converse @ A @ A @ R2 ) ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y ) @ ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% in_listrel1_converse
thf(fact_5487_converse_Ocases,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R2 ) ) ).
% converse.cases
thf(fact_5488_converse_Osimps,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R2 ) )
= ( ? [A5: A,B4: B] :
( ( A1 = B4 )
& ( A22 = A5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R2 ) ) ) ) ).
% converse.simps
thf(fact_5489_converseD,axiom,
! [A: $tType,B: $tType,A4: A,B3: B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( converse @ B @ A @ R2 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A4 ) @ R2 ) ) ).
% converseD
thf(fact_5490_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R2: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R2 ) )
=> ~ ! [X: B,Y2: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y2 @ X ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y2 ) @ R2 ) ) ) ).
% converseE
thf(fact_5491_converseI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B3 @ A4 ) @ ( converse @ A @ B @ R2 ) ) ) ).
% converseI
thf(fact_5492_listrel1__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( listrel1 @ A @ ( converse @ A @ A @ R2 ) )
= ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1_converse
thf(fact_5493_converse__Int,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S3: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S3 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S3 ) ) ) ).
% converse_Int
thf(fact_5494_converse__Un,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S3: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R2 ) @ ( converse @ B @ A @ S3 ) ) ) ).
% converse_Un
thf(fact_5495_converse__subset__swap,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ ( converse @ B @ A @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R2 ) @ S3 ) ) ).
% converse_subset_swap
thf(fact_5496_converse__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ B @ C ),S3: set @ ( product_prod @ C @ A )] :
( ( converse @ B @ A @ ( relcomp @ B @ C @ A @ R2 @ S3 ) )
= ( relcomp @ A @ C @ B @ ( converse @ C @ A @ S3 ) @ ( converse @ B @ C @ R2 ) ) ) ).
% converse_relcomp
thf(fact_5497_rtrancl__converseI,axiom,
! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R2 ) ) ) ) ).
% rtrancl_converseI
thf(fact_5498_rtrancl__converseD,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ).
% rtrancl_converseD
thf(fact_5499_trancl__converseD,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_converseD
thf(fact_5500_trancl__converseI,axiom,
! [A: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R2 ) ) ) ) ).
% trancl_converseI
thf(fact_5501_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_5502_converse__unfold,axiom,
! [A: $tType,B: $tType] :
( ( converse @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [Y5: A,X4: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% converse_unfold
thf(fact_5503_converse__UNION,axiom,
! [B: $tType,A: $tType,C: $tType,R2: C > ( set @ ( product_prod @ B @ A ) ),S: set @ C] :
( ( converse @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: C] : ( converse @ B @ A @ ( R2 @ X4 ) )
@ S ) ) ) ).
% converse_UNION
thf(fact_5504_converse__INTER,axiom,
! [B: $tType,A: $tType,C: $tType,R2: C > ( set @ ( product_prod @ B @ A ) ),S: set @ C] :
( ( converse @ B @ A @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) ) )
= ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: C] : ( converse @ B @ A @ ( R2 @ X4 ) )
@ S ) ) ) ).
% converse_INTER
thf(fact_5505_Image__subset__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A3: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R2 @ A3 ) @ B2 )
= ( ord_less_eq @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ ( converse @ B @ A @ R2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) ) ) ) ) ).
% Image_subset_eq
thf(fact_5506_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_5507_subset__mset_OcSUP__le__iff,axiom,
! [A: $tType,B: $tType,A3: set @ B,F3: B > ( multiset @ A ),U: multiset @ A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) )
=> ( ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( subseteq_mset @ A @ ( F3 @ X4 ) @ U ) ) ) ) ) ) ).
% subset_mset.cSUP_le_iff
thf(fact_5508_subset__mset_OcSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType,A3: set @ B,G: C > ( multiset @ A ),B2: set @ C,F3: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G @ B2 ) )
=> ( ! [N5: B] :
( ( member @ B @ N5 @ A3 )
=> ? [X8: C] :
( ( member @ C @ X8 @ B2 )
& ( subseteq_mset @ A @ ( F3 @ N5 ) @ ( G @ X8 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G @ B2 ) ) ) ) ) ) ).
% subset_mset.cSUP_mono
thf(fact_5509_irrefl__tranclI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A] :
( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R2 ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ).
% irrefl_tranclI
thf(fact_5510_subset__mset_OcSUP__subset__mono,axiom,
! [A: $tType,B: $tType,A3: set @ B,G: B > ( multiset @ A ),B2: set @ B,F3: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B2 )
=> ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( subseteq_mset @ A @ ( F3 @ X ) @ ( G @ X ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ B2 ) ) ) ) ) ) ) ).
% subset_mset.cSUP_subset_mono
thf(fact_5511_subset__Image__Image__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B2: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ A3 ) @ ( image @ A @ A @ R2 @ B2 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R2 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_5512_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ B @ A ),A3: set @ C,B2: C > ( set @ B )] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R2 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( image @ B @ A @ R2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B2 @ A3 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R2 @ ( B2 @ X4 ) )
@ A3 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_5513_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A4 = B3 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_5514_Sup__multiset__in__multiset,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I4: A] :
( ord_less @ nat @ ( zero_zero @ nat )
@ ( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [M10: multiset @ A] : ( count @ A @ M10 @ I4 )
@ A3 ) ) ) ) ) ) ) ).
% Sup_multiset_in_multiset
thf(fact_5515_antisym__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( antisym @ A @ ( converse @ A @ A @ R2 ) )
= ( antisym @ A @ R2 ) ) ).
% antisym_converse
thf(fact_5516_mset__empty__count,axiom,
! [A: $tType,M5: multiset @ A] :
( ( ! [P7: A] :
( ( count @ A @ M5 @ P7 )
= ( zero_zero @ nat ) ) )
= ( M5
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% mset_empty_count
thf(fact_5517_count__greater__zero__iff,axiom,
! [A: $tType,M5: multiset @ A,X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ M5 @ X3 ) )
= ( member @ A @ X3 @ ( set_mset @ A @ M5 ) ) ) ).
% count_greater_zero_iff
thf(fact_5518_count__mset__0__iff,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( count @ A @ ( mset @ A @ Xs ) @ X3 )
= ( zero_zero @ nat ) )
= ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ) ).
% count_mset_0_iff
thf(fact_5519_antisym__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
=> ( ( antisym @ A @ S3 )
=> ( antisym @ A @ R2 ) ) ) ).
% antisym_subset
thf(fact_5520_single__valued__subset,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 )
=> ( ( single_valued @ A @ B @ S3 )
=> ( single_valued @ A @ B @ R2 ) ) ) ).
% single_valued_subset
thf(fact_5521_single__valued__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( single_valued @ A @ B @ R2 )
=> ( ( single_valued @ B @ C @ S3 )
=> ( single_valued @ A @ C @ ( relcomp @ A @ B @ C @ R2 @ S3 ) ) ) ) ).
% single_valued_relcomp
thf(fact_5522_antisym__empty,axiom,
! [A: $tType] : ( antisym @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% antisym_empty
thf(fact_5523_single__valued__empty,axiom,
! [B: $tType,A: $tType] : ( single_valued @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% single_valued_empty
thf(fact_5524_antisym__Id__on,axiom,
! [A: $tType,A3: set @ A] : ( antisym @ A @ ( id_on @ A @ A3 ) ) ).
% antisym_Id_on
thf(fact_5525_antisym__Id,axiom,
! [A: $tType] : ( antisym @ A @ ( id2 @ A ) ) ).
% antisym_Id
thf(fact_5526_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_5527_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_5528_single__valued__Id,axiom,
! [A: $tType] : ( single_valued @ A @ A @ ( id2 @ A ) ) ).
% single_valued_Id
thf(fact_5529_antisymD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R2 )
=> ( A4 = B3 ) ) ) ) ).
% antisymD
thf(fact_5530_antisymI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X ) @ R2 )
=> ( X = Y2 ) ) )
=> ( antisym @ A @ R2 ) ) ).
% antisymI
thf(fact_5531_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A,Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 )
=> ( X4 = Y5 ) ) ) ) ) ).
% antisym_def
thf(fact_5532_single__valuedD,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),X3: A,Y: B,Z2: B] :
( ( single_valued @ A @ B @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z2 ) @ R2 )
=> ( Y = Z2 ) ) ) ) ).
% single_valuedD
thf(fact_5533_single__valuedI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ! [X: A,Y2: B,Z3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z3 ) @ R2 )
=> ( Y2 = Z3 ) ) )
=> ( single_valued @ A @ B @ R2 ) ) ).
% single_valuedI
thf(fact_5534_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
! [X4: A,Y5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 )
=> ! [Z6: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z6 ) @ R5 )
=> ( Y5 = Z6 ) ) ) ) ) ).
% single_valued_def
thf(fact_5535_single__valued__Id__on,axiom,
! [A: $tType,A3: set @ A] : ( single_valued @ A @ A @ ( id_on @ A @ A3 ) ) ).
% single_valued_Id_on
thf(fact_5536_single__valued__confluent,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A,Z2: A] :
( ( single_valued @ A @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R2 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_5537_in__diff__count,axiom,
! [A: $tType,A4: A,M5: multiset @ A,N7: multiset @ A] :
( ( member @ A @ A4 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M5 @ N7 ) ) )
= ( ord_less @ nat @ ( count @ A @ N7 @ A4 ) @ ( count @ A @ M5 @ A4 ) ) ) ).
% in_diff_count
thf(fact_5538_count__ne__remove,axiom,
! [A: $tType,X3: A,T2: A,S: multiset @ A] :
( ( X3 != T2 )
=> ( ( count @ A @ S @ X3 )
= ( count @ A @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ X3 ) ) ) ).
% count_ne_remove
thf(fact_5539_antisym__singleton,axiom,
! [A: $tType,X3: product_prod @ A @ A] : ( antisym @ A @ ( insert3 @ ( product_prod @ A @ A ) @ X3 @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% antisym_singleton
thf(fact_5540_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_5541_set__mset__def,axiom,
! [A: $tType] :
( ( set_mset @ A )
= ( ^ [M10: multiset @ A] :
( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ M10 @ X4 ) ) ) ) ) ).
% set_mset_def
thf(fact_5542_set__mset__diff,axiom,
! [A: $tType,M5: multiset @ A,N7: multiset @ A] :
( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M5 @ N7 ) )
= ( collect @ A
@ ^ [A5: A] : ( ord_less @ nat @ ( count @ A @ N7 @ A5 ) @ ( count @ A @ M5 @ A5 ) ) ) ) ).
% set_mset_diff
thf(fact_5543_count__mset,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( count @ A @ ( mset @ A @ Xs ) @ X3 )
= ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y4: A,Z4: A] : Y4 = Z4
@ X3 )
@ Xs ) ) ) ).
% count_mset
thf(fact_5544_count,axiom,
! [A: $tType,X3: multiset @ A] :
( member @ ( A > nat ) @ ( count @ A @ X3 )
@ ( collect @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) ) ).
% count
thf(fact_5545_count__cases,axiom,
! [A: $tType,Y: A > nat] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
=> ~ ! [X: multiset @ A] :
( Y
!= ( count @ A @ X ) ) ) ).
% count_cases
thf(fact_5546_count__induct,axiom,
! [A: $tType,Y: A > nat,P: ( A > nat ) > $o] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
=> ( ! [X: multiset @ A] : ( P @ ( count @ A @ X ) )
=> ( P @ Y ) ) ) ).
% count_induct
thf(fact_5547_count__mset__gt__0,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ ( mset @ A @ Xs ) @ X3 ) ) ) ).
% count_mset_gt_0
thf(fact_5548_bijective__alt,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R4 )
& ( single_valued @ B @ A @ ( converse @ A @ B @ R4 ) ) ) ) ) ).
% bijective_alt
thf(fact_5549_Image__Int__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),A3: set @ B,B2: set @ B] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R ) )
=> ( ( image @ B @ A @ R @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) )
= ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R @ A3 ) @ ( image @ B @ A @ R @ B2 ) ) ) ) ).
% Image_Int_eq
thf(fact_5550_trans__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( trans @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ).
% trans_diff_Id
thf(fact_5551_distinct__count__atmost__1,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [X4: list @ A] :
! [A5: A] :
( ( ( member @ A @ A5 @ ( set2 @ A @ X4 ) )
=> ( ( count @ A @ ( mset @ A @ X4 ) @ A5 )
= ( one_one @ nat ) ) )
& ( ~ ( member @ A @ A5 @ ( set2 @ A @ X4 ) )
=> ( ( count @ A @ ( mset @ A @ X4 ) @ A5 )
= ( zero_zero @ nat ) ) ) ) ) ) ).
% distinct_count_atmost_1
thf(fact_5552_bdd__above__multiset__imp__finite__support,axiom,
! [A: $tType,A3: set @ ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 )
=> ( finite_finite2 @ A
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( multiset @ A ) @ ( set @ A )
@ ^ [X9: multiset @ A] :
( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ X9 @ X4 ) ) )
@ A3 ) ) ) ) ) ).
% bdd_above_multiset_imp_finite_support
thf(fact_5553_count__image__mset,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: multiset @ B,X3: A] :
( ( count @ A @ ( image_mset @ B @ A @ F3 @ A3 ) @ X3 )
= ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ A3 ) @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( set_mset @ B @ A3 ) ) ) ) ).
% count_image_mset
thf(fact_5554_count__Abs__multiset,axiom,
! [A: $tType,F3: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X4 ) ) ) )
=> ( ( count @ A @ ( abs_multiset @ A @ F3 ) )
= F3 ) ) ).
% count_Abs_multiset
thf(fact_5555_subset__mset_OcSUP__union,axiom,
! [A: $tType,B: $tType,A3: set @ B,F3: B > ( multiset @ A ),B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ B2 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) )
= ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ B2 ) ) ) ) ) ) ) ) ).
% subset_mset.cSUP_union
thf(fact_5556_mset__map__split__orig__le,axiom,
! [B: $tType,A: $tType,F3: B > A,P: multiset @ B,M1: multiset @ A,M22: multiset @ A] :
( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F3 @ P ) @ ( plus_plus @ ( multiset @ A ) @ M1 @ M22 ) )
=> ~ ! [P12: multiset @ B,P23: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P12 @ P23 ) )
=> ( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F3 @ P12 ) @ M1 )
=> ~ ( subseteq_mset @ A @ ( image_mset @ B @ A @ F3 @ P23 ) @ M22 ) ) ) ) ).
% mset_map_split_orig_le
thf(fact_5557_mset__map__id,axiom,
! [B: $tType,A: $tType,F3: B > A,G: A > B,X5: multiset @ A] :
( ! [X: A] :
( ( F3 @ ( G @ X ) )
= X )
=> ( ( image_mset @ B @ A @ F3 @ ( image_mset @ A @ B @ G @ X5 ) )
= X5 ) ) ).
% mset_map_id
thf(fact_5558_mset__map__split__orig,axiom,
! [B: $tType,A: $tType,F3: B > A,P: multiset @ B,M1: multiset @ A,M22: multiset @ A] :
( ( ( image_mset @ B @ A @ F3 @ P )
= ( plus_plus @ ( multiset @ A ) @ M1 @ M22 ) )
=> ~ ! [P12: multiset @ B,P23: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P12 @ P23 ) )
=> ( ( ( image_mset @ B @ A @ F3 @ P12 )
= M1 )
=> ( ( image_mset @ B @ A @ F3 @ P23 )
!= M22 ) ) ) ) ).
% mset_map_split_orig
thf(fact_5559_image__mset__cong__pair,axiom,
! [C: $tType,B: $tType,A: $tType,M5: multiset @ ( product_prod @ A @ B ),F3: A > B > C,G: A > B > C] :
( ! [X: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ ( set_mset @ ( product_prod @ A @ B ) @ M5 ) )
=> ( ( F3 @ X @ Y2 )
= ( G @ X @ Y2 ) ) )
=> ( ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ M5 )
= ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ G ) @ M5 ) ) ) ).
% image_mset_cong_pair
thf(fact_5560_Abs__multiset__inject,axiom,
! [A: $tType,X3: A > nat,Y: A > nat] :
( ( member @ ( A > nat ) @ X3
@ ( collect @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
=> ( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
=> ( ( ( abs_multiset @ A @ X3 )
= ( abs_multiset @ A @ Y ) )
= ( X3 = Y ) ) ) ) ).
% Abs_multiset_inject
thf(fact_5561_Abs__multiset__induct,axiom,
! [A: $tType,P: ( multiset @ A ) > $o,X3: multiset @ A] :
( ! [Y2: A > nat] :
( ( member @ ( A > nat ) @ Y2
@ ( collect @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
=> ( P @ ( abs_multiset @ A @ Y2 ) ) )
=> ( P @ X3 ) ) ).
% Abs_multiset_induct
thf(fact_5562_Abs__multiset__cases,axiom,
! [A: $tType,X3: multiset @ A] :
~ ! [Y2: A > nat] :
( ( X3
= ( abs_multiset @ A @ Y2 ) )
=> ~ ( member @ ( A > nat ) @ Y2
@ ( collect @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) ) ) ).
% Abs_multiset_cases
thf(fact_5563_subset__mset_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType,A3: set @ B,F3: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A3 ) )
=> ( ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A3 ) ) )
= ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A5: B] : ( union_mset @ A @ ( F3 @ A5 ) @ ( G @ A5 ) )
@ A3 ) ) ) ) ) ) ).
% subset_mset.SUP_sup_distrib
thf(fact_5564_Abs__multiset__inverse,axiom,
! [A: $tType,Y: A > nat] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
=> ( ( count @ A @ ( abs_multiset @ A @ Y ) )
= Y ) ) ).
% Abs_multiset_inverse
thf(fact_5565_subset__mset_OcSUP__insert,axiom,
! [A: $tType,B: $tType,A3: set @ B,F3: B > ( multiset @ A ),A4: B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ ( insert3 @ B @ A4 @ A3 ) ) )
= ( union_mset @ A @ ( F3 @ A4 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A3 ) ) ) ) ) ) ).
% subset_mset.cSUP_insert
thf(fact_5566_sum__mset__constant,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [Y: B,A3: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [X4: A] : Y
@ A3 ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( multiset @ A ) @ A3 ) ) @ Y ) ) ) ).
% sum_mset_constant
thf(fact_5567_takeWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( takeWhile @ A
@ ^ [Y5: A] : Y5 != X3
@ ( rev @ A @ Xs ) )
= ( rev @ A
@ ( tl @ A
@ ( dropWhile @ A
@ ^ [Y5: A] : Y5 != X3
@ Xs ) ) ) ) ) ) ).
% takeWhile_neq_rev
thf(fact_5568_type__definition__multiset,axiom,
! [A: $tType] :
( type_definition @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ ( abs_multiset @ A )
@ ( collect @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) ) ).
% type_definition_multiset
thf(fact_5569_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_5570_dropWhile__eq__Nil__conv,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P @ Xs )
= ( nil @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_5571_dropWhile__append1,axiom,
! [A: $tType,X3: A,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_5572_dropWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_5573_dropWhile__replicate,axiom,
! [A: $tType,P: A > $o,X3: A,N: nat] :
( ( ( P @ X3 )
=> ( ( dropWhile @ A @ P @ ( replicate @ A @ N @ X3 ) )
= ( nil @ A ) ) )
& ( ~ ( P @ X3 )
=> ( ( dropWhile @ A @ P @ ( replicate @ A @ N @ X3 ) )
= ( replicate @ A @ N @ X3 ) ) ) ) ).
% dropWhile_replicate
thf(fact_5574_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_5575_sum__mset__replicate__mset,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,A4: A] :
( ( comm_m7189776963980413722m_mset @ A @ ( replicate_mset @ A @ N @ A4 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ A4 ) ) ) ).
% sum_mset_replicate_mset
thf(fact_5576_dropWhile_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( dropWhile @ A @ P @ ( nil @ A ) )
= ( nil @ A ) ) ).
% dropWhile.simps(1)
thf(fact_5577_distinct__dropWhile,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( dropWhile @ A @ P @ Xs ) ) ) ).
% distinct_dropWhile
thf(fact_5578_set__dropWhileD,axiom,
! [A: $tType,X3: A,P: A > $o,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_5579_dropWhile__cong,axiom,
! [A: $tType,L: list @ A,K3: list @ A,P: A > $o,Q: A > $o] :
( ( L = K3 )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ L ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( dropWhile @ A @ P @ L )
= ( dropWhile @ A @ Q @ K3 ) ) ) ) ).
% dropWhile_cong
thf(fact_5580_dropWhile_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( ( P @ X3 )
=> ( ( dropWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( dropWhile @ A @ P @ Xs ) ) )
& ( ~ ( P @ X3 )
=> ( ( dropWhile @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_5581_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_5582_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_5583_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_5584_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_5585_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_5586_dropWhile__last,axiom,
! [A: $tType,X3: A,Xs: list @ A,P: A > $o] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( last @ A @ ( dropWhile @ A @ P @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% dropWhile_last
thf(fact_5587_dropWhile__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: B > A,Xs: list @ B] :
( ( dropWhile @ A @ P @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( dropWhile @ B @ ( comp @ A @ $o @ B @ P @ F3 ) @ Xs ) ) ) ).
% dropWhile_map
thf(fact_5588_remdups__adj__Cons_H,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y5: A] : Y5 = X3
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_5589_sum__mset__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( times @ A )
& ( semiring_0 @ B ) )
=> ! [F3: A > B,A3: multiset @ A,G: C > B,B2: multiset @ C] :
( ( times_times @ B @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ A @ B @ F3 @ A3 ) ) @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ C @ B @ G @ B2 ) ) )
= ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [I4: A] :
( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ C @ B
@ ^ [J2: C] : ( times_times @ B @ ( F3 @ I4 ) @ ( G @ J2 ) )
@ B2 ) )
@ A3 ) ) ) ) ).
% sum_mset_product
thf(fact_5590_sum__mset__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F3: B > A,M5: multiset @ B,C2: A] :
( ( times_times @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F3 @ M5 ) ) @ C2 )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C2 )
@ M5 ) ) ) ) ).
% sum_mset_distrib_right
thf(fact_5591_sum__mset__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F3: B > A,M5: multiset @ B] :
( ( times_times @ A @ C2 @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F3 @ M5 ) ) )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C2 @ ( F3 @ X4 ) )
@ M5 ) ) ) ) ).
% sum_mset_distrib_left
thf(fact_5592_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_5593_sum__mset__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [N7: multiset @ A,M5: multiset @ A] :
( ( subseteq_mset @ A @ N7 @ M5 )
=> ( ( comm_m7189776963980413722m_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M5 @ N7 ) )
= ( minus_minus @ A @ ( comm_m7189776963980413722m_mset @ A @ M5 ) @ ( comm_m7189776963980413722m_mset @ A @ N7 ) ) ) ) ) ).
% sum_mset_diff
thf(fact_5594_takeWhile__eq__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ ( dropWhile @ A @ P @ Xs ) ) )
=> ~ ( P @ X ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( filter2 @ A @ P @ Xs ) ) ) ).
% takeWhile_eq_filter
thf(fact_5595_dropWhile__eq__drop,axiom,
! [A: $tType] :
( ( dropWhile @ A )
= ( ^ [P2: A > $o,Xs3: list @ A] : ( drop @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs3 ) ) @ Xs3 ) ) ) ).
% dropWhile_eq_drop
thf(fact_5596_dropWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) )
& ( ~ ! [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
=> ( P @ X8 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append
thf(fact_5597_length__dropWhile__takeWhile,axiom,
! [A: $tType,X3: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_dropWhile_takeWhile
thf(fact_5598_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
@ ^ [X4: A] : X4 = Y
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_5599_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
@ ^ [Y5: A] :
( Y5
= ( last @ A @ Xs ) )
@ Ys ) ) ) ) ) ).
% remdups_adj_append''
thf(fact_5600_tl__remdups__adj,axiom,
! [A: $tType,Ys: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( tl @ A @ ( remdups_adj @ A @ Ys ) )
= ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X4: A] :
( X4
= ( hd @ A @ Ys ) )
@ ( tl @ A @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_5601_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_5602_dropWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( dropWhile @ A
@ ^ [Y5: A] : Y5 != X3
@ ( rev @ A @ Xs ) )
= ( cons @ A @ X3
@ ( rev @ A
@ ( takeWhile @ A
@ ^ [Y5: A] : Y5 != X3
@ Xs ) ) ) ) ) ) ).
% dropWhile_neq_rev
thf(fact_5603_sum__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C2: A,A3: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( if @ A @ ( Y = X4 ) @ C2 @ ( zero_zero @ A ) )
@ A3 ) )
= ( times_times @ A @ C2 @ ( semiring_1_of_nat @ A @ ( count @ B @ A3 @ Y ) ) ) ) ) ).
% sum_mset_delta'
thf(fact_5604_sum__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C2: A,A3: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( if @ A @ ( X4 = Y ) @ C2 @ ( zero_zero @ A ) )
@ A3 ) )
= ( times_times @ A @ C2 @ ( semiring_1_of_nat @ A @ ( count @ B @ A3 @ Y ) ) ) ) ) ).
% sum_mset_delta
thf(fact_5605_minus__multiset_Oabs__eq,axiom,
! [A: $tType,Xa: A > nat,X3: A > nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) )
@ X3
@ X3 )
=> ( ( minus_minus @ ( multiset @ A ) @ ( abs_multiset @ A @ Xa ) @ ( abs_multiset @ A @ X3 ) )
= ( abs_multiset @ A
@ ^ [A5: A] : ( minus_minus @ nat @ ( Xa @ A5 ) @ ( X3 @ A5 ) ) ) ) ) ) ).
% minus_multiset.abs_eq
thf(fact_5606_relation__of__def,axiom,
! [A: $tType] :
( ( order_relation_of @ A )
= ( ^ [P2: A > A > $o,A7: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 )
@ ( product_Sigma @ A @ A @ A7
@ ^ [Uu3: A] : A7 ) )
& ( P2 @ A5 @ B4 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_5607_stable__sort__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord3483353639454293061rt_key @ B @ A )
= ( ^ [Sk: ( B > A ) > ( list @ B ) > ( list @ B )] :
! [F7: B > A,Xs3: list @ B,K4: A] :
( ( filter2 @ B
@ ^ [Y5: B] :
( ( F7 @ Y5 )
= K4 )
@ ( Sk @ F7 @ Xs3 ) )
= ( filter2 @ B
@ ^ [Y5: B] :
( ( F7 @ Y5 )
= K4 )
@ Xs3 ) ) ) ) ) ).
% stable_sort_key_def
thf(fact_5608_zero__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) )
@ ^ [A5: A] : ( zero_zero @ nat )
@ ^ [A5: A] : ( zero_zero @ nat ) ) ).
% zero_multiset.rsp
thf(fact_5609_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_5610_add__mset_Oabs__eq,axiom,
! [A: $tType,X3: A > nat,Xa: A] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) )
@ X3
@ X3 )
=> ( ( add_mset @ A @ Xa @ ( abs_multiset @ A @ X3 ) )
= ( abs_multiset @ A
@ ^ [B4: A] : ( if @ nat @ ( B4 = Xa ) @ ( suc @ ( X3 @ B4 ) ) @ ( X3 @ B4 ) ) ) ) ) ).
% add_mset.abs_eq
thf(fact_5611_plus__multiset_Oabs__eq,axiom,
! [A: $tType,Xa: A > nat,X3: A > nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) )
@ X3
@ X3 )
=> ( ( plus_plus @ ( multiset @ A ) @ ( abs_multiset @ A @ Xa ) @ ( abs_multiset @ A @ X3 ) )
= ( abs_multiset @ A
@ ^ [A5: A] : ( plus_plus @ nat @ ( Xa @ A5 ) @ ( X3 @ A5 ) ) ) ) ) ) ).
% plus_multiset.abs_eq
thf(fact_5612_repeat__mset_Oabs__eq,axiom,
! [A: $tType,X3: A > nat,Xa: nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) )
@ X3
@ X3 )
=> ( ( repeat_mset @ A @ Xa @ ( abs_multiset @ A @ X3 ) )
= ( abs_multiset @ A
@ ^ [A5: A] : ( times_times @ nat @ Xa @ ( X3 @ A5 ) ) ) ) ) ).
% repeat_mset.abs_eq
thf(fact_5613_Inf__multiset_Oabs__eq,axiom,
! [A: $tType,X3: set @ ( A > nat )] :
( ( bNF_rel_set @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
@ X3
@ X3 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ ( A > nat ) @ ( multiset @ A ) @ ( abs_multiset @ A ) @ X3 ) )
= ( abs_multiset @ A
@ ^ [I4: A] :
( if @ nat
@ ( X3
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F7: A > nat] : ( F7 @ I4 )
@ X3 ) ) ) ) ) ) ).
% Inf_multiset.abs_eq
thf(fact_5614_filter__mset_Oabs__eq,axiom,
! [A: $tType,X3: A > nat,Xa: A > $o] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) )
@ X3
@ X3 )
=> ( ( filter_mset @ A @ Xa @ ( abs_multiset @ A @ X3 ) )
= ( abs_multiset @ A
@ ^ [X4: A] : ( if @ nat @ ( Xa @ X4 ) @ ( X3 @ X4 ) @ ( zero_zero @ nat ) ) ) ) ) ).
% filter_mset.abs_eq
thf(fact_5615_Inf__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( set @ ( A > nat ) ) @ ( set @ ( A > nat ) ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_rel_set @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
@ ^ [A7: set @ ( A > nat ),I4: A] :
( if @ nat
@ ( A7
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F7: A > nat] : ( F7 @ I4 )
@ A7 ) ) )
@ ^ [A7: set @ ( A > nat ),I4: A] :
( if @ nat
@ ( A7
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F7: A > nat] : ( F7 @ I4 )
@ A7 ) ) ) ) ).
% Inf_multiset.rsp
thf(fact_5616_empty__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_set @ A @ B @ A3 @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ B ) ) ) ).
% empty_transfer
thf(fact_5617_shuffles_Opelims,axiom,
! [A: $tType,X3: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Xa ) )
=> ( ( ( X3
= ( nil @ A ) )
=> ( ( Y
= ( insert3 @ ( 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
= ( insert3 @ ( list @ A ) @ X3 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) ) ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y2: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ Y2 @ Ys5 ) )
=> ( ( Y
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y2 @ Ys5 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y2 ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs2 ) @ Ys5 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys5 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_5618_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 ) )
=> ( ! [Ys5: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys5 ) )
=> ( P @ ( nil @ A ) @ Ys5 ) )
=> ( ! [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 ) ) )
=> ( ! [X: A,Xs2: list @ A,Y2: A,Ys5: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys5 ) ) )
=> ( ( P @ Xs2 @ ( cons @ A @ Y2 @ Ys5 ) )
=> ( ( P @ ( cons @ A @ X @ Xs2 ) @ Ys5 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys5 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_5619_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R: A > B > $o] :
( ( R @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R @ ( bNF_rel_fun @ A @ B @ A @ B @ R @ R ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ R )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_5620_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_mult @ B )
& ( monoid_mult @ A ) )
=> ! [A3: A > B > $o] :
( ( A3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A3 @ ( bNF_rel_fun @ A @ B @ A @ B @ A3 @ A3 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A3 ) @ A3 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% prod_list_transfer
thf(fact_5621_foldr__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A3: A > C > $o,B2: 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 ) @ A3 @ ( bNF_rel_fun @ B @ D @ B @ D @ B2 @ B2 ) ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ C ) @ ( B > B ) @ ( D > D ) @ ( list_all2 @ A @ C @ A3 ) @ ( bNF_rel_fun @ B @ D @ B @ D @ B2 @ B2 ) ) @ ( foldr @ A @ B ) @ ( foldr @ C @ D ) ) ).
% foldr_transfer
thf(fact_5622_foldl__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,B2: A > C > $o,A3: 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 ) @ B2 @ ( bNF_rel_fun @ B @ D @ A @ C @ A3 @ B2 ) ) @ ( bNF_rel_fun @ A @ C @ ( ( list @ B ) > A ) @ ( ( list @ D ) > C ) @ B2 @ ( bNF_rel_fun @ ( list @ B ) @ ( list @ D ) @ A @ C @ ( list_all2 @ B @ D @ A3 ) @ B2 ) ) @ ( foldl @ A @ B ) @ ( foldl @ C @ D ) ) ).
% foldl_transfer
thf(fact_5623_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A3: A > B > $o,B2: C > D > $o] :
( ( A3 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A3 @ ( bNF_rel_fun @ A @ B @ A @ B @ A3 @ A3 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A3 @ ( bNF_rel_fun @ A @ B @ A @ B @ A3 @ A3 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B2 @ A3 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A3 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B2 ) @ A3 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_5624_list_Omap__transfer,axiom,
! [A: $tType,B: $tType,F6: $tType,E: $tType,Rb2: A > E > $o,Sd: B > F6 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F6 ) @ ( ( list @ A ) > ( list @ B ) ) @ ( ( list @ E ) > ( list @ F6 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F6 @ Rb2 @ Sd ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ E ) @ ( list @ B ) @ ( list @ F6 ) @ ( list_all2 @ A @ E @ Rb2 ) @ ( list_all2 @ B @ F6 @ Sd ) ) @ ( map @ A @ B ) @ ( map @ E @ F6 ) ) ).
% list.map_transfer
thf(fact_5625_List_Ofold__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A3: A > C > $o,B2: 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 ) @ A3 @ ( bNF_rel_fun @ B @ D @ B @ D @ B2 @ B2 ) ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ C ) @ ( B > B ) @ ( D > D ) @ ( list_all2 @ A @ C @ A3 ) @ ( bNF_rel_fun @ B @ D @ B @ D @ B2 @ B2 ) ) @ ( fold @ A @ B ) @ ( fold @ C @ D ) ) ).
% List.fold_transfer
thf(fact_5626_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 )
= ( insert3 @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(1)
thf(fact_5627_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 ) )
= ( insert3 @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(2)
thf(fact_5628_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X3: 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 @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) )
=> ( ( shuffles @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs ) @ Ys ) ) ) ) ) ).
% shuffles.psimps(3)
thf(fact_5629_revg_Opelims,axiom,
! [A: $tType,X3: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Xa ) )
=> ( ( ( X3
= ( 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] :
( ( X3
= ( 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_5630_merge_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( merge @ A @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Xa ) )
=> ( ( ( X3
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ( ! [V3: A,Va: list @ A] :
( ( X3
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( cons @ A @ V3 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X12: A,L1: list @ A] :
( ( X3
= ( cons @ A @ X12 @ L1 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ( ( ( ( ord_less @ A @ X12 @ X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L1 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( ( X12 = X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L1 @ L22 ) ) ) )
& ( ( X12 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X12 @ L1 ) @ L22 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ L1 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ) ).
% merge.pelims
thf(fact_5631_merge__list_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X3 @ Xa ) )
=> ( ( ( X3
= ( 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 ) ) ) ) ) ) )
=> ( ( ( X3
= ( nil @ ( list @ A ) ) )
=> ! [L3: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y = L3 )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( X3
= ( 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 )] :
( ( X3
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ! [L3: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ~ ! [L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) )
=> ( ( Y
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ X3 ) @ Ls ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X3 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
thf(fact_5632_shuffles__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( list @ A ) > ( set @ ( list @ A ) ) ) @ ( ( list @ B ) > ( set @ ( list @ B ) ) ) @ ( list_all2 @ A @ B @ A3 ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( set @ ( list @ A ) ) @ ( set @ ( list @ B ) ) @ ( list_all2 @ A @ B @ A3 ) @ ( bNF_rel_set @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) ) ) @ ( shuffles @ A ) @ ( shuffles @ B ) ) ).
% shuffles_transfer
thf(fact_5633_set__Cons__transfer,axiom,
! [A: $tType,B: $tType,A3: 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 @ A3 ) @ ( 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 @ A3 ) ) @ ( bNF_rel_set @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) ) ) @ ( set_Cons @ A ) @ ( set_Cons @ B ) ) ).
% set_Cons_transfer
thf(fact_5634_nths__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( set @ nat ) > ( list @ A ) ) @ ( ( set @ nat ) > ( list @ B ) ) @ ( list_all2 @ A @ B @ A3 )
@ ( bNF_rel_fun @ ( set @ nat ) @ ( set @ nat ) @ ( list @ A ) @ ( list @ B )
@ ( bNF_rel_set @ nat @ nat
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4 )
@ ( list_all2 @ A @ B @ A3 ) )
@ ( nths @ A )
@ ( nths @ B ) ) ).
% nths_transfer
thf(fact_5635_Lifting__Set_Oinsert__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( ( set @ A ) > ( set @ A ) ) @ ( ( set @ B ) > ( set @ B ) ) @ A3 @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A3 ) @ ( bNF_rel_set @ A @ B @ A3 ) ) @ ( insert3 @ A ) @ ( insert3 @ B ) ) ).
% Lifting_Set.insert_transfer
thf(fact_5636_dropWhile__transfer,axiom,
! [A: $tType,B: $tType,A3: 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 @ A3
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) )
@ ( dropWhile @ A )
@ ( dropWhile @ B ) ) ).
% dropWhile_transfer
thf(fact_5637_tl__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) @ ( tl @ A ) @ ( tl @ B ) ) ).
% tl_transfer
thf(fact_5638_less__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( ord_less @ nat )
@ ( ord_less @ nat ) ) ).
% less_natural.rsp
thf(fact_5639_rev__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) @ ( rev @ A ) @ ( rev @ B ) ) ).
% rev_transfer
thf(fact_5640_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_5641_List_Ofilter__transfer,axiom,
! [A: $tType,B: $tType,A3: 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 @ A3
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) )
@ ( filter2 @ A )
@ ( filter2 @ B ) ) ).
% List.filter_transfer
thf(fact_5642_takeWhile__transfer,axiom,
! [A: $tType,B: $tType,A3: 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 @ A3
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) )
@ ( takeWhile @ A )
@ ( takeWhile @ B ) ) ).
% takeWhile_transfer
thf(fact_5643_replicate__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( A > ( list @ A ) ) @ ( B > ( list @ B ) )
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ ( bNF_rel_fun @ A @ B @ ( list @ A ) @ ( list @ B ) @ A3 @ ( list_all2 @ A @ B @ A3 ) )
@ ( replicate @ A )
@ ( replicate @ B ) ) ).
% replicate_transfer
thf(fact_5644_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_5645_append__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ ( list_all2 @ A @ B @ A3 ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) ) @ ( append @ A ) @ ( append @ B ) ) ).
% append_transfer
thf(fact_5646_rotate__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) )
@ ( rotate @ A )
@ ( rotate @ B ) ) ).
% rotate_transfer
thf(fact_5647_list__update__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( nat > A > ( list @ A ) ) @ ( nat > B > ( list @ B ) ) @ ( list_all2 @ A @ B @ A3 )
@ ( bNF_rel_fun @ nat @ nat @ ( A > ( list @ A ) ) @ ( B > ( list @ B ) )
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ ( bNF_rel_fun @ A @ B @ ( list @ A ) @ ( list @ B ) @ A3 @ ( list_all2 @ A @ B @ A3 ) ) )
@ ( list_update @ A )
@ ( list_update @ B ) ) ).
% list_update_transfer
thf(fact_5648_list__ex__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > $o ) @ ( ( list @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A3 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( list_ex @ A )
@ ( list_ex @ B ) ) ).
% list_ex_transfer
thf(fact_5649_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
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 ) )
@ ( 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 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 ) )
@ ( list_all2 @ A @ B )
@ ( list_all2 @ C @ D ) ) ).
% list.rel_transfer
thf(fact_5650_length__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ nat @ nat @ ( list_all2 @ A @ B @ A3 )
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ ( size_size @ ( list @ A ) )
@ ( size_size @ ( list @ B ) ) ) ).
% length_transfer
thf(fact_5651_take__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) )
@ ( take @ A )
@ ( take @ B ) ) ).
% take_transfer
thf(fact_5652_drop__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ nat @ nat @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) )
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) )
@ ( drop @ A )
@ ( drop @ B ) ) ).
% drop_transfer
thf(fact_5653_concat__transfer,axiom,
! [A: $tType,B: $tType,A3: 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 @ A3 ) ) @ ( list_all2 @ A @ B @ A3 ) @ ( concat @ A ) @ ( concat @ B ) ) ).
% concat_transfer
thf(fact_5654_butlast__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) @ ( butlast @ A ) @ ( butlast @ B ) ) ).
% butlast_transfer
thf(fact_5655_filter__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y4: A > $o,Z4: A > $o] : Y4 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
@ ^ [P2: A > $o,M10: A > nat,X4: A] : ( if @ nat @ ( P2 @ X4 ) @ ( M10 @ X4 ) @ ( zero_zero @ nat ) )
@ ^ [P2: A > $o,M10: A > nat,X4: A] : ( if @ nat @ ( P2 @ X4 ) @ ( M10 @ X4 ) @ ( zero_zero @ nat ) ) ) ).
% filter_mset.rsp
thf(fact_5656_rotate1__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) @ ( rotate1 @ A ) @ ( rotate1 @ B ) ) ).
% rotate1_transfer
thf(fact_5657_subseqs__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ ( list @ A ) ) @ ( list @ ( list @ B ) ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) ) @ ( subseqs @ A ) @ ( subseqs @ B ) ) ).
% subseqs_transfer
thf(fact_5658_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 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4
@ ^ [List2: list @ A] :
( List2
= ( nil @ A ) )
@ ^ [List2: list @ B] :
( List2
= ( nil @ B ) ) ) ).
% list.disc_transfer(1)
thf(fact_5659_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 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4
@ ^ [List2: list @ A] :
( List2
!= ( nil @ A ) )
@ ^ [List2: list @ B] :
( List2
!= ( nil @ B ) ) ) ).
% list.disc_transfer(2)
thf(fact_5660_product__lists__transfer,axiom,
! [A: $tType,B: $tType,A3: 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 @ A3 ) ) @ ( list_all2 @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) ) @ ( product_lists @ A ) @ ( product_lists @ B ) ) ).
% product_lists_transfer
thf(fact_5661_add__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ A @ A @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y4: A,Z4: A] : Y4 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
@ ^ [A5: A,M10: A > nat,B4: A] : ( if @ nat @ ( B4 = A5 ) @ ( suc @ ( M10 @ B4 ) ) @ ( M10 @ B4 ) )
@ ^ [A5: A,M10: A > nat,B4: A] : ( if @ nat @ ( B4 = A5 ) @ ( suc @ ( M10 @ B4 ) ) @ ( M10 @ B4 ) ) ) ).
% add_mset.rsp
thf(fact_5662_plus__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
@ ^ [M10: A > nat,N9: A > nat,A5: A] : ( plus_plus @ nat @ ( M10 @ A5 ) @ ( N9 @ A5 ) )
@ ^ [M10: A > nat,N9: A > nat,A5: A] : ( plus_plus @ nat @ ( M10 @ A5 ) @ ( N9 @ A5 ) ) ) ).
% plus_multiset.rsp
thf(fact_5663_minus__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
@ ^ [M10: A > nat,N9: A > nat,A5: A] : ( minus_minus @ nat @ ( M10 @ A5 ) @ ( N9 @ A5 ) )
@ ^ [M10: A > nat,N9: A > nat,A5: A] : ( minus_minus @ nat @ ( M10 @ A5 ) @ ( N9 @ A5 ) ) ) ).
% minus_multiset.rsp
thf(fact_5664_repeat__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) )
@ ^ [N2: nat,M10: A > nat,A5: A] : ( times_times @ nat @ N2 @ ( M10 @ A5 ) )
@ ^ [N2: nat,M10: A > nat,A5: A] : ( times_times @ nat @ N2 @ ( M10 @ A5 ) ) ) ).
% repeat_mset.rsp
thf(fact_5665_null__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A3 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4
@ ( null @ A )
@ ( null @ B ) ) ).
% null_transfer
thf(fact_5666_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_5667_lists__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ ( list @ A ) ) @ ( set @ ( list @ B ) ) @ ( bNF_rel_set @ A @ B @ A3 ) @ ( bNF_rel_set @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) ) @ ( lists @ A ) @ ( lists @ B ) ) ).
% lists_transfer
thf(fact_5668_listset__transfer,axiom,
! [A: $tType,B: $tType,A3: 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 @ A3 ) ) @ ( bNF_rel_set @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) ) @ ( listset @ A ) @ ( listset @ B ) ) ).
% listset_transfer
thf(fact_5669_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_5670_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_5671_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_5672_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_5673_merge__list_Opinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A0: list @ ( list @ A ),A1: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ A0 @ A1 ) )
=> ( ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( P @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [L3: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A ),L3: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) ) )
=> ( ! [Acc2: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc2 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) )
=> ( ( P @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ Acc2 ) @ Ls )
=> ( P @ Acc2 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ).
% merge_list.pinduct
thf(fact_5674_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc22: list @ ( list @ A ),L12: list @ A,L23: list @ A,Ls3: 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 @ Ls3 ) ) ) )
=> ( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls3 ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ Acc22 ) @ Ls3 ) ) ) ) ).
% merge_list.psimps(5)
thf(fact_5675_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_5676_times__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U3 ) @ ( times_times @ nat @ Y5 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V4 ) @ ( times_times @ nat @ Y5 @ U3 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_5677_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 ) )
=> ( ! [Ys5: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys5 ) )
=> ( P @ ( nil @ A ) @ Ys5 ) )
=> ( ! [X: A,Xs2: list @ A,Ys5: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Ys5 ) )
=> ( ( P @ Ys5 @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ Ys5 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_5678_list_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F23: A > ( list @ A ) > C > C] :
( ( rec_list @ C @ A @ F1 @ F23 @ ( nil @ A ) )
= F1 ) ).
% list.simps(6)
thf(fact_5679_list_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F23: A > ( list @ A ) > C > C,X21: A,X222: list @ A] :
( ( rec_list @ C @ A @ F1 @ F23 @ ( cons @ A @ X21 @ X222 ) )
= ( F23 @ X21 @ X222 @ ( rec_list @ C @ A @ F1 @ F23 @ X222 ) ) ) ).
% list.simps(7)
thf(fact_5680_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_5681_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ pcr_int
@ ^ [N2: nat] : ( product_Pair @ nat @ nat @ N2 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_5682_uminus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X4 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_5683_list_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: C,Ga: B > ( list @ B ) > C > C,F3: A > B] :
( ( comp @ ( list @ B ) @ C @ ( list @ A ) @ ( rec_list @ C @ B @ G @ Ga ) @ ( map @ A @ B @ F3 ) )
= ( rec_list @ C @ A @ G
@ ^ [X4: A,Xa5: list @ A] : ( Ga @ ( F3 @ X4 ) @ ( map @ A @ B @ F3 @ Xa5 ) ) ) ) ).
% list.rec_o_map
thf(fact_5684_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_5685_of__int_Otransfer,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
@ ^ [Y4: A,Z4: A] : Y4 = Z4
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J2 ) ) )
@ ( ring_1_of_int @ A ) ) ) ).
% of_int.transfer
thf(fact_5686_set__rec,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( rec_list @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X4: A,Uu3: list @ A] : ( insert3 @ A @ X4 ) ) ) ).
% set_rec
thf(fact_5687_map__rec,axiom,
! [A: $tType,B: $tType] :
( ( map @ B @ A )
= ( ^ [F7: B > A] :
( rec_list @ ( list @ A ) @ B @ ( nil @ A )
@ ^ [X4: B,Uu3: list @ B] : ( cons @ A @ ( F7 @ X4 ) ) ) ) ) ).
% map_rec
thf(fact_5688_less__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ U3 @ Y5 ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_5689_plus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U3 ) @ ( plus_plus @ nat @ Y5 @ V4 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_5690_minus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ Y5 @ U3 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_5691_splice_Opelims,axiom,
! [A: $tType,X3: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Xa ) )
=> ( ( ( X3
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ Xs2 ) )
=> ( ( Y
= ( cons @ A @ X @ ( splice @ A @ Xa @ Xs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_5692_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_5693_splice_Opsimps_I2_J,axiom,
! [A: $tType,X3: 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 @ X3 @ Xs ) @ Ys ) )
=> ( ( splice @ A @ ( cons @ A @ X3 @ Xs ) @ Ys )
= ( cons @ A @ X3 @ ( splice @ A @ Ys @ Xs ) ) ) ) ).
% splice.psimps(2)
thf(fact_5694_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_5695_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_5696_splice__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( member @ ( list @ A ) @ ( splice @ A @ Xs @ Ys ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% splice_in_shuffles
thf(fact_5697_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_5698_splice__replicate,axiom,
! [A: $tType,M2: nat,X3: A,N: nat] :
( ( splice @ A @ ( replicate @ A @ M2 @ X3 ) @ ( replicate @ A @ N @ X3 ) )
= ( replicate @ A @ ( plus_plus @ nat @ M2 @ N ) @ X3 ) ) ).
% splice_replicate
thf(fact_5699_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_5700_splice_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys: list @ A] :
( ( splice @ A @ ( cons @ A @ X3 @ Xs ) @ Ys )
= ( cons @ A @ X3 @ ( splice @ A @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_5701_splice_Oelims,axiom,
! [A: $tType,X3: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X3 @ Xa )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ Xs2 ) )
=> ( Y
!= ( cons @ A @ X @ ( splice @ A @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_5702_splice__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ ( list_all2 @ A @ B @ A3 ) @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) ) @ ( splice @ A ) @ ( splice @ B ) ) ).
% splice_transfer
thf(fact_5703_times__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U3 ) @ ( times_times @ nat @ Y5 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V4 ) @ ( times_times @ nat @ Y5 @ U3 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U3 ) @ ( times_times @ nat @ Y5 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V4 ) @ ( times_times @ nat @ Y5 @ U3 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_5704_funpow__inj__finite,axiom,
! [A: $tType,P4: A > A,X3: A] :
( ( inj_on @ A @ A @ P4 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y5: A] :
? [N2: nat] :
( Y5
= ( compow @ ( A > A ) @ N2 @ P4 @ X3 ) ) ) )
=> ~ ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( compow @ ( A > A ) @ N5 @ P4 @ X3 )
!= X3 ) ) ) ) ).
% funpow_inj_finite
thf(fact_5705_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I3: set @ A,F: A > ( set @ B ),F3: B > C,G5: D > ( set @ C ),J4: set @ D] :
( ( I3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I5: A] :
( ( member @ A @ I5 @ I3 )
=> ! [J3: A] :
( ( member @ A @ J3 @ I3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F @ I5 ) @ ( F @ J3 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F @ J3 ) @ ( F @ I5 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F3
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J2: D] : ( principal @ C @ ( G5 @ J2 ) )
@ J4 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I4: A] : ( principal @ B @ ( F @ I4 ) )
@ I3 ) ) )
= ( ! [X4: D] :
( ( member @ D @ X4 @ J4 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ I3 )
& ! [Z6: B] :
( ( member @ B @ Z6 @ ( F @ Y5 ) )
=> ( member @ C @ ( F3 @ Z6 ) @ ( G5 @ X4 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_5706_inj__on__rev,axiom,
! [A: $tType,A3: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ A3 ) ).
% inj_on_rev
thf(fact_5707_inj__on__empty,axiom,
! [B: $tType,A: $tType,F3: A > B] : ( inj_on @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_5708_inj__map__eq__map,axiom,
! [B: $tType,A: $tType,F3: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ F3 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_5709_intrel__iff,axiom,
! [X3: nat,Y: nat,U: nat,V2: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X3 @ Y ) @ ( product_Pair @ nat @ nat @ U @ V2 ) )
= ( ( plus_plus @ nat @ X3 @ V2 )
= ( plus_plus @ nat @ U @ Y ) ) ) ).
% intrel_iff
thf(fact_5710_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A4: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A4 ) @ ( top_top @ ( set @ A ) ) )
= ( A4
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_5711_inj__mapI,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F3 ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ) ).
% inj_mapI
thf(fact_5712_inj__map,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F3 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
= ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_map
thf(fact_5713_inj__on__insert,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: A,A3: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert3 @ A @ A4 @ A3 ) )
= ( ( inj_on @ A @ B @ F3 @ A3 )
& ~ ( member @ B @ ( F3 @ A4 ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_5714_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ ( list @ A )] :
( ( inj_on @ A @ B @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A3 ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F3 ) @ A3 ) ) ).
% inj_on_mapI
thf(fact_5715_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X3: B,B2: set @ B] :
( ( inj_on @ A @ B @ F3 @ A3 )
=> ( ~ ( member @ B @ X3 @ B2 )
=> ( ( ( insert3 @ B @ X3 @ B2 )
= ( image2 @ A @ B @ F3 @ A3 ) )
=> ~ ! [X2: A,A12: set @ A] :
( ~ ( member @ A @ X2 @ A12 )
=> ( ( A3
= ( insert3 @ A @ X2 @ A12 ) )
=> ( ( X3
= ( F3 @ X2 ) )
=> ( B2
!= ( image2 @ A @ B @ F3 @ A12 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_5716_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,B2: set @ A,A3: set @ A] :
( ( inj_on @ A @ B @ F3 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A3 ) @ ( image2 @ A @ B @ F3 @ B2 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_5717_map__injective,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ B @ A @ F3 @ Ys ) )
=> ( ( inj_on @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_5718_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F3: A > B] :
( ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( F3 @ X )
!= ( F3 @ Y2 ) ) )
=> ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_5719_inj__on__Cons1,axiom,
! [A: $tType,X3: A,A3: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ A3 ) ).
% inj_on_Cons1
thf(fact_5720_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X4: A] : ( product_Pair @ B @ A @ ( C2 @ X4 ) @ X4 )
@ S ) ).
% inj_Pair(2)
thf(fact_5721_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( C2 @ X4 ) )
@ S ) ).
% inj_Pair(1)
thf(fact_5722_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F3: A > B,X5: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( F3 @ X4 ) )
@ X5 ) ).
% inj_on_convol_ident
thf(fact_5723_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X4: A] : X4
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_5724_filterlim__ident,axiom,
! [A: $tType,F: filter @ A] :
( filterlim @ A @ A
@ ^ [X4: A] : X4
@ F
@ F ) ).
% filterlim_ident
thf(fact_5725_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G: A > B,F32: filter @ B,F22: filter @ A,F3: C > A,F12: filter @ C] :
( ( filterlim @ A @ B @ G @ F32 @ F22 )
=> ( ( filterlim @ C @ A @ F3 @ F22 @ F12 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( G @ ( F3 @ X4 ) )
@ F32
@ F12 ) ) ) ).
% filterlim_compose
thf(fact_5726_filterlim__sequentially__Suc,axiom,
! [A: $tType,F3: nat > A,F: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X4: nat] : ( F3 @ ( suc @ X4 ) )
@ F
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F3 @ F @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_5727_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_5728_folding__insort__key_Oinj__on,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( inj_on @ B @ A @ F3 @ S ) ) ).
% folding_insort_key.inj_on
thf(fact_5729_inj__on__diff,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ A3 )
=> ( inj_on @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% inj_on_diff
thf(fact_5730_filterlim__top,axiom,
! [B: $tType,A: $tType,F3: A > B,F: filter @ A] : ( filterlim @ A @ B @ F3 @ ( top_top @ ( filter @ B ) ) @ F ) ).
% filterlim_top
thf(fact_5731_filterlim__sup,axiom,
! [B: $tType,A: $tType,F3: A > B,F: filter @ B,F12: filter @ A,F22: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F @ F12 )
=> ( ( filterlim @ A @ B @ F3 @ F @ F22 )
=> ( filterlim @ A @ B @ F3 @ F @ ( sup_sup @ ( filter @ A ) @ F12 @ F22 ) ) ) ) ).
% filterlim_sup
thf(fact_5732_filterlim__inf,axiom,
! [B: $tType,A: $tType,F3: A > B,F22: filter @ B,F32: filter @ B,F12: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( inf_inf @ ( filter @ B ) @ F22 @ F32 ) @ F12 )
= ( ( filterlim @ A @ B @ F3 @ F22 @ F12 )
& ( filterlim @ A @ B @ F3 @ F32 @ F12 ) ) ) ).
% filterlim_inf
thf(fact_5733_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A4: A,A3: set @ A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A4 ) @ A3 ) ) ) ).
% inj_on_mult
thf(fact_5734_inj__mapD,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F3 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
=> ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_mapD
thf(fact_5735_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,F3: A > B] :
( ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( ( F3 @ X )
!= ( F3 @ Y2 ) ) ) ) )
=> ( ! [X: A,Y2: A] :
( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X ) ) ) )
=> ( inj_on @ A @ B @ F3 @ A3 ) ) ) ) ).
% linorder_inj_onI
thf(fact_5736_filterlim__mono,axiom,
! [B: $tType,A: $tType,F3: A > B,F22: filter @ B,F12: filter @ A,F25: filter @ B,F14: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F22 @ F12 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F22 @ F25 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F14 @ F12 )
=> ( filterlim @ A @ B @ F3 @ F25 @ F14 ) ) ) ) ).
% filterlim_mono
thf(fact_5737_inj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A] :
( inj_on @ A @ A
@ ^ [B4: A] : ( minus_minus @ A @ B4 @ A4 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_diff_right
thf(fact_5738_inj__singleton,axiom,
! [A: $tType,A3: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X4: A] : ( insert3 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) ).
% inj_singleton
thf(fact_5739_finite__Collect,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: B > A] :
( ( finite_finite2 @ A @ S )
=> ( ( inj_on @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [A5: B] : ( member @ A @ ( F3 @ A5 ) @ S ) ) ) ) ) ).
% finite_Collect
thf(fact_5740_swap__inj__on,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I4: A,J2: B] : ( product_Pair @ B @ A @ J2 @ I4 ) )
@ A3 ) ).
% swap_inj_on
thf(fact_5741_zero__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ).
% zero_int.rsp
thf(fact_5742_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,N2: A] : ( cons @ A @ N2 @ Xs3 ) )
@ X5 ) ).
% inj_split_Cons
thf(fact_5743_filterlim__atMost__at__top,axiom,
filterlim @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( finite5375528669736107172at_top @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( at_top @ nat ) ).
% filterlim_atMost_at_top
thf(fact_5744_filterlim__lessThan__at__top,axiom,
filterlim @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( finite5375528669736107172at_top @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( at_top @ nat ) ).
% filterlim_lessThan_at_top
thf(fact_5745_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A3: set @ A,A8: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F7: A > B] :
( ( inj_on @ A @ B @ F7 @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F7 @ A3 ) @ A8 ) ) )
= ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ A8 )
= A3 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_5746_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F3: A > B,C3: set @ A,A3: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A3 ) @ ( image2 @ A @ B @ F3 @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_5747_image__set__diff,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A3 ) @ ( image2 @ A @ B @ F3 @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_5748_pigeonhole,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ ( finite_card @ B @ A3 ) )
=> ~ ( inj_on @ B @ A @ F3 @ A3 ) ) ).
% pigeonhole
thf(fact_5749_map__inj__on,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ B @ A @ F3 @ Ys ) )
=> ( ( inj_on @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_5750_inj__on__map__eq__map,axiom,
! [B: $tType,A: $tType,F3: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) )
=> ( ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ F3 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_5751_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X3: A,A3: set @ A,F3: B > C,F: filter @ C,G5: A > ( filter @ B )] :
( ( member @ A @ X3 @ A3 )
=> ( ( filterlim @ B @ C @ F3 @ F @ ( G5 @ X3 ) )
=> ( filterlim @ B @ C @ F3 @ F @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ G5 @ A3 ) ) ) ) ) ).
% filterlim_INF'
thf(fact_5752_filterlim__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,G5: C > ( filter @ B ),B2: set @ C,F: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ G5 @ B2 ) ) @ F )
= ( ! [X4: C] :
( ( member @ C @ X4 @ B2 )
=> ( filterlim @ A @ B @ F3 @ ( G5 @ X4 ) @ F ) ) ) ) ).
% filterlim_INF
thf(fact_5753_distinct__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
= ( ( distinct @ B @ Xs )
& ( inj_on @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) ) ) ).
% distinct_map
thf(fact_5754_remdups__adj__map__injective,axiom,
! [B: $tType,A: $tType,F3: A > B,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( remdups_adj @ B @ ( map @ A @ B @ F3 @ Xs ) )
= ( map @ A @ B @ F3 @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% remdups_adj_map_injective
thf(fact_5755_filterlim__If,axiom,
! [B: $tType,A: $tType,F3: A > B,G5: filter @ B,F: filter @ A,P: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F3 @ G5 @ ( inf_inf @ ( filter @ A ) @ F @ ( principal @ A @ ( collect @ A @ P ) ) ) )
=> ( ( filterlim @ A @ B @ G @ G5
@ ( inf_inf @ ( filter @ A ) @ F
@ ( principal @ A
@ ( collect @ A
@ ^ [X4: A] :
~ ( P @ X4 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( if @ B @ ( P @ X4 ) @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ G5
@ F ) ) ) ).
% filterlim_If
thf(fact_5756_map__removeAll__inj,axiom,
! [B: $tType,A: $tType,F3: A > B,X3: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( map @ A @ B @ F3 @ ( removeAll @ A @ X3 @ Xs ) )
= ( removeAll @ B @ ( F3 @ X3 ) @ ( map @ A @ B @ F3 @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_5757_sort__key__eq__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,Ys: list @ B,F3: B > A] :
( ( ( mset @ B @ Xs )
= ( mset @ B @ Ys ) )
=> ( ( inj_on @ B @ A @ F3 @ ( set2 @ B @ Xs ) )
=> ( ( linorder_sort_key @ B @ A @ F3 @ Xs )
= ( linorder_sort_key @ B @ A @ F3 @ Ys ) ) ) ) ) ).
% sort_key_eq_sort_key
thf(fact_5758_uminus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X4 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X4 ) ) ) ).
% uminus_int.rsp
thf(fact_5759_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,Y: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert3 @ A @ Y @ ( set2 @ A @ Xs ) ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
( ( F3 @ Y )
= ( F3 @ X4 ) )
@ Xs )
= ( filter2 @ A
@ ( ^ [Y4: A,Z4: A] : Y4 = Z4
@ Y )
@ Xs ) ) ) ).
% inj_on_filter_key_eq
thf(fact_5760_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I3: set @ A,F3: B > C,A3: A > ( set @ B )] :
( ( I3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I5: A] :
( ( member @ A @ I5 @ I3 )
=> ( inj_on @ B @ C @ F3 @ ( A3 @ I5 ) ) )
=> ( inj_on @ B @ C @ F3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A3 @ I3 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_5761_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [N5: nat,F5: nat > A] :
( ( A3
= ( image2 @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N5 ) ) ) )
& ( inj_on @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N5 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_5762_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [F5: A > nat,N5: nat] :
( ( ( image2 @ A @ nat @ F5 @ A3 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N5 ) ) )
& ( inj_on @ A @ nat @ F5 @ A3 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_5763_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ~ ! [F5: A > nat] :
( ? [N5: nat] :
( ( image2 @ A @ nat @ F5 @ A3 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N5 ) ) )
=> ~ ( inj_on @ A @ nat @ F5 @ A3 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_5764_one__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ).
% one_int.rsp
thf(fact_5765_inj__on__nth,axiom,
! [A: $tType,Xs: list @ A,I3: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ! [X: nat] :
( ( member @ nat @ X @ I3 )
=> ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I3 ) ) ) ).
% inj_on_nth
thf(fact_5766_of__int_Orsp,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
@ ^ [Y4: A,Z4: A] : Y4 = Z4
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J2 ) ) )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J2: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J2 ) ) ) ) ) ).
% of_int.rsp
thf(fact_5767_map__removeAll__inj__on,axiom,
! [B: $tType,A: $tType,F3: A > B,X3: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert3 @ A @ X3 @ ( set2 @ A @ Xs ) ) )
=> ( ( map @ A @ B @ F3 @ ( removeAll @ A @ X3 @ Xs ) )
= ( removeAll @ B @ ( F3 @ X3 ) @ ( map @ A @ B @ F3 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_5768_filterlim__base,axiom,
! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J4: set @ A,I: A > C,I3: set @ C,F: C > ( set @ D ),F3: D > E,G5: A > ( set @ E )] :
( ! [M6: A,X: B] :
( ( member @ A @ M6 @ J4 )
=> ( member @ C @ ( I @ M6 ) @ I3 ) )
=> ( ! [M6: A,X: D] :
( ( member @ A @ M6 @ J4 )
=> ( ( member @ D @ X @ ( F @ ( I @ M6 ) ) )
=> ( member @ E @ ( F3 @ X ) @ ( G5 @ M6 ) ) ) )
=> ( filterlim @ D @ E @ F3
@ ( complete_Inf_Inf @ ( filter @ E )
@ ( image2 @ A @ ( filter @ E )
@ ^ [J2: A] : ( principal @ E @ ( G5 @ J2 ) )
@ J4 ) )
@ ( complete_Inf_Inf @ ( filter @ D )
@ ( image2 @ C @ ( filter @ D )
@ ^ [I4: C] : ( principal @ D @ ( F @ I4 ) )
@ I3 ) ) ) ) ) ).
% filterlim_base
thf(fact_5769_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ A,G: A > B,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ A3 )
=> ( ( inj_on @ A @ B @ G @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A3 ) @ ( image2 @ A @ B @ G @ B2 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ A3 ) @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_5770_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F3: A > A,S3: A] :
( ( ( compow @ ( A > A ) @ N @ F3 @ S3 )
= S3 )
=> ( ! [M6: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M6 )
=> ( ( ord_less @ nat @ M6 @ N )
=> ( ( compow @ ( A > A ) @ M6 @ F3 @ S3 )
!= S3 ) ) )
=> ( inj_on @ nat @ A
@ ^ [K4: nat] : ( compow @ ( A > A ) @ K4 @ F3 @ S3 )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% inj_on_funpow_least
thf(fact_5771_inj__on__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ A,B2: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ( inj_on @ A @ B @ F3 @ A3 )
& ( inj_on @ A @ B @ F3 @ B2 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_5772_sort__key__conv__fold,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( inj_on @ B @ A @ F3 @ ( set2 @ B @ Xs ) )
=> ( ( linorder_sort_key @ B @ A @ F3 @ Xs )
= ( fold @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 ) @ Xs @ ( nil @ B ) ) ) ) ) ).
% sort_key_conv_fold
thf(fact_5773_less__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ U3 @ Y5 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ U3 @ Y5 ) ) ) ) ) ).
% less_int.rsp
thf(fact_5774_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,Ys: list @ B] :
( ( inj_on @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Ys ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Ys ) )
=> ( ( ( set2 @ B @ Xs )
= ( set2 @ B @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_5775_minus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ Y5 @ U3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V4 ) @ ( plus_plus @ nat @ Y5 @ U3 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_5776_plus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U3 ) @ ( plus_plus @ nat @ Y5 @ V4 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U3 ) @ ( plus_plus @ nat @ Y5 @ V4 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_5777_sort__key__inj__key__eq,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,Ys: list @ B,F3: B > A] :
( ( ( mset @ B @ Xs )
= ( mset @ B @ Ys ) )
=> ( ( inj_on @ B @ A @ F3 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Ys ) )
=> ( ( linorder_sort_key @ B @ A @ F3 @ Xs )
= Ys ) ) ) ) ) ).
% sort_key_inj_key_eq
thf(fact_5778_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F3: A > B,G: C > B,A3: set @ A,B2: set @ C] :
( ( inj_on @ ( product_prod @ A @ C ) @ B
@ ( product_case_prod @ A @ C @ B
@ ^ [A5: A,B4: C] : ( times_times @ B @ ( F3 @ A5 ) @ ( G @ B4 ) ) )
@ ( product_Sigma @ A @ C @ A3
@ ^ [Uu3: A] : B2 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
@ ( collect @ B
@ ^ [Uu3: B] :
? [A5: A,B4: C] :
( ( Uu3
= ( times_times @ B @ ( F3 @ A5 ) @ ( G @ B4 ) ) )
& ( member @ A @ A5 @ A3 )
& ( member @ C @ B4 @ B2 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_5779_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L: list @ A,A3: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L ) @ A3 )
=> ( ( inj_on @ A @ B @ F3 @ A3 )
=> ( ( map @ B @ A @ ( inv_on @ A @ B @ F3 @ A3 ) @ ( map @ A @ B @ F3 @ L ) )
= L ) ) ) ).
% inj_on_map_inv_f
thf(fact_5780_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ A,A4: B] :
( ( inj_on @ A @ B @ F3 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) @ A3 )
@ ( insert3 @ A
@ ( the @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ( F3 @ X4 )
= A4 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_5781_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F3: A > B,A4: B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert3 @ A
@ ( the @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= A4 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_5782_trivial__limit__sequentially,axiom,
( ( at_top @ nat )
!= ( bot_bot @ ( filter @ nat ) ) ) ).
% trivial_limit_sequentially
thf(fact_5783_The__split__eq,axiom,
! [A: $tType,B: $tType,X3: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X10: A,Y8: B] :
( ( X3 = X10 )
& ( Y = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X3 @ Y ) ) ).
% The_split_eq
thf(fact_5784_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ A,X3: A] :
( ( inj_on @ A @ B @ F3 @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( inv_on @ A @ B @ F3 @ A3 @ ( F3 @ X3 ) )
= X3 ) ) ) ).
% inv_on_f_f
thf(fact_5785_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y: A,F3: B > A,A3: set @ B] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( member @ B @ ( inv_on @ B @ A @ F3 @ A3 @ Y ) @ A3 ) ) ).
% inv_on_f_range
thf(fact_5786_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y: A,F3: B > A,A3: set @ B] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( F3 @ ( inv_on @ B @ A @ F3 @ A3 @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_5787_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P2: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
& ! [Y5: A] :
( ( P2 @ Y5 )
=> ( ord_less_eq @ A @ X4 @ Y5 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_5788_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X9: set @ A] :
( the @ A
@ ^ [X4: A] :
( X9
= ( insert3 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_5789_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P2: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
& ! [Y5: A] :
( ( P2 @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_5790_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq2: A > A > $o,P2: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
& ! [Y5: A] :
( ( P2 @ Y5 )
=> ( Less_eq2 @ X4 @ Y5 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_5791_card__quotient__disjoint,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A] : ( equiv_quotient @ A @ ( insert3 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A3 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A3 @ R2 ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_5792_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_5793_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ! [X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_5794_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A] :
( ( P @ X3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ( order_Greatest @ A @ P )
= X3 ) ) ) ) ).
% Greatest_equality
thf(fact_5795_quotient__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( bot_bot @ ( set @ A ) ) @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% quotient_empty
thf(fact_5796_quotient__is__empty,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( equiv_quotient @ A @ A3 @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty
thf(fact_5797_quotient__is__empty2,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( bot_bot @ ( set @ ( set @ A ) ) )
= ( equiv_quotient @ A @ A3 @ R2 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty2
thf(fact_5798_quotientE,axiom,
! [A: $tType,X5: set @ A,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ~ ! [X: A] :
( ( X5
= ( image @ A @ A @ R2 @ ( insert3 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ~ ( member @ A @ X @ A3 ) ) ) ).
% quotientE
thf(fact_5799_quotientI,axiom,
! [A: $tType,X3: A,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X3 @ A3 )
=> ( member @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( equiv_quotient @ A @ A3 @ R2 ) ) ) ).
% quotientI
thf(fact_5800_quotient__diff1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,A4: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A5: A] : ( equiv_quotient @ A @ ( insert3 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A3 )
=> ( ( member @ A @ A4 @ A3 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ R2 )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A3 @ R2 ) @ ( equiv_quotient @ A @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) ) ) ) ) ).
% quotient_diff1
thf(fact_5801_singleton__quotient,axiom,
! [A: $tType,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( insert3 @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% singleton_quotient
thf(fact_5802_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A] : ( insert3 @ ( set @ A ) @ ( image @ A @ A @ R5 @ ( insert3 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A7 ) ) ) ) ).
% quotient_def
thf(fact_5803_list__all__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( list @ A ) > $o ) @ ( ( list @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A3 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( list_all @ A )
@ ( list_all @ B ) ) ).
% list_all_transfer
thf(fact_5804_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
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( list_all @ A )
@ ( list_all @ B ) ) ).
% list.pred_transfer
thf(fact_5805_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A3 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert3 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite2 @ A @ A3 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_5806_list__all__simps_I1_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( list_all @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( ( P @ X3 )
& ( list_all @ A @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_5807_list_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A4: A,Aa4: list @ A] :
( ( list_all @ A @ P @ ( cons @ A @ A4 @ Aa4 ) )
= ( ( P @ A4 )
& ( list_all @ A @ P @ Aa4 ) ) ) ).
% list.pred_inject(2)
thf(fact_5808_list__all__simps_I2_J,axiom,
! [A: $tType,P: A > $o] : ( list_all @ A @ P @ ( nil @ A ) ) ).
% list_all_simps(2)
thf(fact_5809_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_5810_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_5811_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_5812_list_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X3: list @ A,Pa: A > $o] :
( ( list_all @ A @ P @ X3 )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X3 ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( list_all @ A @ Pa @ X3 ) ) ) ).
% list.pred_mono_strong
thf(fact_5813_list__all__cong,axiom,
! [A: $tType,X3: list @ A,Ya: list @ A,P: A > $o,Pa: A > $o] :
( ( X3 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ Ya ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( list_all @ A @ P @ X3 )
= ( list_all @ A @ Pa @ Ya ) ) ) ) ).
% list_all_cong
thf(fact_5814_list_Opred__True,axiom,
! [A: $tType] :
( ( list_all @ A
@ ^ [Uu3: A] : $true )
= ( ^ [Uu3: list @ A] : $true ) ) ).
% list.pred_True
thf(fact_5815_list_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( list_all @ A @ P @ ( nil @ A ) ) ).
% list.pred_inject(1)
thf(fact_5816_list_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X3: list @ A,Ya: list @ A,F3: A > B,G: A > B] :
( ( X3 = Ya )
=> ( ( list_all @ A
@ ^ [Z6: A] :
( ( F3 @ Z6 )
= ( G @ Z6 ) )
@ Ya )
=> ( ( map @ A @ B @ F3 @ X3 )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_5817_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_5818_prop__matchD,axiom,
! [A: $tType,Al: list @ A,E3: A,Bl: list @ A,Al2: list @ A,E4: A,Bl2: list @ A,P: A > $o] :
( ( ( append @ A @ Al @ ( cons @ A @ E3 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E4 @ Bl2 ) ) )
=> ( ( list_all @ A @ P @ Al )
=> ( ~ ( P @ E3 )
=> ( ~ ( P @ E4 )
=> ( ( list_all @ A @ P @ Bl )
=> ( ( Al = Al2 )
& ( E3 = E4 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_matchD
thf(fact_5819_prop__match,axiom,
! [A: $tType,P: A > $o,Al: list @ A,E3: A,E4: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( list_all @ A @ P @ Al )
=> ( ~ ( P @ E3 )
=> ( ~ ( P @ E4 )
=> ( ( list_all @ A @ P @ Bl )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E3 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E4 @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( E3 = E4 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_match
thf(fact_5820_Ball__set__list__all,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) ) )
= ( list_all @ A @ P @ Xs ) ) ).
% Ball_set_list_all
thf(fact_5821_list__all__iff,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P2: A > $o,X4: list @ A] :
! [Y5: A] :
( ( member @ A @ Y5 @ ( set2 @ A @ X4 ) )
=> ( P2 @ Y5 ) ) ) ) ).
% list_all_iff
thf(fact_5822_list_Opred__set,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P2: A > $o,X4: list @ A] :
! [Y5: A] :
( ( member @ A @ Y5 @ ( set2 @ A @ X4 ) )
=> ( P2 @ Y5 ) ) ) ) ).
% list.pred_set
thf(fact_5823_list_Opred__rel,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P2: A > $o,X4: list @ A] : ( list_all2 @ A @ A @ ( bNF_eq_onp @ A @ P2 ) @ X4 @ X4 ) ) ) ).
% list.pred_rel
thf(fact_5824_list_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( list_all2 @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( list @ A ) @ ( list_all @ A @ P ) ) ) ).
% list.rel_eq_onp
thf(fact_5825_dvd__gcd__list__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B3: A,Xs: list @ A] :
( ( dvd_dvd @ A @ B3 @ ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( dvd_dvd @ A @ B3 @ X4 ) ) ) ) ) ).
% dvd_gcd_list_iff
thf(fact_5826_gcd__list__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Bs: list @ A,A4: A] :
( ! [B5: A] :
( ( member @ A @ B5 @ ( set2 @ A @ Bs ) )
=> ( dvd_dvd @ A @ A4 @ B5 ) )
=> ( dvd_dvd @ A @ A4 @ ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Bs ) ) ) ) ) ).
% gcd_list_greatest
thf(fact_5827_list_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F3: A > B,X3: list @ A] :
( ( list_all @ B @ Q @ ( map @ A @ B @ F3 @ X3 ) )
= ( list_all @ A @ ( comp @ B @ $o @ A @ Q @ F3 ) @ X3 ) ) ).
% list.pred_map
thf(fact_5828_list__all__length,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P2: A > $o,Xs3: list @ A] :
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P2 @ ( nth @ A @ Xs3 @ N2 ) ) ) ) ) ).
% list_all_length
thf(fact_5829_list__ex1__simps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( list_ex1 @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( ( ( P @ X3 )
=> ( list_all @ A
@ ^ [Y5: A] :
( ~ ( P @ Y5 )
| ( X3 = Y5 ) )
@ Xs ) )
& ( ~ ( P @ X3 )
=> ( list_ex1 @ A @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_5830_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_5831_init__seg__of__def,axiom,
! [A: $tType] :
( ( init_seg_of @ A )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ $o
@ ^ [R5: set @ ( product_prod @ A @ A ),S7: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S7 )
& ! [A5: A,B4: A,C6: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ S7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C6 ) @ R5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R5 ) ) ) ) ) ) ).
% init_seg_of_def
thf(fact_5832_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_5833_trans__init__seg__of,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) @ ( init_seg_of @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S3 @ T2 ) @ ( init_seg_of @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ T2 ) @ ( init_seg_of @ A ) ) ) ) ).
% trans_init_seg_of
thf(fact_5834_antisym__init__seg__of,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) @ ( init_seg_of @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S3 @ R2 ) @ ( init_seg_of @ A ) )
=> ( R2 = S3 ) ) ) ).
% antisym_init_seg_of
thf(fact_5835_refl__on__init__seg__of,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R2 ) @ ( init_seg_of @ A ) ) ).
% refl_on_init_seg_of
thf(fact_5836_list__ex1__iff,axiom,
! [A: $tType] :
( ( list_ex1 @ A )
= ( ^ [P2: A > $o,Xs3: list @ A] :
? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs3 ) )
& ( P2 @ X4 )
& ! [Y5: A] :
( ( ( member @ A @ Y5 @ ( set2 @ A @ Xs3 ) )
& ( P2 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_5837_map__of__eqI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
= ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_of @ A @ B @ Xs @ X )
= ( map_of @ A @ B @ Ys @ X ) ) )
=> ( ( map_of @ A @ B @ Xs )
= ( map_of @ A @ B @ Ys ) ) ) ) ).
% map_of_eqI
thf(fact_5838_map__of__zip__inject,axiom,
! [B: $tType,A: $tType,Ys: list @ A,Xs: list @ B,Zs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys )
= ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Ys ) )
= ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Zs ) ) )
=> ( Ys = Zs ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_5839_map__of__eq__dom,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( map_of @ A @ B @ Xs )
= ( map_of @ A @ B @ Ys ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Ys ) ) ) ) ).
% map_of_eq_dom
thf(fact_5840_initial__segment__of__Diff,axiom,
! [A: $tType,P4: set @ ( product_prod @ A @ A ),Q4: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ Q4 ) @ ( init_seg_of @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ S3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ Q4 @ S3 ) ) @ ( init_seg_of @ A ) ) ) ).
% initial_segment_of_Diff
thf(fact_5841_map__of__inject__set,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( 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 ) @ Ys ) )
=> ( ( ( map_of @ A @ B @ Xs )
= ( map_of @ A @ B @ Ys ) )
= ( ( set2 @ ( product_prod @ A @ B ) @ Xs )
= ( set2 @ ( product_prod @ A @ B ) @ Ys ) ) ) ) ) ).
% map_of_inject_set
thf(fact_5842_can__select__set__list__ex1,axiom,
! [A: $tType,P: A > $o,A3: list @ A] :
( ( can_select @ A @ P @ ( set2 @ A @ A3 ) )
= ( list_ex1 @ A @ P @ A3 ) ) ).
% can_select_set_list_ex1
thf(fact_5843_image__mset__map__of,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( image_mset @ A @ B
@ ^ [I4: A] : ( the2 @ B @ ( map_of @ A @ B @ Xs @ I4 ) )
@ ( mset @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
= ( mset @ B @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs ) ) ) ) ).
% image_mset_map_of
thf(fact_5844_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_5845_can__select__def,axiom,
! [A: $tType] :
( ( can_select @ A )
= ( ^ [P2: A > $o,A7: set @ A] :
? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( P2 @ X4 )
& ! [Y5: A] :
( ( ( member @ A @ Y5 @ A7 )
& ( P2 @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_5846_ran__map__of__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ran @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( set2 @ B @ Ys ) ) ) ) ).
% ran_map_of_zip
thf(fact_5847_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_5848_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_5849_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_5850_map__of__distinct__upd4,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X3 @ ( 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 @ X3 @ Y ) @ Ys ) ) ) @ X3 @ ( none @ B ) ) ) ) ) ).
% map_of_distinct_upd4
thf(fact_5851_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X4: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_5852_map__to__set__empty,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B
@ ^ [X4: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% map_to_set_empty
thf(fact_5853_map__of__zip__is__None,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X3 )
= ( none @ B ) )
= ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_5854_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
= ( ^ [X4: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(1)
thf(fact_5855_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
= ( ^ [X4: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(2)
thf(fact_5856_map__of__None__filterD,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),X3: B,P: ( product_prod @ B @ A ) > $o] :
( ( ( map_of @ B @ A @ Xs @ X3 )
= ( none @ A ) )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ P @ Xs ) @ X3 )
= ( none @ A ) ) ) ).
% map_of_None_filterD
thf(fact_5857_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_5858_map__of__eq__None__iff,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A ),X3: B] :
( ( ( map_of @ B @ A @ Xys @ X3 )
= ( none @ A ) )
= ( ~ ( member @ B @ X3 @ ( image2 @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xys ) ) ) ) ) ).
% map_of_eq_None_iff
thf(fact_5859_map__to__set__ran,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M: B > ( option @ A )] : ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map_to_set @ B @ A @ M ) ) ) ) ).
% map_to_set_ran
thf(fact_5860_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
( ( rel_of @ A @ B
@ ^ [X4: A] : ( none @ B )
@ P )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% rel_of_empty
thf(fact_5861_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F7: A > ( option @ B ),Xs3: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F7 )
@ ( filter2 @ A
@ ^ [X4: A] :
( ( F7 @ X4 )
!= ( none @ B ) )
@ Xs3 ) ) ) ) ).
% map_filter_def
thf(fact_5862_map__of__distinct__upd3,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B,Y7: B] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X3 @ ( 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 @ X3 @ 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 @ X3 @ Y7 ) @ Ys ) ) ) @ X3 @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd3
thf(fact_5863_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V2: option @ ( product_prod @ A @ B )] :
( ( ! [X4: A,Y5: B] :
( V2
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) ) )
= ( V2
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_5864_map__update__eta__repair_I2_J,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K3: A,V2: B] :
( ( ( M2 @ K3 )
= ( none @ B ) )
=> ( ( ran @ A @ B
@ ^ [X4: A] : ( if @ ( option @ B ) @ ( X4 = K3 ) @ ( some @ B @ V2 ) @ ( M2 @ X4 ) ) )
= ( insert3 @ B @ V2 @ ( ran @ A @ B @ M2 ) ) ) ) ).
% map_update_eta_repair(2)
thf(fact_5865_ran__map__upd,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A4: B,B3: A] :
( ( ( M2 @ A4 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ A4 @ ( some @ A @ B3 ) ) )
= ( insert3 @ A @ B3 @ ( ran @ B @ A @ M2 ) ) ) ) ).
% ran_map_upd
thf(fact_5866_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ran @ nat @ A
@ ^ [I4: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I4 ) ) @ ( none @ A ) ) )
= ( set2 @ A @ L ) ) ).
% ran_nth_set_encoding_conv
thf(fact_5867_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X3: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
=> ( ( map_of @ A @ B @ Xys @ X3 )
= ( some @ B @ Y ) ) ) ) ).
% map_of_is_SomeI
thf(fact_5868_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y: B,X3: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( some @ B @ Y )
= ( map_of @ A @ B @ Xys @ X3 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_5869_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X3: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( map_of @ A @ B @ Xys @ X3 )
= ( some @ B @ Y ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_5870_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K3: A,V2: B] :
( ( map_to_set @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K3 @ ( some @ B @ V2 ) ) )
= ( insert3 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V2 )
@ ( 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 @ K3 @ V5 ) ) ) ) ) ) ).
% map_to_set_upd
thf(fact_5871_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F3: B > A,P: B > $o,Xs: list @ B] :
( ( map @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs ) )
= ( map_filter @ B @ A
@ ^ [X4: B] : ( if @ ( option @ A ) @ ( P @ X4 ) @ ( some @ A @ ( F3 @ X4 ) ) @ ( none @ A ) )
@ Xs ) ) ).
% map_filter_map_filter
thf(fact_5872_le__some__optE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M2: A,X3: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M2 ) @ X3 )
=> ~ ! [M11: A] :
( ( X3
= ( some @ A @ M11 ) )
=> ~ ( ord_less_eq @ A @ M2 @ M11 ) ) ) ) ).
% le_some_optE
thf(fact_5873_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M: A > ( option @ B ),P2: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K4: A,V4: B] :
( ( ( M @ K4 )
= ( some @ B @ V4 ) )
& ( P2 @ ( product_Pair @ A @ B @ K4 @ V4 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_5874_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
@ ^ [Y5: A] :
? [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs ) )
& ( ( G @ X4 )
= ( some @ A @ Y5 ) ) ) ) ) ).
% set_map_filter
thf(fact_5875_map__filter__simps_I2_J,axiom,
! [B: $tType,A: $tType,F3: B > ( option @ A )] :
( ( map_filter @ B @ A @ F3 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% map_filter_simps(2)
thf(fact_5876_map__to__set__def,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B )
= ( ^ [M: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K4: A,V4: B] :
( ( M @ K4 )
= ( some @ B @ V4 ) ) ) ) ) ) ).
% map_to_set_def
thf(fact_5877_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K3: B,Y: A] :
( ( ( map_of @ B @ A @ Xs @ K3 )
= ( some @ A @ Y ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K3 @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% map_of_SomeD
thf(fact_5878_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K3: A,X3: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ X3 ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X: B] :
( ( map_of @ A @ B @ L @ K3 )
= ( some @ B @ X ) ) ) ).
% weak_map_of_SomeI
thf(fact_5879_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K3: B,V2: C,Ps3: list @ ( product_prod @ B @ C )] :
( ( ( L = K3 )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps3 ) @ K3 )
= ( some @ C @ V2 ) ) )
& ( ( L != K3 )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps3 ) @ K3 )
= ( map_of @ B @ C @ Ps3 @ K3 ) ) ) ) ).
% map_of_Cons_code(2)
thf(fact_5880_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
= ( ? [Y5: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X3 )
= ( some @ B @ Y5 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_5881_map__of__zip__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ ( map @ A @ B @ F3 @ Xs ) ) )
= ( ^ [X4: A] : ( if @ ( option @ B ) @ ( member @ A @ X4 @ ( set2 @ A @ Xs ) ) @ ( some @ B @ ( F3 @ X4 ) ) @ ( none @ B ) ) ) ) ).
% map_of_zip_map
thf(fact_5882_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys: list @ B,Xs: list @ A,Zs: list @ B,X3: A,Y: B,Z2: B] :
( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( size_size @ ( list @ B ) @ Zs )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) @ X3 @ ( some @ B @ Y ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) @ X3 @ ( some @ B @ Z2 ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_5883_map__of__Some__split,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K3: B,V2: A] :
( ( ( map_of @ B @ A @ Xs @ K3 )
= ( some @ A @ V2 ) )
=> ? [Ys5: list @ ( product_prod @ B @ A ),Zs2: list @ ( product_prod @ B @ A )] :
( ( Xs
= ( append @ ( product_prod @ B @ A ) @ Ys5 @ ( cons @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K3 @ V2 ) @ Zs2 ) ) )
& ( ( map_of @ B @ A @ Ys5 @ K3 )
= ( none @ A ) ) ) ) ).
% map_of_Some_split
thf(fact_5884_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ ( nth @ A @ Xs @ I ) )
= ( some @ B @ ( nth @ B @ Ys @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_5885_set__map__of__compr,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( set2 @ ( product_prod @ A @ B ) @ Xs )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K4: A,V4: B] :
( ( map_of @ A @ B @ Xs @ K4 )
= ( some @ B @ V4 ) ) ) ) ) ) ).
% set_map_of_compr
thf(fact_5886_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,T2: list @ ( product_prod @ A @ C ),K3: A,X3: C] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T2 @ K3 )
= ( some @ C @ X3 ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K4: A] : ( product_Pair @ B @ C @ ( F3 @ K4 ) ) )
@ T2 )
@ ( F3 @ K3 ) )
= ( some @ C @ X3 ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_5887_map__of__Some__filter__not__in,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K3: B,V2: A,P: ( product_prod @ B @ A ) > $o] :
( ( ( map_of @ B @ A @ Xs @ K3 )
= ( some @ A @ V2 ) )
=> ( ~ ( P @ ( product_Pair @ B @ A @ K3 @ V2 ) )
=> ( ( distinct @ B @ ( map @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ Xs ) )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ P @ Xs ) @ K3 )
= ( none @ A ) ) ) ) ) ).
% map_of_Some_filter_not_in
thf(fact_5888_map__of__distinct__lookup,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X3 @ ( 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 @ X3 @ Y ) @ Ys ) ) @ X3 )
= ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_lookup
thf(fact_5889_map__of__distinct__upd2,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X3 @ ( 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 @ X3 @ Y ) @ Ys ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys ) ) @ X3 @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd2
thf(fact_5890_less__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ ( option @ A ) @ ( some @ A @ X3 ) @ ( some @ A @ Y ) )
= ( ord_less @ A @ X3 @ Y ) ) ) ).
% less_option_Some
thf(fact_5891_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F7: A > ( option @ B ),A7: set @ A] :
( collect @ B
@ ^ [B4: B] :
? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( ( F7 @ X4 )
= ( some @ B @ B4 ) ) ) ) ) ) ).
% map_project_def
thf(fact_5892_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ B )
=> ! [A3: set @ A,F3: A > B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A3 ) ) )
= ( complete_Sup_Sup @ ( option @ B )
@ ( image2 @ A @ ( option @ B )
@ ^ [X4: A] : ( some @ B @ ( F3 @ X4 ) )
@ A3 ) ) ) ) ) ).
% Some_SUP
thf(fact_5893_Some__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ A @ ( complete_Sup_Sup @ A @ A3 ) )
= ( complete_Sup_Sup @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) ) ) ) ).
% Some_Sup
thf(fact_5894_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,M2: A > ( option @ B ),X3: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ Xs @ Ys ) @ X3 @ ( some @ B @ ( nth @ B @ Ys @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_5895_set__to__map__insert,axiom,
! [B: $tType,A: $tType,Kv: product_prod @ A @ B,S: set @ ( product_prod @ A @ B )] :
( ~ ( member @ A @ ( product_fst @ A @ B @ Kv ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) )
=> ( ( set_to_map @ A @ B @ ( insert3 @ ( 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_5896_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P2: 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 ) ) ) )
@ ^ [Y5: 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 @ (~) @ P2 ) @ Xs3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y5 @ Ys3 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ Xs3 ) ) ) ) ).
% extract_def
thf(fact_5897_map__upds__apply__nontin,axiom,
! [B: $tType,A: $tType,X3: A,Xs: list @ A,F3: A > ( option @ B ),Ys: list @ B] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( map_upds @ A @ B @ F3 @ Xs @ Ys @ X3 )
= ( F3 @ X3 ) ) ) ).
% map_upds_apply_nontin
thf(fact_5898_fun__upds__append2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,M2: A > ( option @ B ),Zs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ Xs @ ( append @ B @ Ys @ Zs ) )
= ( map_upds @ A @ B @ M2 @ Xs @ Ys ) ) ) ).
% fun_upds_append2_drop
thf(fact_5899_fun__upds__append__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,M2: A > ( option @ B ),Zs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs @ Zs ) @ Ys )
= ( map_upds @ A @ B @ M2 @ Xs @ Ys ) ) ) ).
% fun_upds_append_drop
thf(fact_5900_set__to__map__empty,axiom,
! [B: $tType,A: $tType] :
( ( set_to_map @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% set_to_map_empty
thf(fact_5901_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,M2: A > ( option @ B ),Ys: list @ B,Y: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( map_upds @ A @ B @ M2 @ Xs @ ( list_update @ B @ Ys @ I @ Y ) )
= ( map_upds @ A @ B @ M2 @ Xs @ Ys ) ) ) ).
% map_upds_list_update2_drop
thf(fact_5902_map__upds__twist,axiom,
! [A: $tType,B: $tType,A4: A,As2: list @ A,M2: A > ( option @ B ),B3: B,Bs: list @ B] :
( ~ ( member @ A @ A4 @ ( set2 @ A @ As2 ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A4 @ ( some @ B @ B3 ) ) @ As2 @ Bs )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ As2 @ Bs ) @ A4 @ ( some @ B @ B3 ) ) ) ) ).
% map_upds_twist
thf(fact_5903_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 ) ) ) ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% extract_None_iff
thf(fact_5904_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_5905_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_5906_set__to__map__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B )] :
( ( ( set_to_map @ A @ B @ S )
= ( ^ [X4: A] : ( none @ B ) ) )
= ( S
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(1)
thf(fact_5907_set__to__map__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: 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_5908_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X3: A,Ys: list @ B,Xs: list @ A,F3: A > ( option @ B ),Y: B] :
( ( ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X3 @ ( some @ B @ Y ) ) @ Xs @ Ys )
= ( map_upds @ A @ B @ F3 @ Xs @ Ys ) ) )
& ( ~ ( member @ A @ X3 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X3 @ ( some @ B @ Y ) ) @ Xs @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F3 @ Xs @ Ys ) @ X3 @ ( some @ B @ Y ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_5909_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_5910_set__to__map__simp,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),K3: A,V2: B] :
( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S )
=> ( ( ( set_to_map @ A @ B @ S @ K3 )
= ( some @ B @ V2 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V2 ) @ S ) ) ) ).
% set_to_map_simp
thf(fact_5911_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_5912_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 )
& ~ ? [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Ys ) )
& ( P @ X8 ) ) ) ) ).
% extract_SomeE
thf(fact_5913_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 )
& ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
& ( P @ X4 ) ) ) ) ).
% extract_Some_iff
thf(fact_5914_extract__Cons__code,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( ( P @ X3 )
=> ( ( extract @ A @ P @ ( cons @ A @ X3 @ 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 ) @ X3 @ Xs ) ) ) ) )
& ( ~ ( P @ X3 )
=> ( ( extract @ A @ P @ ( cons @ A @ X3 @ 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 ) ) ) )
@ ^ [Y5: A,Zs3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X3 @ Ys3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y5 @ Zs3 ) ) ) ) )
@ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_5915_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P2: A > $o,Xs3: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X4: A,Xa5: list @ A] : ( some @ A @ X4 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ Xs3 ) ) ) ) ).
% find_dropWhile
thf(fact_5916_graph__map__upd,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),K3: A,V2: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K3 @ ( some @ B @ V2 ) ) )
= ( insert3 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V2 ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K3 @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_5917_find__SomeD_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X3: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X3 ) )
=> ( P @ X3 ) ) ).
% find_SomeD(1)
thf(fact_5918_find__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( find @ A @ P @ Xs )
= ( find @ A @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_5919_in__graphI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),K3: B,V2: A] :
( ( ( M2 @ K3 )
= ( some @ A @ V2 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K3 @ V2 ) @ ( graph @ B @ A @ M2 ) ) ) ).
% in_graphI
thf(fact_5920_in__graphD,axiom,
! [A: $tType,B: $tType,K3: A,V2: B,M2: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V2 ) @ ( graph @ A @ B @ M2 ) )
=> ( ( M2 @ K3 )
= ( some @ B @ V2 ) ) ) ).
% in_graphD
thf(fact_5921_find_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: list @ A] :
( ( ( P @ X3 )
=> ( ( find @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( some @ A @ X3 ) ) )
& ( ~ ( P @ X3 )
=> ( ( find @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( find @ A @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_5922_find_Osimps_I1_J,axiom,
! [A: $tType,Uu: A > $o] :
( ( find @ A @ Uu @ ( nil @ A ) )
= ( none @ A ) ) ).
% find.simps(1)
thf(fact_5923_find__SomeD_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X3: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X3 ) )
=> ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ).
% find_SomeD(2)
thf(fact_5924_find__None__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( find @ A @ P @ Xs )
= ( none @ A ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% find_None_iff
thf(fact_5925_find__None__iff2,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( none @ A )
= ( find @ A @ P @ Xs ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) ) ) ).
% find_None_iff2
thf(fact_5926_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A5: A,B4: B] :
( ( Uu3
= ( product_Pair @ A @ B @ A5 @ B4 ) )
& ( ( M @ A5 )
= ( some @ B @ B4 ) ) ) ) ) ) ).
% graph_def
thf(fact_5927_find__Some__iff2,axiom,
! [A: $tType,X3: A,P: A > $o,Xs: list @ A] :
( ( ( some @ A @ X3 )
= ( find @ A @ P @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I4 ) )
& ( X3
= ( nth @ A @ Xs @ I4 ) )
& ! [J2: nat] :
( ( ord_less @ nat @ J2 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_5928_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X3: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X3 ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I4 ) )
& ( X3
= ( nth @ A @ Xs @ I4 ) )
& ! [J2: nat] :
( ( ord_less @ nat @ J2 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_5929_graph__map__of__if__distinct__dom,axiom,
! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Al ) )
=> ( ( graph @ A @ B @ ( map_of @ A @ B @ Al ) )
= ( set2 @ ( product_prod @ A @ B ) @ Al ) ) ) ).
% graph_map_of_if_distinct_dom
thf(fact_5930_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
& ( P @ X8 ) )
=> ( ( find @ A @ P @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_5931_map__of__distinct__upd,axiom,
! [A: $tType,B: $tType,X3: A,Xs: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_add @ A @ B
@ ( fun_upd @ A @ ( option @ B )
@ ^ [X4: A] : ( none @ B )
@ X3
@ ( some @ B @ Y ) )
@ ( map_of @ A @ B @ Xs ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Xs ) @ X3 @ ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_upd
thf(fact_5932_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X3: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X3 @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_5933_restrict__map__upds,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,D2: set @ A,M2: A > ( option @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ D2 )
=> ( ( restrict_map @ A @ B @ ( map_upds @ A @ B @ M2 @ Xs @ Ys ) @ D2 )
= ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D2 @ ( set2 @ A @ Xs ) ) ) @ Xs @ Ys ) ) ) ) ).
% restrict_map_upds
thf(fact_5934_restrict__map__UNIV,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
= F3 ) ).
% restrict_map_UNIV
thf(fact_5935_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_5936_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X3: A,D2: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( member @ A @ X3 @ D2 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D2 ) @ X3 @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D2 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_5937_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X3: A,D2: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( ( member @ A @ X3 @ D2 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X3 @ Y ) @ D2 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D2 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) )
& ( ~ ( member @ A @ X3 @ D2 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X3 @ Y ) @ D2 )
= ( restrict_map @ A @ B @ M2 @ D2 ) ) ) ) ).
% restrict_fun_upd
thf(fact_5938_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X3: A,D2: set @ A,M2: A > ( option @ B )] :
( ( ( member @ A @ X3 @ D2 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D2 ) @ X3 @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D2 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X3 @ D2 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D2 ) @ X3 @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ D2 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_5939_map__add__first__le,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M2: A > ( option @ B ),M9: A > ( option @ B ),N: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M9 )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M2 @ N ) @ ( map_add @ A @ B @ M9 @ N ) ) ) ) ).
% map_add_first_le
thf(fact_5940_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_5941_restrict__map__subset__eq,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),R: set @ A,M9: A > ( option @ B ),R8: set @ A] :
( ( ( restrict_map @ A @ B @ M2 @ R )
= M9 )
=> ( ( ord_less_eq @ ( set @ A ) @ R8 @ R )
=> ( ( restrict_map @ A @ B @ M2 @ R8 )
= ( restrict_map @ A @ B @ M9 @ R8 ) ) ) ) ).
% restrict_map_subset_eq
thf(fact_5942_map__add__left__None,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),K3: B,G: B > ( option @ A )] :
( ( ( F3 @ K3 )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F3 @ G @ K3 )
= ( G @ K3 ) ) ) ).
% map_add_left_None
thf(fact_5943_map__add__find__left,axiom,
! [A: $tType,B: $tType,G: B > ( option @ A ),K3: B,F3: B > ( option @ A )] :
( ( ( G @ K3 )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F3 @ G @ K3 )
= ( F3 @ K3 ) ) ) ).
% map_add_find_left
thf(fact_5944_restrict__map__eq_I2_J,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: set @ B,K3: B,V2: A] :
( ( ( restrict_map @ B @ A @ M2 @ A3 @ K3 )
= ( some @ A @ V2 ) )
= ( ( ( M2 @ K3 )
= ( some @ A @ V2 ) )
& ( member @ B @ K3 @ A3 ) ) ) ).
% restrict_map_eq(2)
thf(fact_5945_restrict__map__insert,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),A4: A,A3: set @ A] :
( ( restrict_map @ A @ B @ F3 @ ( insert3 @ A @ A4 @ A3 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F3 @ A3 ) @ A4 @ ( F3 @ A4 ) ) ) ).
% restrict_map_insert
thf(fact_5946_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K3: A,V2: B,M2: A > ( option @ B ),A3: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V2 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A3 ) ) )
=> ( member @ A @ K3 @ A3 ) ) ).
% graph_restrictD(1)
thf(fact_5947_map__restrict__insert__none__simp,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),X3: B,S3: set @ B] :
( ( ( M2 @ X3 )
= ( none @ A ) )
=> ( ( restrict_map @ B @ A @ M2 @ ( uminus_uminus @ ( set @ B ) @ ( insert3 @ B @ X3 @ S3 ) ) )
= ( restrict_map @ B @ A @ M2 @ ( uminus_uminus @ ( set @ B ) @ S3 ) ) ) ) ).
% map_restrict_insert_none_simp
thf(fact_5948_restrict__map__upd,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),S: set @ A,K3: A,V2: B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F3 @ S ) @ K3 @ ( some @ B @ V2 ) )
= ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ K3 @ ( some @ B @ V2 ) ) @ ( insert3 @ A @ K3 @ S ) ) ) ).
% restrict_map_upd
thf(fact_5949_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K3: A,V2: B,M2: A > ( option @ B ),A3: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K3 @ V2 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A3 ) ) )
=> ( ( M2 @ K3 )
= ( some @ B @ V2 ) ) ) ).
% graph_restrictD(2)
thf(fact_5950_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),D2: set @ A,X3: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D2 ) @ X3 @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D2 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X3 @ Y ) ) ).
% fun_upd_restrict
thf(fact_5951_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 ),F7: A > ( option @ B )] : ( map_add @ A @ B @ F7 @ ( map_of @ A @ B @ Xs3 ) )
@ Xss2
@ ^ [X4: A] : ( none @ B ) ) ) ).
% map_of_concat
thf(fact_5952_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X3: A] :
( ( fun_upd @ A @ ( option @ B ) @ M2 @ X3 @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% map_upd_eq_restrict
thf(fact_5953_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X3: A] :
( ( restrict_map @ A @ B @ F3 @ ( uminus_uminus @ ( set @ A ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F3 @ X3 @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_5954_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F3: A > B,Ks: list @ A] :
( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K4: A] : ( product_Pair @ A @ B @ K4 @ ( F3 @ K4 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F3 ) @ ( set2 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_5955_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),X3: B,Y: A,Z2: A] :
( ( ( M2 @ X3 )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M2 @ ( dom @ B @ A @ M2 ) )
=> ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M2 ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ X3 @ ( some @ A @ Z2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M2 ) @ ( insert3 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert3 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_5956_Eps__Opt__eq__None,axiom,
! [A: $tType,P: A > $o] :
( ( ( eps_Opt @ A @ P )
= ( none @ A ) )
= ( ~ ? [X9: A] : ( P @ X9 ) ) ) ).
% Eps_Opt_eq_None
thf(fact_5957_Chains__init__seg__of__Union,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) ),R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) @ R @ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) )
=> ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R ) ) @ ( init_seg_of @ A ) ) ) ) ).
% Chains_init_seg_of_Union
thf(fact_5958_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_5959_some__opt__sym__eq__trivial,axiom,
! [A: $tType,X3: A] :
( ( eps_Opt @ A
@ ( ^ [Y4: A,Z4: A] : Y4 = Z4
@ X3 ) )
= ( some @ A @ X3 ) ) ).
% some_opt_sym_eq_trivial
thf(fact_5960_some__opt__eq__trivial,axiom,
! [A: $tType,X3: A] :
( ( eps_Opt @ A
@ ^ [Y5: A] : Y5 = X3 )
= ( some @ A @ X3 ) ) ).
% some_opt_eq_trivial
thf(fact_5961_some__opt__false__trivial,axiom,
! [A: $tType] :
( ( eps_Opt @ A
@ ^ [Uu3: A] : $false )
= ( none @ A ) ) ).
% some_opt_false_trivial
thf(fact_5962_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B )] :
( ( ( dom @ A @ B @ F3 )
= ( bot_bot @ ( set @ A ) ) )
= ( F3
= ( ^ [X4: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_5963_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X4: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_5964_map__update__eta__repair_I1_J,axiom,
! [B: $tType,A: $tType,K3: A,V2: B,M2: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X4: A] : ( if @ ( option @ B ) @ ( X4 = K3 ) @ ( some @ B @ V2 ) @ ( M2 @ X4 ) ) )
= ( insert3 @ A @ K3 @ ( dom @ A @ B @ M2 ) ) ) ).
% map_update_eta_repair(1)
thf(fact_5965_dom__const_H,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( dom @ A @ B
@ ^ [X4: A] : ( some @ B @ ( F3 @ X4 ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const'
thf(fact_5966_restrict__map__inv,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F3 @ ( uminus_uminus @ ( set @ A ) @ ( dom @ A @ B @ F3 ) ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% restrict_map_inv
thf(fact_5967_dom__map__of__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( dom @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( set2 @ A @ Xs ) ) ) ).
% dom_map_of_zip
thf(fact_5968_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option @ B,F3: A > ( option @ B ),X3: A] :
( ( ( Y
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X3 @ Y ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F3 ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X3 @ Y ) )
= ( insert3 @ A @ X3 @ ( dom @ A @ B @ F3 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_5969_dom__map__upds,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),Xs: list @ A,Ys: list @ B] :
( ( dom @ A @ B @ ( map_upds @ A @ B @ M2 @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) @ ( dom @ A @ B @ M2 ) ) ) ).
% dom_map_upds
thf(fact_5970_ran__add,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),G: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F3 ) @ ( dom @ A @ B @ G ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ F3 @ G ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ F3 ) @ ( ran @ A @ B @ G ) ) ) ) ).
% ran_add
thf(fact_5971_ran__map__add,axiom,
! [B: $tType,A: $tType,M12: A > ( option @ B ),M23: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M12 ) @ ( dom @ A @ B @ M23 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ M12 @ M23 ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ M12 ) @ ( ran @ A @ B @ M23 ) ) ) ) ).
% ran_map_add
thf(fact_5972_insert__dom,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X3: B,Y: A] :
( ( ( F3 @ X3 )
= ( some @ A @ Y ) )
=> ( ( insert3 @ B @ X3 @ ( dom @ B @ A @ F3 ) )
= ( dom @ B @ A @ F3 ) ) ) ).
% insert_dom
thf(fact_5973_nempty__dom,axiom,
! [B: $tType,A: $tType,E3: A > ( option @ B )] :
( ( E3
!= ( ^ [X4: A] : ( none @ B ) ) )
=> ~ ! [M6: A] :
~ ( member @ A @ M6 @ ( dom @ A @ B @ E3 ) ) ) ).
% nempty_dom
thf(fact_5974_le__map__dom__mono,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M2: A > ( option @ B ),M9: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M9 )
=> ( ord_less_eq @ ( set @ A ) @ ( dom @ A @ B @ M2 ) @ ( dom @ A @ B @ M9 ) ) ) ) ).
% le_map_dom_mono
thf(fact_5975_map__dom__ran__finite,axiom,
! [B: $tType,A: $tType,M5: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M5 ) )
=> ( finite_finite2 @ B @ ( ran @ A @ B @ M5 ) ) ) ).
% map_dom_ran_finite
thf(fact_5976_Eps__Opt__eq__Some__implies,axiom,
! [A: $tType,P: A > $o,X3: A] :
( ( ( eps_Opt @ A @ P )
= ( some @ A @ X3 ) )
=> ( P @ X3 ) ) ).
% Eps_Opt_eq_Some_implies
thf(fact_5977_Eps__Opt__eq__Some,axiom,
! [A: $tType,P: A > $o,X3: A] :
( ! [X2: A] :
( ( P @ X3 )
=> ( ( P @ X2 )
=> ( X2 = X3 ) ) )
=> ( ( ( eps_Opt @ A @ P )
= ( some @ A @ X3 ) )
= ( P @ X3 ) ) ) ).
% Eps_Opt_eq_Some
thf(fact_5978_dom__minus,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X3: B,A3: set @ B] :
( ( ( F3 @ X3 )
= ( none @ A ) )
=> ( ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F3 ) @ ( insert3 @ B @ X3 @ A3 ) )
= ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F3 ) @ A3 ) ) ) ).
% dom_minus
thf(fact_5979_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A3: A > ( option @ B ),B2: A > ( option @ B ),C3: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ A3 ) @ ( dom @ A @ B @ B2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ A3 @ ( map_add @ A @ B @ B2 @ C3 ) )
= ( map_add @ A @ B @ B2 @ ( map_add @ A @ B @ A3 @ C3 ) ) ) ) ).
% map_add_left_comm
thf(fact_5980_map__add__comm,axiom,
! [B: $tType,A: $tType,M12: A > ( option @ B ),M23: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M12 ) @ ( dom @ A @ B @ M23 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ M12 @ M23 )
= ( map_add @ A @ B @ M23 @ M12 ) ) ) ).
% map_add_comm
thf(fact_5981_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M2: A > ( option @ B ),M9: A > ( option @ B ),N: A > ( option @ B ),N6: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M9 )
=> ( ( ord_less_eq @ ( A > ( option @ B ) ) @ N @ N6 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M9 ) @ ( dom @ A @ B @ N6 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M2 @ N ) @ ( map_add @ A @ B @ M9 @ N6 ) ) ) ) ) ) ).
% map_add_distinct_le
thf(fact_5982_restrict__map__eq_I1_J,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: set @ B,K3: B] :
( ( ( restrict_map @ B @ A @ M2 @ A3 @ K3 )
= ( none @ A ) )
= ( ~ ( member @ B @ K3 @ ( inf_inf @ ( set @ B ) @ ( dom @ B @ A @ M2 ) @ A3 ) ) ) ) ).
% restrict_map_eq(1)
thf(fact_5983_map__card__eq__iff,axiom,
! [B: $tType,A: $tType,M5: A > ( option @ B ),X3: A,Y: A] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M5 ) )
=> ( ( ( finite_card @ A @ ( dom @ A @ B @ M5 ) )
= ( finite_card @ B @ ( ran @ A @ B @ M5 ) ) )
=> ( ( member @ A @ X3 @ ( dom @ A @ B @ M5 ) )
=> ( ( ( M5 @ X3 )
= ( M5 @ Y ) )
= ( X3 = Y ) ) ) ) ) ).
% map_card_eq_iff
thf(fact_5984_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_5985_map__to__set__dom,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M: A > ( option @ B )] : ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M ) ) ) ) ).
% map_to_set_dom
thf(fact_5986_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_5987_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_5988_dom__map__of__conv__image__fst,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( dom @ A @ B @ ( map_of @ A @ B @ Xys ) )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ).
% dom_map_of_conv_image_fst
thf(fact_5989_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_5990_ran__is__image,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M10: B > ( option @ A )] : ( image2 @ B @ A @ ( comp @ ( option @ A ) @ A @ B @ ( the2 @ A ) @ M10 ) @ ( dom @ B @ A @ M10 ) ) ) ) ).
% ran_is_image
thf(fact_5991_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( the2 @ B @ ( M @ X4 ) ) )
@ ( dom @ A @ B @ M ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_5992_Chains__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( set @ A )
@ ^ [C4: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ C4 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ C4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% Chains_def
thf(fact_5993_graph__map__add,axiom,
! [B: $tType,A: $tType,M12: A > ( option @ B ),M23: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M12 ) @ ( dom @ A @ B @ M23 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( graph @ A @ B @ ( map_add @ A @ B @ M12 @ M23 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( graph @ A @ B @ M12 ) @ ( graph @ A @ B @ M23 ) ) ) ) ).
% graph_map_add
thf(fact_5994_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X3: A] :
( ( ( dom @ A @ B @ F3 )
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V4: B] :
( F3
= ( fun_upd @ A @ ( option @ B )
@ ^ [X4: A] : ( none @ B )
@ X3
@ ( some @ B @ V4 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_5995_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list @ A,M2: A > ( option @ B )] :
( ( ( set2 @ A @ Xs )
= ( dom @ A @ B @ M2 ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K4: A] : ( product_Pair @ A @ B @ K4 @ ( the2 @ B @ ( M2 @ K4 ) ) )
@ Xs ) )
= M2 ) ) ).
% map_of_map_keys
thf(fact_5996_set__to__map__def,axiom,
! [A: $tType,B: $tType] :
( ( set_to_map @ B @ A )
= ( ^ [S2: set @ ( product_prod @ B @ A ),K4: B] :
( eps_Opt @ A
@ ^ [V4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K4 @ V4 ) @ S2 ) ) ) ) ).
% set_to_map_def
thf(fact_5997_dom__override__on,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),G: A > ( option @ B ),A3: set @ A] :
( ( dom @ A @ B @ ( override_on @ A @ ( option @ B ) @ F3 @ G @ A3 ) )
= ( sup_sup @ ( set @ A )
@ ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F3 )
@ ( collect @ A
@ ^ [A5: A] : ( member @ A @ A5 @ ( minus_minus @ ( set @ A ) @ A3 @ ( dom @ A @ B @ G ) ) ) ) )
@ ( collect @ A
@ ^ [A5: A] : ( member @ A @ A5 @ ( inf_inf @ ( set @ A ) @ A3 @ ( dom @ A @ B @ G ) ) ) ) ) ) ).
% dom_override_on
thf(fact_5998_Chains__subset_H,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R2 ) ) )
@ ( chains @ A @ R2 ) ) ) ).
% Chains_subset'
thf(fact_5999_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F3: A > B,G: A > B] :
( ( override_on @ A @ B @ F3 @ G @ ( bot_bot @ ( set @ A ) ) )
= F3 ) ).
% override_on_emptyset
thf(fact_6000_subset__Zorn,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ! [C9: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C9 )
=> ? [X8: set @ A] :
( ( member @ ( set @ A ) @ X8 @ A3 )
& ! [Xa4: set @ A] :
( ( member @ ( set @ A ) @ Xa4 @ C9 )
=> ( ord_less_eq @ ( set @ A ) @ Xa4 @ X8 ) ) ) )
=> ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A3 )
& ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X @ Xa3 )
=> ( Xa3 = X ) ) ) ) ) ).
% subset_Zorn
thf(fact_6001_subset_Ochain__def,axiom,
! [A: $tType,A3: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C3 @ A3 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C3 )
=> ! [Y5: set @ A] :
( ( member @ ( set @ A ) @ Y5 @ C3 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z4: set @ A] : Y4 = Z4
@ X4
@ Y5 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z4: set @ A] : Y4 = Z4
@ Y5
@ X4 ) ) ) ) ) ) ).
% subset.chain_def
thf(fact_6002_subset_OchainI,axiom,
! [A: $tType,C3: set @ ( set @ A ),A3: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C3 @ A3 )
=> ( ! [X: set @ A,Y2: set @ A] :
( ( member @ ( set @ A ) @ X @ C3 )
=> ( ( member @ ( set @ A ) @ Y2 @ C3 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z4: set @ A] : Y4 = Z4
@ X
@ Y2 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z4: set @ A] : Y4 = Z4
@ Y2
@ X ) ) ) )
=> ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 ) ) ) ).
% subset.chainI
thf(fact_6003_subset_Ochain__total,axiom,
! [A: $tType,A3: set @ ( set @ A ),C3: set @ ( set @ A ),X3: set @ A,Y: set @ A] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( member @ ( set @ A ) @ X3 @ C3 )
=> ( ( member @ ( set @ A ) @ Y @ C3 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z4: set @ A] : Y4 = Z4
@ X3
@ Y )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z4: set @ A] : Y4 = Z4
@ Y
@ X3 ) ) ) ) ) ).
% subset.chain_total
thf(fact_6004_pred__on_Ochain__empty,axiom,
! [A: $tType,A3: set @ A,P: A > A > $o] : ( pred_chain @ A @ A3 @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pred_on.chain_empty
thf(fact_6005_subset_Ochain__empty,axiom,
! [A: $tType,A3: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% subset.chain_empty
thf(fact_6006_subset__Zorn_H,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ! [C9: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C9 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C9 ) @ A3 ) )
=> ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A3 )
& ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X @ Xa3 )
=> ( Xa3 = X ) ) ) ) ) ).
% subset_Zorn'
thf(fact_6007_subset__chain__def,axiom,
! [A: $tType,A17: set @ ( set @ A ),C10: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A17 @ ( ord_less @ ( set @ A ) ) @ C10 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C10 @ A17 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C10 )
=> ! [Y5: set @ A] :
( ( member @ ( set @ A ) @ Y5 @ C10 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y5 )
| ( ord_less_eq @ ( set @ A ) @ Y5 @ X4 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_6008_subset__chain__insert,axiom,
! [A: $tType,A17: set @ ( set @ A ),B2: set @ A,B16: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A17 @ ( ord_less @ ( set @ A ) ) @ ( insert3 @ ( set @ A ) @ B2 @ B16 ) )
= ( ( member @ ( set @ A ) @ B2 @ A17 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ B16 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ B2 )
| ( ord_less_eq @ ( set @ A ) @ B2 @ X4 ) ) )
& ( pred_chain @ ( set @ A ) @ A17 @ ( ord_less @ ( set @ A ) ) @ B16 ) ) ) ).
% subset_chain_insert
thf(fact_6009_subset__Zorn__nonempty,axiom,
! [A: $tType,A17: set @ ( set @ A )] :
( ( A17
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [C11: set @ ( set @ A )] :
( ( C11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A17 @ ( ord_less @ ( set @ A ) ) @ C11 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C11 ) @ A17 ) ) )
=> ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A17 )
& ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ A17 )
=> ( ( ord_less_eq @ ( set @ A ) @ X @ Xa3 )
=> ( Xa3 = X ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_6010_override__on__insert_H,axiom,
! [B: $tType,A: $tType,F3: A > B,G: A > B,X3: A,X5: set @ A] :
( ( override_on @ A @ B @ F3 @ G @ ( insert3 @ A @ X3 @ X5 ) )
= ( override_on @ A @ B @ ( fun_upd @ A @ B @ F3 @ X3 @ ( G @ X3 ) ) @ G @ X5 ) ) ).
% override_on_insert'
thf(fact_6011_override__on__insert,axiom,
! [B: $tType,A: $tType,F3: A > B,G: A > B,X3: A,X5: set @ A] :
( ( override_on @ A @ B @ F3 @ G @ ( insert3 @ A @ X3 @ X5 ) )
= ( fun_upd @ A @ B @ ( override_on @ A @ B @ F3 @ G @ X5 ) @ X3 @ ( G @ X3 ) ) ) ).
% override_on_insert
thf(fact_6012_Union__in__chain,axiom,
! [A: $tType,B16: set @ ( set @ A ),A17: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B16 )
=> ( ( B16
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A17 @ ( ord_less @ ( set @ A ) ) @ B16 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B16 ) @ B16 ) ) ) ) ).
% Union_in_chain
thf(fact_6013_Inter__in__chain,axiom,
! [A: $tType,B16: set @ ( set @ A ),A17: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B16 )
=> ( ( B16
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A17 @ ( ord_less @ ( set @ A ) ) @ B16 )
=> ( member @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B16 ) @ B16 ) ) ) ) ).
% Inter_in_chain
thf(fact_6014_subset_Ochain__extend,axiom,
! [A: $tType,A3: set @ ( set @ A ),C3: set @ ( set @ A ),Z2: set @ A] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( member @ ( set @ A ) @ Z2 @ A3 )
=> ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ C3 )
=> ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y4: set @ A,Z4: set @ A] : Y4 = Z4
@ X
@ Z2 ) )
=> ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert3 @ ( set @ A ) @ Z2 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C3 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_6015_pred__on_Ochain__extend,axiom,
! [A: $tType,A3: set @ A,P: A > A > $o,C3: set @ A,Z2: A] :
( ( pred_chain @ A @ A3 @ P @ C3 )
=> ( ( member @ A @ Z2 @ A3 )
=> ( ! [X: A] :
( ( member @ A @ X @ C3 )
=> ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y4: A,Z4: A] : Y4 = Z4
@ X
@ Z2 ) )
=> ( pred_chain @ A @ A3 @ P @ ( sup_sup @ ( set @ A ) @ ( insert3 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) @ C3 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_6016_finite__subset__Union__chain,axiom,
! [A: $tType,A3: set @ A,B16: set @ ( set @ A ),A17: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B16 ) )
=> ( ( B16
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A17 @ ( ord_less @ ( set @ A ) ) @ B16 )
=> ~ ! [B6: set @ A] :
( ( member @ ( set @ A ) @ B6 @ B16 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A3 @ B6 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_6017_Chains__alt__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( ( chains @ A @ R2 )
= ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R2 ) ) ) ) ) ).
% Chains_alt_def
thf(fact_6018_Chains__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R2 )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R2 ) ) ) ) ).
% Chains_subset
thf(fact_6019_chain__subset__alt__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( pred_chain @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) @ ( ord_less @ ( set @ A ) ) ) ) ).
% chain_subset_alt_def
thf(fact_6020_eq__f__restr__ss__eq,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: ( A > ( option @ B ) ) > A > ( option @ B ),A3: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( dom @ A @ B @ ( F3 @ A3 ) ) )
=> ( ( A3
= ( restrict_map @ A @ B @ ( F3 @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ S3 ) ) )
= ( ( map_le @ A @ B @ A3 @ ( F3 @ A3 ) )
& ( S3
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F3 @ A3 ) ) @ ( dom @ A @ B @ A3 ) ) ) ) ) ) ).
% eq_f_restr_ss_eq
thf(fact_6021_map__leD,axiom,
! [A: $tType,B: $tType,M12: A > ( option @ B ),M23: A > ( option @ B ),K3: A,V2: B] :
( ( map_le @ A @ B @ M12 @ M23 )
=> ( ( ( M12 @ K3 )
= ( some @ B @ V2 ) )
=> ( ( M23 @ K3 )
= ( some @ B @ V2 ) ) ) ) ).
% map_leD
thf(fact_6022_map__leI,axiom,
! [B: $tType,A: $tType,M12: A > ( option @ B ),M23: A > ( option @ B )] :
( ! [X: A,V3: B] :
( ( ( M12 @ X )
= ( some @ B @ V3 ) )
=> ( ( M23 @ X )
= ( some @ B @ V3 ) ) )
=> ( map_le @ A @ B @ M12 @ M23 ) ) ).
% map_leI
thf(fact_6023_eq__f__restr__conv,axiom,
! [B: $tType,A: $tType,S3: set @ A,F3: ( A > ( option @ B ) ) > A > ( option @ B ),A3: A > ( option @ B )] :
( ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( dom @ A @ B @ ( F3 @ A3 ) ) )
& ( A3
= ( restrict_map @ A @ B @ ( F3 @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ S3 ) ) ) )
= ( ( map_le @ A @ B @ A3 @ ( F3 @ A3 ) )
& ( S3
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F3 @ A3 ) ) @ ( dom @ A @ B @ A3 ) ) ) ) ) ).
% eq_f_restr_conv
thf(fact_6024_le__map__mmupd__not__dom,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),K6: set @ A,V2: B] : ( map_le @ A @ B @ M2 @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K6 @ ( dom @ A @ B @ M2 ) ) @ V2 ) ) ).
% le_map_mmupd_not_dom
thf(fact_6025_map__mmupd__update__less,axiom,
! [A: $tType,B: $tType,K6: set @ A,K9: set @ A,M2: A > ( option @ B ),V2: B] :
( ( ord_less_eq @ ( set @ A ) @ K6 @ K9 )
=> ( map_le @ A @ B @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K6 @ ( dom @ A @ B @ M2 ) ) @ V2 ) @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K9 @ ( dom @ A @ B @ M2 ) ) @ V2 ) ) ) ).
% map_mmupd_update_less
thf(fact_6026_mmupd__notin__upd,axiom,
! [B: $tType,A: $tType,K3: A,K6: set @ A,M2: A > ( option @ B ),V2: B] :
( ~ ( member @ A @ K3 @ K6 )
=> ( ( map_mmupd @ A @ B @ M2 @ K6 @ V2 @ K3 )
= ( M2 @ K3 ) ) ) ).
% mmupd_notin_upd
thf(fact_6027_map__mmupd__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),V2: B] :
( ( map_mmupd @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) @ V2 )
= M2 ) ).
% map_mmupd_empty
thf(fact_6028_mmupd__in__upd,axiom,
! [A: $tType,B: $tType,K3: A,K6: set @ A,M2: A > ( option @ B ),V2: B] :
( ( member @ A @ K3 @ K6 )
=> ( ( map_mmupd @ A @ B @ M2 @ K6 @ V2 @ K3 )
= ( some @ B @ V2 ) ) ) ).
% mmupd_in_upd
thf(fact_6029_dom__mmupd,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K6: set @ A,V2: B] :
( ( dom @ A @ B @ ( map_mmupd @ A @ B @ M2 @ K6 @ V2 ) )
= ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ M2 ) @ K6 ) ) ).
% dom_mmupd
thf(fact_6030_map__mmupd__def,axiom,
! [A: $tType,B: $tType] :
( ( map_mmupd @ B @ A )
= ( ^ [M: B > ( option @ A ),K8: set @ B,V4: A,K4: B] : ( if @ ( option @ A ) @ ( member @ B @ K4 @ K8 ) @ ( some @ A @ V4 ) @ ( M @ K4 ) ) ) ) ).
% map_mmupd_def
thf(fact_6031_map__mmupdE,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),K6: set @ B,V2: A,K3: B,X3: A] :
( ( ( map_mmupd @ B @ A @ M2 @ K6 @ V2 @ K3 )
= ( some @ A @ X3 ) )
=> ( ( ~ ( member @ B @ K3 @ K6 )
=> ( ( M2 @ K3 )
!= ( some @ A @ X3 ) ) )
=> ~ ( ( member @ B @ K3 @ K6 )
=> ( X3 != V2 ) ) ) ) ).
% map_mmupdE
thf(fact_6032_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_6033_the__dflt__None__empty,axiom,
! [A: $tType] :
( ( dflt_None_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( none @ ( set @ A ) ) ) ).
% the_dflt_None_empty
thf(fact_6034_dflt__None__set__def,axiom,
! [A: $tType] :
( ( dflt_None_set @ A )
= ( ^ [S2: set @ A] :
( if @ ( option @ ( set @ A ) )
@ ( S2
= ( bot_bot @ ( set @ A ) ) )
@ ( none @ ( set @ A ) )
@ ( some @ ( set @ A ) @ S2 ) ) ) ) ).
% dflt_None_set_def
thf(fact_6035_the__dflt__None__set,axiom,
! [A: $tType,X3: set @ A] :
( ( the_default @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( dflt_None_set @ A @ X3 ) )
= X3 ) ).
% the_dflt_None_set
thf(fact_6036_Func__empty,axiom,
! [B: $tType,A: $tType,B2: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B2 )
= ( insert3 @ ( A > B )
@ ^ [X4: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_6037_Func__non__emp,axiom,
! [A: $tType,B: $tType,B2: set @ A,A3: set @ B] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_6038_Func__is__emp,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A3 @ B2 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_6039_the__default_Osimps_I1_J,axiom,
! [A: $tType,Uu: A,X3: A] :
( ( the_default @ A @ Uu @ ( some @ A @ X3 ) )
= X3 ) ).
% the_default.simps(1)
thf(fact_6040_the__default_Osimps_I2_J,axiom,
! [A: $tType,X3: A] :
( ( the_default @ A @ X3 @ ( none @ A ) )
= X3 ) ).
% the_default.simps(2)
thf(fact_6041_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A18: set @ B,B17: set @ A,F23: C > D,B24: set @ C,A24: set @ D] :
( ( ( image2 @ B @ A @ F1 @ A18 )
= B17 )
=> ( ( inj_on @ C @ D @ F23 @ B24 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image2 @ C @ D @ F23 @ B24 ) @ A24 )
=> ( ( ( B24
= ( bot_bot @ ( set @ C ) ) )
=> ( A24
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B24 @ B17 )
= ( image2 @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B24 @ F1 @ F23 ) @ ( bNF_Wellorder_Func @ D @ B @ A24 @ A18 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_6042_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: A > B,C3: set @ A,B2: set @ A,X3: A] :
( ( inj_on @ A @ B @ G @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ B2 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I4: B] : ( if @ A @ ( member @ B @ I4 @ ( image2 @ A @ B @ G @ C3 ) ) @ ( the_inv_into @ A @ B @ C3 @ G @ I4 ) @ X3 )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B2 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_6043_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B4: A,A7: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert3 @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) @ B4
@ ( the @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert3 @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_6044_insort__insert__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord329482645794927042rt_key @ B @ A )
= ( ^ [F7: B > A,X4: B,Xs3: list @ B] : ( if @ ( list @ B ) @ ( member @ A @ ( F7 @ X4 ) @ ( image2 @ B @ A @ F7 @ ( set2 @ B @ Xs3 ) ) ) @ Xs3 @ ( linorder_insort_key @ B @ A @ F7 @ X4 @ Xs3 ) ) ) ) ) ).
% insort_insert_key_def
thf(fact_6045_distinct__insort__insert,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F3: B > A,X3: B] :
( ( distinct @ B @ Xs )
=> ( distinct @ B @ ( linord329482645794927042rt_key @ B @ A @ F3 @ X3 @ Xs ) ) ) ) ).
% distinct_insort_insert
thf(fact_6046_insort__insert__triv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs: list @ A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs )
= Xs ) ) ) ).
% insort_insert_triv
thf(fact_6047_insort__insert__key__triv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs: list @ B] :
( ( member @ A @ ( F3 @ X3 ) @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) )
=> ( ( linord329482645794927042rt_key @ B @ A @ F3 @ X3 @ Xs )
= Xs ) ) ) ).
% insort_insert_key_triv
thf(fact_6048_set__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs: list @ A] :
( ( set2 @ A
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs ) )
= ( insert3 @ A @ X3 @ ( set2 @ A @ Xs ) ) ) ) ).
% set_insort_insert
thf(fact_6049_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,X3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs ) ) ) ) ).
% sorted_insort_insert
thf(fact_6050_insort__insert__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Xs: list @ A] :
( ~ ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X3
@ Xs ) ) ) ) ).
% insort_insert_insort
thf(fact_6051_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,X3: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linord329482645794927042rt_key @ B @ A @ F3 @ X3 @ Xs ) ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_6052_insort__insert__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X3: B,Xs: list @ B] :
( ~ ( member @ A @ ( F3 @ X3 ) @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) )
=> ( ( linord329482645794927042rt_key @ B @ A @ F3 @ X3 @ Xs )
= ( linorder_insort_key @ B @ A @ F3 @ X3 @ Xs ) ) ) ) ).
% insort_insert_insort_key
thf(fact_6053_maps__def,axiom,
! [B: $tType,A: $tType] :
( ( maps @ A @ B )
= ( ^ [F7: A > ( list @ B ),Xs3: list @ A] : ( concat @ B @ ( map @ A @ ( list @ B ) @ F7 @ Xs3 ) ) ) ) ).
% maps_def
thf(fact_6054_concat__map__maps,axiom,
! [A: $tType,B: $tType,F3: B > ( list @ A ),Xs: list @ B] :
( ( concat @ A @ ( map @ B @ ( list @ A ) @ F3 @ Xs ) )
= ( maps @ B @ A @ F3 @ Xs ) ) ).
% concat_map_maps
thf(fact_6055_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F3: B > ( list @ A )] :
( ( maps @ B @ A @ F3 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_6056_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F3: B > ( list @ A ),X3: B,Xs: list @ B] :
( ( maps @ B @ A @ F3 @ ( cons @ B @ X3 @ Xs ) )
= ( append @ A @ ( F3 @ X3 ) @ ( maps @ B @ A @ F3 @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_6057_mset__set__Union,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( mset_set @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset_set @ A @ A3 ) @ ( mset_set @ A @ B2 ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_6058_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A4 = B3 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_6059_mset__set_Oempty,axiom,
! [A: $tType] :
( ( mset_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% mset_set.empty
thf(fact_6060_mset__set_Oinsert,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( mset_set @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( add_mset @ A @ X3 @ ( mset_set @ A @ A3 ) ) ) ) ) ).
% mset_set.insert
thf(fact_6061_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_6062_mset__set__empty__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ( mset_set @ A @ A3 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A3 ) ) ) ).
% mset_set_empty_iff
thf(fact_6063_infinite__set__mset__mset__set,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( set_mset @ A @ ( mset_set @ A @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_set_mset_mset_set
thf(fact_6064_mset__set__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( mset_set @ A @ ( set2 @ A @ Xs ) )
= ( mset @ A @ Xs ) ) ) ).
% mset_set_set
thf(fact_6065_mset__set__Diff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( minus_minus @ ( multiset @ A ) @ ( mset_set @ A @ A3 ) @ ( mset_set @ A @ B2 ) ) ) ) ) ).
% mset_set_Diff
thf(fact_6066_count__mset__set__finite__iff,axiom,
! [A: $tType,S: set @ A,A4: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( member @ A @ A4 @ S )
=> ( ( count @ A @ ( mset_set @ A @ S ) @ A4 )
= ( one_one @ nat ) ) )
& ( ~ ( member @ A @ A4 @ S )
=> ( ( count @ A @ ( mset_set @ A @ S ) @ A4 )
= ( zero_zero @ nat ) ) ) ) ) ).
% count_mset_set_finite_iff
thf(fact_6067_Zorns__po__lemma,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ! [C9: set @ A] :
( ( member @ ( set @ A ) @ C9 @ ( chains @ A @ R2 ) )
=> ? [X8: A] :
( ( member @ A @ X8 @ ( field2 @ A @ R2 ) )
& ! [Xa4: A] :
( ( member @ A @ Xa4 @ C9 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa4 @ X8 ) @ R2 ) ) ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R2 ) )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Xa3 ) @ R2 )
=> ( Xa3 = X ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_6068_mset__set_Oremove,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( mset_set @ A @ A3 )
= ( add_mset @ A @ X3 @ ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% mset_set.remove
thf(fact_6069_mset__set_Oinsert__remove,axiom,
! [A: $tType,A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( mset_set @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( add_mset @ A @ X3 @ ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% mset_set.insert_remove
thf(fact_6070_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_6071_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert3 @ A @ X3 @ A3 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_6072_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Sup_fin.singleton
thf(fact_6073_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_6074_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ A @ X4 @ X3 ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_6075_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ A6 @ X3 ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X3 ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_6076_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ X3 )
=> ! [A15: A] :
( ( member @ A @ A15 @ A3 )
=> ( ord_less_eq @ A @ A15 @ X3 ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_6077_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_6078_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A3 )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_6079_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_6080_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H2: A > A,N7: set @ A] :
( ! [X: A,Y2: A] :
( ( H2 @ ( sup_sup @ A @ X @ Y2 ) )
= ( sup_sup @ A @ ( H2 @ X ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N7 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_6081_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B2 ) @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_6082_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A,Y2: A] : ( member @ A @ ( sup_sup @ A @ X @ Y2 ) @ ( insert3 @ A @ X @ ( insert3 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_6083_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( sup_sup @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_6084_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_6085_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X3 @ A3 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_6086_Sup__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,Xs: list @ A] :
( ( lattic5882676163264333800up_fin @ A @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs @ X3 ) ) ) ).
% Sup_fin.set_eq_fold
thf(fact_6087_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X3 @ ( lattic5882676163264333800up_fin @ A @ A3 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A] :
( ( Uu3
= ( inf_inf @ A @ X3 @ A5 ) )
& ( member @ A @ A5 @ A3 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_6088_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ B2 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A,B4: A] :
( ( Uu3
= ( inf_inf @ A @ A5 @ B4 ) )
& ( member @ A @ A5 @ A3 )
& ( member @ A @ B4 @ B2 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_6089_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert3 @ A @ X3 @ A3 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_6090_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_6091_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% Inf_fin.singleton
thf(fact_6092_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_6093_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
=> ! [A15: A] :
( ( member @ A @ A15 @ A3 )
=> ( ord_less_eq @ A @ X3 @ A15 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_6094_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( ord_less_eq @ A @ X3 @ A6 ) )
=> ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_6095_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ A @ X3 @ X4 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_6096_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A3 )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_6097_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_6098_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B2 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_6099_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H2: A > A,N7: set @ A] :
( ! [X: A,Y2: A] :
( ( H2 @ ( inf_inf @ A @ X @ Y2 ) )
= ( inf_inf @ A @ ( H2 @ X ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N7 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_6100_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B2 ) @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_6101_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( inf_inf @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_6102_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A,Y2: A] : ( member @ A @ ( inf_inf @ A @ X @ Y2 ) @ ( insert3 @ A @ X @ ( insert3 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ A3 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_6103_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic7752659483105999362nf_fin @ A @ B2 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_6104_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert3 @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X3 @ A3 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_6105_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic5882676163264333800up_fin @ A @ A3 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_6106_Inf__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Xs: list @ A] :
( ( lattic7752659483105999362nf_fin @ A @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs @ X3 ) ) ) ).
% Inf_fin.set_eq_fold
thf(fact_6107_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,X3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X3 @ ( lattic7752659483105999362nf_fin @ A @ A3 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A] :
( ( Uu3
= ( sup_sup @ A @ X3 @ A5 ) )
& ( member @ A @ A5 @ A3 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_6108_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A3 ) @ ( lattic7752659483105999362nf_fin @ A @ B2 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A5: A,B4: A] :
( ( Uu3
= ( sup_sup @ A @ A5 @ B4 ) )
& ( member @ A @ A5 @ A3 )
& ( member @ A @ B4 @ B2 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_6109_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B2: set @ A,F: A > ( filter @ B ),P: B > $o] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ B2 )
=> ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ? [X8: A] :
( ( member @ A @ X8 @ B2 )
& ( ord_less_eq @ ( filter @ B ) @ ( F @ X8 ) @ ( inf_inf @ ( filter @ B ) @ ( F @ A6 ) @ ( F @ B5 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F @ B2 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ B2 )
& ( eventually @ B @ P @ ( F @ X4 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_6110_prod__mset_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: multiset @ B,B2: multiset @ B,G: B > A] :
( ( ( inter_mset @ B @ A3 @ B2 )
= ( zero_zero @ ( multiset @ B ) ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ ( union_mset @ B @ A3 @ B2 ) ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ A3 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% prod_mset.union_disjoint
thf(fact_6111_eventually__top,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( top_top @ ( filter @ A ) ) )
= ( ! [X9: A] : ( P @ X9 ) ) ) ).
% eventually_top
thf(fact_6112_eventually__const,axiom,
! [A: $tType,F: filter @ A,P: $o] :
( ( F
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X4: A] : P
@ F )
= P ) ) ).
% eventually_const
thf(fact_6113_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_6114_eventually__sequentially__seg,axiom,
! [P: nat > $o,K3: nat] :
( ( eventually @ nat
@ ^ [N2: nat] : ( P @ ( plus_plus @ nat @ N2 @ K3 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_6115_prod__mset_Oadd__mset,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [X3: A,N7: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( add_mset @ A @ X3 @ N7 ) )
= ( times_times @ A @ X3 @ ( comm_m9189036328036947845d_mset @ A @ N7 ) ) ) ) ).
% prod_mset.add_mset
thf(fact_6116_prod__mset__Un,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: multiset @ A,B2: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ A3 ) @ ( comm_m9189036328036947845d_mset @ A @ B2 ) ) ) ) ).
% prod_mset_Un
thf(fact_6117_prod__mset_Ounion,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M5: multiset @ A,N7: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M5 @ N7 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ M5 ) @ ( comm_m9189036328036947845d_mset @ A @ N7 ) ) ) ) ).
% prod_mset.union
thf(fact_6118_eventually__finite__subsets__at__top__weakI,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ! [X11: set @ A] :
( ( finite_finite2 @ A @ X11 )
=> ( ( ord_less_eq @ ( set @ A ) @ X11 @ A3 )
=> ( P @ X11 ) ) )
=> ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) ) ) ).
% eventually_finite_subsets_at_top_weakI
thf(fact_6119_prod__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,X3: B,A3: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ ( add_mset @ B @ X3 @ A3 ) ) )
= ( times_times @ A @ ( G @ X3 ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ A3 ) ) ) ) ) ).
% prod_mset.insert
thf(fact_6120_trivial__limit__def,axiom,
! [A: $tType,F: filter @ A] :
( ( F
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X4: A] : $false
@ F ) ) ).
% trivial_limit_def
thf(fact_6121_eventually__const__iff,axiom,
! [A: $tType,P: $o,F: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] : P
@ F )
= ( P
| ( F
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_6122_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X: A] :
~ ( P @ X )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_6123_eventually__happens_H,axiom,
! [A: $tType,F: filter @ A,P: A > $o] :
( ( F
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F )
=> ? [X_1: A] : ( P @ X_1 ) ) ) ).
% eventually_happens'
thf(fact_6124_eventually__happens,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ( eventually @ A @ P @ Net )
=> ( ( Net
= ( bot_bot @ ( filter @ A ) ) )
| ? [X_1: A] : ( P @ X_1 ) ) ) ).
% eventually_happens
thf(fact_6125_eventually__bot,axiom,
! [A: $tType,P: A > $o] : ( eventually @ A @ P @ ( bot_bot @ ( filter @ A ) ) ) ).
% eventually_bot
thf(fact_6126_eventually__finite__subsets__at__top__finite,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) )
= ( P @ A3 ) ) ) ).
% eventually_finite_subsets_at_top_finite
thf(fact_6127_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A3: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( eventually @ B
@ ^ [Y5: B] : ( P @ Y5 @ X )
@ Net ) )
=> ( eventually @ B
@ ^ [X4: B] :
! [Y5: A] :
( ( member @ A @ Y5 @ A3 )
=> ( P @ X4 @ Y5 ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_6128_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ B
@ ^ [X4: B] :
! [Y5: A] :
( ( member @ A @ Y5 @ A3 )
=> ( P @ X4 @ Y5 ) )
@ Net )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( eventually @ B
@ ^ [Y5: B] : ( P @ Y5 @ X4 )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_6129_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y2: B] :
( eventually @ A
@ ^ [X4: A] : ( P @ X4 @ Y2 )
@ Net )
=> ( eventually @ A
@ ^ [X4: A] :
! [X9: B] : ( P @ X4 @ X9 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_6130_eventually__ex,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
? [X9: B] : ( P @ X4 @ X9 )
@ F )
= ( ? [Y9: A > B] :
( eventually @ A
@ ^ [X4: A] : ( P @ X4 @ ( Y9 @ X4 ) )
@ F ) ) ) ).
% eventually_ex
thf(fact_6131_eventually__principal,axiom,
! [A: $tType,P: A > $o,S: set @ A] :
( ( eventually @ A @ P @ ( principal @ A @ S ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( P @ X4 ) ) ) ) ).
% eventually_principal
thf(fact_6132_eventually__inf,axiom,
! [A: $tType,P: A > $o,F: filter @ A,F2: filter @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F @ F2 ) )
= ( ? [Q2: A > $o,R4: A > $o] :
( ( eventually @ A @ Q2 @ F )
& ( eventually @ A @ R4 @ F2 )
& ! [X4: A] :
( ( ( Q2 @ X4 )
& ( R4 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_inf
thf(fact_6133_always__eventually,axiom,
! [A: $tType,P: A > $o,F: filter @ A] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ F ) ) ).
% always_eventually
thf(fact_6134_not__eventuallyD,axiom,
! [A: $tType,P: A > $o,F: filter @ A] :
( ~ ( eventually @ A @ P @ F )
=> ? [X: A] :
~ ( P @ X ) ) ).
% not_eventuallyD
thf(fact_6135_eventually__mono,axiom,
! [A: $tType,P: A > $o,F: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F )
=> ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( eventually @ A @ Q @ F ) ) ) ).
% eventually_mono
thf(fact_6136_filter__eq__iff,axiom,
! [A: $tType] :
( ( ^ [Y4: filter @ A,Z4: filter @ A] : Y4 = Z4 )
= ( ^ [F4: filter @ A,F8: filter @ A] :
! [P2: A > $o] :
( ( eventually @ A @ P2 @ F4 )
= ( eventually @ A @ P2 @ F8 ) ) ) ) ).
% filter_eq_iff
thf(fact_6137_eventuallyI,axiom,
! [A: $tType,P: A > $o,F: filter @ A] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ F ) ) ).
% eventuallyI
thf(fact_6138_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X4: A] : X4 != C2
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_6139_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F )
=> ( ~ ( eventually @ A @ Q @ F )
=> ~ ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F ) ) ) ).
% not_eventually_impI
thf(fact_6140_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) )
@ F )
= ( ( eventually @ A @ P @ F )
& ( eventually @ A @ Q @ F ) ) ) ).
% eventually_conj_iff
thf(fact_6141_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F )
=> ( eventually @ A @ Q @ F ) ) ) ).
% eventually_rev_mp
thf(fact_6142_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F: filter @ A] :
( ( eventually @ A
@ ^ [N2: A] :
( ( P @ N2 )
= ( Q @ N2 ) )
@ F )
=> ( ( eventually @ A @ P @ F )
= ( eventually @ A @ Q @ F ) ) ) ).
% eventually_subst
thf(fact_6143_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F: filter @ A,Q: A > $o,R: A > $o] :
( ( eventually @ A @ P @ F )
=> ( ( eventually @ A @ Q @ F )
=> ( ! [I5: A] :
( ( P @ I5 )
=> ( ( Q @ I5 )
=> ( R @ I5 ) ) )
=> ( eventually @ A @ R @ F ) ) ) ) ).
% eventually_elim2
thf(fact_6144_eventually__conj,axiom,
! [A: $tType,P: A > $o,F: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F )
=> ( ( eventually @ A @ Q @ F )
=> ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) )
@ F ) ) ) ).
% eventually_conj
thf(fact_6145_eventually__True,axiom,
! [A: $tType,F: filter @ A] :
( eventually @ A
@ ^ [X4: A] : $true
@ F ) ).
% eventually_True
thf(fact_6146_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F )
=> ( ( eventually @ A @ P @ F )
=> ( eventually @ A @ Q @ F ) ) ) ).
% eventually_mp
thf(fact_6147_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C3: $o,F: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
| C3 )
@ F )
= ( ( eventually @ A @ P @ F )
| C3 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_6148_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C3: $o,P: A > $o,F: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( C3
| ( P @ X4 ) )
@ F )
= ( C3
| ( eventually @ A @ P @ F ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_6149_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C3: $o,P: A > $o,F: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( C3
=> ( P @ X4 ) )
@ F )
= ( C3
=> ( eventually @ A @ P @ F ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_6150_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N2: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_6151_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X4: A] : X4 != C2
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_6152_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F: filter @ A,F3: B > A,G5: filter @ B] :
( ( eventually @ A @ P @ F )
=> ( ( filterlim @ B @ A @ F3 @ F @ G5 )
=> ( eventually @ B
@ ^ [X4: B] : ( P @ ( F3 @ X4 ) )
@ G5 ) ) ) ).
% eventually_compose_filterlim
thf(fact_6153_filterlim__cong,axiom,
! [A: $tType,B: $tType,F12: filter @ A,F14: filter @ A,F22: filter @ B,F25: filter @ B,F3: B > A,G: B > A] :
( ( F12 = F14 )
=> ( ( F22 = F25 )
=> ( ( eventually @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= ( G @ X4 ) )
@ F22 )
=> ( ( filterlim @ B @ A @ F3 @ F12 @ F22 )
= ( filterlim @ B @ A @ G @ F14 @ F25 ) ) ) ) ) ).
% filterlim_cong
thf(fact_6154_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F7: A > B,F26: filter @ B,F15: filter @ A] :
! [P2: B > $o] :
( ( eventually @ B @ P2 @ F26 )
=> ( eventually @ A
@ ^ [X4: A] : ( P2 @ ( F7 @ X4 ) )
@ F15 ) ) ) ) ).
% filterlim_iff
thf(fact_6155_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N9: A] :
! [N2: A] :
( ( ord_less @ A @ N9 @ N2 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_6156_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N9: A] :
! [N2: A] :
( ( ord_less @ A @ N2 @ N9 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_6157_eventually__Sup,axiom,
! [A: $tType,P: A > $o,S: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Sup_Sup @ ( filter @ A ) @ S ) )
= ( ! [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ S )
=> ( eventually @ A @ P @ X4 ) ) ) ) ).
% eventually_Sup
thf(fact_6158_eventually__sequentiallyI,axiom,
! [C2: nat,P: nat > $o] :
( ! [X: nat] :
( ( ord_less_eq @ nat @ C2 @ X )
=> ( P @ X ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_6159_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N9: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N9 @ N2 )
=> ( P @ N2 ) ) ) ) ).
% eventually_sequentially
thf(fact_6160_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,P: A > $o] :
( ! [X: A] :
( ( ord_less_eq @ A @ C2 @ X )
=> ( P @ X ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_6161_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N9: A] :
! [N2: A] :
( ( ord_less_eq @ A @ N9 @ N2 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_6162_le__filter__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( filter @ A ) )
= ( ^ [F4: filter @ A,F8: filter @ A] :
! [P2: A > $o] :
( ( eventually @ A @ P2 @ F8 )
=> ( eventually @ A @ P2 @ F4 ) ) ) ) ).
% le_filter_def
thf(fact_6163_filter__leI,axiom,
! [A: $tType,F2: filter @ A,F: filter @ A] :
( ! [P6: A > $o] :
( ( eventually @ A @ P6 @ F2 )
=> ( eventually @ A @ P6 @ F ) )
=> ( ord_less_eq @ ( filter @ A ) @ F @ F2 ) ) ).
% filter_leI
thf(fact_6164_filter__leD,axiom,
! [A: $tType,F: filter @ A,F2: filter @ A,P: A > $o] :
( ( ord_less_eq @ ( filter @ A ) @ F @ F2 )
=> ( ( eventually @ A @ P @ F2 )
=> ( eventually @ A @ P @ F ) ) ) ).
% filter_leD
thf(fact_6165_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N9: A] :
! [N2: A] :
( ( ord_less_eq @ A @ N2 @ N9 )
=> ( P @ N2 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_6166_eventually__sup,axiom,
! [A: $tType,P: A > $o,F: filter @ A,F2: filter @ A] :
( ( eventually @ A @ P @ ( sup_sup @ ( filter @ A ) @ F @ F2 ) )
= ( ( eventually @ A @ P @ F )
& ( eventually @ A @ P @ F2 ) ) ) ).
% eventually_sup
thf(fact_6167_eventually__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] : ( eventually @ A @ ( ord_less_eq @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_ge_at_top
thf(fact_6168_eventually__gt__at__top,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] : ( eventually @ A @ ( ord_less @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_gt_at_top
thf(fact_6169_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_6170_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_6171_le__sequentially,axiom,
! [F: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F @ ( at_top @ nat ) )
= ( ! [N9: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N9 ) @ F ) ) ) ).
% le_sequentially
thf(fact_6172_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F3: A > B,F: filter @ B,G5: filter @ A,F2: filter @ B,G6: filter @ A,F16: A > B] :
( ( filterlim @ A @ B @ F3 @ F @ G5 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F @ F2 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G6 @ G5 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= ( F16 @ X4 ) )
@ G6 )
=> ( filterlim @ A @ B @ F16 @ F2 @ G6 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_6173_filterlim__principal,axiom,
! [B: $tType,A: $tType,F3: A > B,S: set @ B,F: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( principal @ B @ S ) @ F )
= ( eventually @ A
@ ^ [X4: A] : ( member @ B @ ( F3 @ X4 ) @ S )
@ F ) ) ).
% filterlim_principal
thf(fact_6174_le__principal,axiom,
! [A: $tType,F: filter @ A,A3: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F @ ( principal @ A @ A3 ) )
= ( eventually @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 )
@ F ) ) ).
% le_principal
thf(fact_6175_eventually__INF1,axiom,
! [B: $tType,A: $tType,I: A,I3: set @ A,P: B > $o,F: A > ( filter @ B )] :
( ( member @ A @ I @ I3 )
=> ( ( eventually @ B @ P @ ( F @ I ) )
=> ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F @ I3 ) ) ) ) ) ).
% eventually_INF1
thf(fact_6176_eventually__inf__principal,axiom,
! [A: $tType,P: A > $o,F: filter @ A,S3: set @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F @ ( principal @ A @ S3 ) ) )
= ( eventually @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( P @ X4 ) )
@ F ) ) ).
% eventually_inf_principal
thf(fact_6177_prod__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H2: B > A,A3: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( G @ X4 ) @ ( H2 @ X4 ) )
@ A3 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ A3 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ H2 @ A3 ) ) ) ) ) ).
% prod_mset.distrib
thf(fact_6178_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P: B > $o,G: B > A] :
( ! [X: A,Y2: A] :
( ( Q @ X )
=> ( ( Q @ Y2 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ) )
=> ( ! [X: B] :
( ( P @ X )
=> ( ( F3 @ ( G @ X ) )
= X ) )
=> ( ! [X: B] :
( ( P @ X )
=> ( Q @ ( G @ X ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_6179_eventually__finite__subsets__at__top,axiom,
! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A3 ) )
= ( ? [X9: set @ A] :
( ( finite_finite2 @ A @ X9 )
& ( ord_less_eq @ ( set @ A ) @ X9 @ A3 )
& ! [Y9: set @ A] :
( ( ( finite_finite2 @ A @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ X9 @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ Y9 @ A3 ) )
=> ( P @ Y9 ) ) ) ) ) ).
% eventually_finite_subsets_at_top
thf(fact_6180_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X4: A] :
! [Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( P @ Y5 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_6181_eventually__Inf__base,axiom,
! [A: $tType,B2: set @ ( filter @ A ),P: A > $o] :
( ( B2
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F9: filter @ A] :
( ( member @ ( filter @ A ) @ F9 @ B2 )
=> ! [G7: filter @ A] :
( ( member @ ( filter @ A ) @ G7 @ B2 )
=> ? [X8: filter @ A] :
( ( member @ ( filter @ A ) @ X8 @ B2 )
& ( ord_less_eq @ ( filter @ A ) @ X8 @ ( inf_inf @ ( filter @ A ) @ F9 @ G7 ) ) ) ) )
=> ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B2 ) )
= ( ? [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ B2 )
& ( eventually @ A @ P @ X4 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_6182_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,F: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F3 @ ( at_top @ A ) @ F )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ F )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_6183_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z9 @ ( F3 @ X4 ) )
@ F ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_6184_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z9 @ ( F3 @ X4 ) )
@ F ) ) ) ) ).
% filterlim_at_top
thf(fact_6185_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ Z9 @ ( F3 @ X4 ) )
@ F ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_6186_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: B > $o,F: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F @ A3 ) ) )
= ( ? [Q2: A > B > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( eventually @ B @ ( Q2 @ X4 ) @ ( F @ X4 ) ) )
& ! [Y5: B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( Q2 @ X4 @ Y5 ) )
=> ( P @ Y5 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_6187_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z9 )
@ F ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_6188_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z9 )
@ F ) ) ) ) ).
% filterlim_at_bot
thf(fact_6189_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F3: A > B,F: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F3 @ X4 ) @ Z9 )
@ F ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_6190_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F )
= ( ! [Z9: B] :
( ( ord_less @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z9 @ ( F3 @ X4 ) )
@ F ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_6191_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > ( filter @ A ),B2: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F @ B2 ) ) )
= ( ? [X9: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X9 @ B2 )
& ( finite_finite2 @ B @ X9 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F @ X9 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_6192_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F )
= ( ! [Z9: B] :
( ( ord_less @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z9 )
@ F ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_6193_eventually__Inf,axiom,
! [A: $tType,P: A > $o,B2: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B2 ) )
= ( ? [X9: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X9 @ B2 )
& ( finite_finite2 @ ( filter @ A ) @ X9 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ X9 ) ) ) ) ) ).
% eventually_Inf
thf(fact_6194_filterlim__finite__subsets__at__top,axiom,
! [A: $tType,B: $tType,F3: A > ( set @ B ),A3: set @ B,F: filter @ A] :
( ( filterlim @ A @ ( set @ B ) @ F3 @ ( finite5375528669736107172at_top @ B @ A3 ) @ F )
= ( ! [X9: set @ B] :
( ( ( finite_finite2 @ B @ X9 )
& ( ord_less_eq @ ( set @ B ) @ X9 @ A3 ) )
=> ( eventually @ A
@ ^ [Y5: A] :
( ( finite_finite2 @ B @ ( F3 @ Y5 ) )
& ( ord_less_eq @ ( set @ B ) @ X9 @ ( F3 @ Y5 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F3 @ Y5 ) @ A3 ) )
@ F ) ) ) ) ).
% filterlim_finite_subsets_at_top
thf(fact_6195_prod__mset_Oremove,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [X3: A,A3: multiset @ A] :
( ( member @ A @ X3 @ ( set_mset @ A @ A3 ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ A3 )
= ( times_times @ A @ X3 @ ( comm_m9189036328036947845d_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A3 @ ( add_mset @ A @ X3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% prod_mset.remove
thf(fact_6196_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: B > A,Y6: set @ B,X5: set @ A,F: filter @ B,F3: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G @ Y6 ) @ X5 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ B @ X4 @ Y6 )
@ F )
=> ( ( map_filter_on @ A @ C @ X5 @ F3 @ ( map_filter_on @ B @ A @ Y6 @ G @ F ) )
= ( map_filter_on @ B @ C @ Y6 @ ( comp @ A @ C @ B @ F3 @ G ) @ F ) ) ) ) ).
% map_filter_on_comp
thf(fact_6197_Sup__filter__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( filter @ A ) )
= ( ^ [S2: set @ ( filter @ A )] :
( abs_filter @ A
@ ^ [P2: A > $o] :
! [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ S2 )
=> ( eventually @ A @ P2 @ X4 ) ) ) ) ) ).
% Sup_filter_def
thf(fact_6198_map__filter__on__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B )
= ( ^ [X9: set @ A,F7: A > B,F4: filter @ A] :
( abs_filter @ B
@ ^ [P2: B > $o] :
( eventually @ A
@ ^ [X4: A] :
( ( P2 @ ( F7 @ X4 ) )
& ( member @ A @ X4 @ X9 ) )
@ F4 ) ) ) ) ).
% map_filter_on_def
thf(fact_6199_bot__filter__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P2: A > $o] : $true ) ) ).
% bot_filter_def
thf(fact_6200_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X5: set @ A,F: filter @ A,P: B > $o,F3: A > B] :
( ( eventually @ A
@ ^ [X4: A] : ( member @ A @ X4 @ X5 )
@ F )
=> ( ( eventually @ B @ P @ ( map_filter_on @ A @ B @ X5 @ F3 @ F ) )
= ( eventually @ A
@ ^ [X4: A] :
( ( P @ ( F3 @ X4 ) )
& ( member @ A @ X4 @ X5 ) )
@ F ) ) ) ).
% eventually_map_filter_on
thf(fact_6201_sup__filter__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( filter @ A ) )
= ( ^ [F4: filter @ A,F8: filter @ A] :
( abs_filter @ A
@ ^ [P2: A > $o] :
( ( eventually @ A @ P2 @ F4 )
& ( eventually @ A @ P2 @ F8 ) ) ) ) ) ).
% sup_filter_def
thf(fact_6202_principal__def,axiom,
! [A: $tType] :
( ( principal @ A )
= ( ^ [S2: set @ A] : ( abs_filter @ A @ ( ball @ A @ S2 ) ) ) ) ).
% principal_def
thf(fact_6203_top__filter__def,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P5: A > $o] :
! [X6: A] : ( P5 @ X6 ) ) ) ).
% top_filter_def
thf(fact_6204_inf__filter__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( filter @ A ) )
= ( ^ [F4: filter @ A,F8: filter @ A] :
( abs_filter @ A
@ ^ [P2: A > $o] :
? [Q2: A > $o,R4: A > $o] :
( ( eventually @ A @ Q2 @ F4 )
& ( eventually @ A @ R4 @ F8 )
& ! [X4: A] :
( ( ( Q2 @ X4 )
& ( R4 @ X4 ) )
=> ( P2 @ X4 ) ) ) ) ) ) ).
% inf_filter_def
thf(fact_6205_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_6206_bit__concat__bit__iff,axiom,
! [M2: nat,K3: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M2 @ K3 @ L ) @ N )
= ( ( ( ord_less @ nat @ N @ M2 )
& ( bit_se5641148757651400278ts_bit @ int @ K3 @ N ) )
| ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_6207_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A4 )
=> ( A4 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_6208_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( A4 != Top )
= ( Less @ A4 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_6209_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A4 )
= ( A4 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_6210_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A4 ) ) ).
% ordering_top.extremum_strict
thf(fact_6211_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( Less_eq @ A4 @ Top ) ) ).
% ordering_top.extremum
thf(fact_6212_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_6213_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X4: nat,Y5: nat] : ( ord_less_eq @ nat @ Y5 @ X4 )
@ ^ [X4: nat,Y5: nat] : ( ord_less @ nat @ Y5 @ X4 )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_6214_filtercomap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtercomap @ A @ B )
= ( ^ [F7: A > B,F4: filter @ B] :
( abs_filter @ A
@ ^ [P2: A > $o] :
? [Q2: B > $o] :
( ( eventually @ B @ Q2 @ F4 )
& ! [X4: A] :
( ( Q2 @ ( F7 @ X4 ) )
=> ( P2 @ X4 ) ) ) ) ) ) ).
% filtercomap_def
thf(fact_6215_multp__def,axiom,
! [A: $tType] :
( ( multp @ A )
= ( ^ [R5: A > A > $o,M10: multiset @ A,N9: multiset @ A] : ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M10 @ N9 ) @ ( mult @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% multp_def
thf(fact_6216_filterlim__filtercomap,axiom,
! [A: $tType,B: $tType,F3: A > B,F: filter @ B] : ( filterlim @ A @ B @ F3 @ F @ ( filtercomap @ A @ B @ F3 @ F ) ) ).
% filterlim_filtercomap
thf(fact_6217_filtercomap__bot,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( filtercomap @ A @ B @ F3 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtercomap_bot
thf(fact_6218_eventually__filtercomapI,axiom,
! [B: $tType,A: $tType,P: A > $o,F: filter @ A,F3: B > A] :
( ( eventually @ A @ P @ F )
=> ( eventually @ B
@ ^ [X4: B] : ( P @ ( F3 @ X4 ) )
@ ( filtercomap @ B @ A @ F3 @ F ) ) ) ).
% eventually_filtercomapI
thf(fact_6219_filtercomap__top,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( filtercomap @ A @ B @ F3 @ ( top_top @ ( filter @ B ) ) )
= ( top_top @ ( filter @ A ) ) ) ).
% filtercomap_top
thf(fact_6220_filtercomap__principal,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ B] :
( ( filtercomap @ A @ B @ F3 @ ( principal @ B @ A3 ) )
= ( principal @ A @ ( vimage @ A @ B @ F3 @ A3 ) ) ) ).
% filtercomap_principal
thf(fact_6221_less__multiset__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ ( multiset @ A ) )
= ( multp @ A @ ( ord_less @ A ) ) ) ) ).
% less_multiset_def
thf(fact_6222_filtercomap__ident,axiom,
! [A: $tType,F: filter @ A] :
( ( filtercomap @ A @ A
@ ^ [X4: A] : X4
@ F )
= F ) ).
% filtercomap_ident
thf(fact_6223_filtercomap__filtercomap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,G: B > C,F: filter @ C] :
( ( filtercomap @ A @ B @ F3 @ ( filtercomap @ B @ C @ G @ F ) )
= ( filtercomap @ A @ C
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) )
@ F ) ) ).
% filtercomap_filtercomap
thf(fact_6224_filtercomap__inf,axiom,
! [A: $tType,B: $tType,F3: A > B,F12: filter @ B,F22: filter @ B] :
( ( filtercomap @ A @ B @ F3 @ ( inf_inf @ ( filter @ B ) @ F12 @ F22 ) )
= ( inf_inf @ ( filter @ A ) @ ( filtercomap @ A @ B @ F3 @ F12 ) @ ( filtercomap @ A @ B @ F3 @ F22 ) ) ) ).
% filtercomap_inf
thf(fact_6225_filtercomap__mono,axiom,
! [B: $tType,A: $tType,F: filter @ A,F2: filter @ A,F3: B > A] :
( ( ord_less_eq @ ( filter @ A ) @ F @ F2 )
=> ( ord_less_eq @ ( filter @ B ) @ ( filtercomap @ B @ A @ F3 @ F ) @ ( filtercomap @ B @ A @ F3 @ F2 ) ) ) ).
% filtercomap_mono
thf(fact_6226_eventually__filtercomap,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: A > B,F: filter @ B] :
( ( eventually @ A @ P @ ( filtercomap @ A @ B @ F3 @ F ) )
= ( ? [Q2: B > $o] :
( ( eventually @ B @ Q2 @ F )
& ! [X4: A] :
( ( Q2 @ ( F3 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap
thf(fact_6227_filterlim__filtercomap__iff,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B,G: B > C,G5: filter @ C,F: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( filtercomap @ B @ C @ G @ G5 ) @ F )
= ( filterlim @ A @ C @ ( comp @ B @ C @ A @ G @ F3 ) @ G5 @ F ) ) ).
% filterlim_filtercomap_iff
thf(fact_6228_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F: filter @ A,F3: B > A] :
( ! [P6: A > $o] :
( ( eventually @ A @ P6 @ F )
=> ? [X8: B] : ( P6 @ ( F3 @ X8 ) ) )
=> ( ( filtercomap @ B @ A @ F3 @ F )
!= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtercomap_neq_bot
thf(fact_6229_filterlim__iff__le__filtercomap,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F7: A > B,F4: filter @ B,G8: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ G8 @ ( filtercomap @ A @ B @ F7 @ F4 ) ) ) ) ).
% filterlim_iff_le_filtercomap
thf(fact_6230_filtercomap__sup,axiom,
! [A: $tType,B: $tType,F3: A > B,F12: filter @ B,F22: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( sup_sup @ ( filter @ A ) @ ( filtercomap @ A @ B @ F3 @ F12 ) @ ( filtercomap @ A @ B @ F3 @ F22 ) ) @ ( filtercomap @ A @ B @ F3 @ ( sup_sup @ ( filter @ B ) @ F12 @ F22 ) ) ) ).
% filtercomap_sup
thf(fact_6231_filtercomap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,F: C > ( filter @ B ),B2: set @ C] :
( ( filtercomap @ A @ B @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F @ B2 ) ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B4: C] : ( filtercomap @ A @ B @ F3 @ ( F @ B4 ) )
@ B2 ) ) ) ).
% filtercomap_INF
thf(fact_6232_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F: filter @ A,F3: B > A] :
( ( F
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( filtercomap @ B @ A @ F3 @ F )
!= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% filtercomap_neq_bot_surj
thf(fact_6233_eventually__filtercomap__at__top__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F3: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F3 @ ( at_top @ A ) ) )
= ( ? [N9: A] :
! [X4: B] :
( ( ord_less_eq @ A @ N9 @ ( F3 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_top_linorder
thf(fact_6234_eventually__filtercomap__at__top__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: B > $o,F3: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F3 @ ( at_top @ A ) ) )
= ( ? [N9: A] :
! [X4: B] :
( ( ord_less @ A @ N9 @ ( F3 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_top_dense
thf(fact_6235_eventually__filtercomap__at__bot__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F3: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F3 @ ( at_bot @ A ) ) )
= ( ? [N9: A] :
! [X4: B] :
( ( ord_less_eq @ A @ ( F3 @ X4 ) @ N9 )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
thf(fact_6236_eventually__filtercomap__at__bot__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: B > $o,F3: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F3 @ ( at_bot @ A ) ) )
= ( ? [N9: A] :
! [X4: B] :
( ( ord_less @ A @ ( F3 @ X4 ) @ N9 )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_dense
thf(fact_6237_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > C,F: B > ( filter @ C ),B2: set @ B] :
( ord_less_eq @ ( filter @ A )
@ ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [B4: B] : ( filtercomap @ A @ C @ F3 @ ( F @ B4 ) )
@ B2 ) )
@ ( filtercomap @ A @ C @ F3 @ ( complete_Sup_Sup @ ( filter @ C ) @ ( image2 @ B @ ( filter @ C ) @ F @ B2 ) ) ) ) ).
% filtercomap_SUP
thf(fact_6238_pair__lessI2,axiom,
! [A4: nat,B3: nat,S3: nat,T2: nat] :
( ( ord_less_eq @ nat @ A4 @ B3 )
=> ( ( ord_less @ nat @ S3 @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A4 @ S3 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_6239_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_6240_euclidean__size__greater__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
= ( B3
!= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_greater_0_iff
thf(fact_6241_pair__less__iff1,axiom,
! [X3: nat,Y: nat,Z2: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ Y ) @ ( product_Pair @ nat @ nat @ X3 @ Z2 ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y @ Z2 ) ) ).
% pair_less_iff1
thf(fact_6242_euclidean__size__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A4: A,B3: A] :
( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ nat @ ( euclid6346220572633701492n_size @ A @ A4 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ).
% euclidean_size_mult
thf(fact_6243_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A4 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% size_mult_mono
thf(fact_6244_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A4 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B3 @ A4 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_6245_euclidean__size__times__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ).
% euclidean_size_times_unit
thf(fact_6246_dvd__proper__imp__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ~ ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A4 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_6247_mod__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B3: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ ( modulo_modulo @ A @ A4 @ B3 ) ) @ ( euclid6346220572633701492n_size @ A @ B3 ) ) ) ) ).
% mod_size_less
thf(fact_6248_pair__lessI1,axiom,
! [A4: nat,B3: nat,S3: nat,T2: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A4 @ S3 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_6249_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,A4: A] :
( ( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
=> ( A4
!= ( times_times @ A @ ( divide_divide @ A @ A4 @ B3 ) @ B3 ) ) ) )
=> ( ( ( B3
!= ( zero_zero @ A ) )
=> ! [Q6: A,R3: A] :
( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A4 @ B3 )
= Q6 )
=> ( ( ( modulo_modulo @ A @ A4 @ B3 )
= R3 )
=> ( A4
!= ( plus_plus @ A @ ( times_times @ A @ Q6 @ B3 ) @ R3 ) ) ) ) ) ) ) )
=> ( B3
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_6250_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q4: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B3 ) @ R2 )
= A4 )
=> ( ( modulo_modulo @ A @ A4 @ B3 )
= R2 ) ) ) ) ) ) ).
% mod_eqI
thf(fact_6251_division__segment__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A4 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A4 ) ) )
= A4 ) ) ).
% division_segment_euclidean_size
thf(fact_6252_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A4 ) @ ( euclid7384307370059645450egment @ A @ B3 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_6253_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A4: A,B3: A] :
( ( ( euclid7384307370059645450egment @ A @ A4 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ( divide_divide @ A @ A4 @ B3 )
= ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A4 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
| ( B3
= ( zero_zero @ A ) ) ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_6254_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q4: A,A4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B3 ) @ R2 )
= A4 )
=> ( ( divide_divide @ A @ A4 @ B3 )
= Q4 ) ) ) ) ) ) ).
% div_eqI
thf(fact_6255_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B3: A,R2: A,Q4: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B3 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B3 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q4 @ B3 ) @ R2 ) @ B3 )
= Q4 ) ) ) ) ) ).
% div_bounded
thf(fact_6256_smin__insertI,axiom,
! [X3: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X3 @ XS2 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_min_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ ( insert3 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_6257_pair__leqI2,axiom,
! [A4: nat,B3: nat,S3: nat,T2: nat] :
( ( ord_less_eq @ nat @ A4 @ B3 )
=> ( ( ord_less_eq @ nat @ S3 @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A4 @ S3 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_6258_smin__emptyI,axiom,
! [X5: set @ ( product_prod @ nat @ nat )] :
( ( X5
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X5 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_6259_pair__leqI1,axiom,
! [A4: nat,B3: nat,S3: nat,T2: nat] :
( ( ord_less @ nat @ A4 @ B3 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A4 @ S3 ) @ ( product_Pair @ nat @ nat @ B3 @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_6260_wmax__insertI,axiom,
! [Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X3: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ YS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_max_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert3 @ ( product_prod @ nat @ nat ) @ X3 @ XS2 ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_6261_wmin__insertI,axiom,
! [X3: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X3 @ XS2 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_min_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ ( insert3 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_6262_wmax__emptyI,axiom,
! [X5: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ X5 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_6263_wmin__emptyI,axiom,
! [X5: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X5 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_6264_min__rpair__set,axiom,
fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict @ fun_min_weak ) ).
% min_rpair_set
thf(fact_6265_max__weak__def,axiom,
( fun_max_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert3 @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% max_weak_def
thf(fact_6266_min__weak__def,axiom,
( fun_min_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert3 @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% min_weak_def
thf(fact_6267_smax__insertI,axiom,
! [Y: product_prod @ nat @ nat,Y6: set @ ( product_prod @ nat @ nat ),X3: product_prod @ nat @ nat,X5: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y6 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X5 @ Y6 ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert3 @ ( product_prod @ nat @ nat ) @ X3 @ X5 ) @ Y6 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_6268_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_6269_distinct__adj__Cons__Cons,axiom,
! [B: $tType,X3: B,Y: B,Xs: list @ B] :
( ( distinct_adj @ B @ ( cons @ B @ X3 @ ( cons @ B @ Y @ Xs ) ) )
= ( ( X3 != Y )
& ( distinct_adj @ B @ ( cons @ B @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_6270_distinct__adj__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct_adj @ A @ ( rev @ A @ Xs ) )
= ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_rev
thf(fact_6271_distinct__adj__mapD,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( distinct_adj @ A @ ( map @ B @ A @ F3 @ Xs ) )
=> ( distinct_adj @ B @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_6272_distinct__adj__Nil,axiom,
! [A: $tType] : ( distinct_adj @ A @ ( nil @ A ) ) ).
% distinct_adj_Nil
thf(fact_6273_distinct__adj__altdef,axiom,
! [A: $tType] :
( ( distinct_adj @ A )
= ( ^ [Xs3: list @ A] :
( ( remdups_adj @ A @ Xs3 )
= Xs3 ) ) ) ).
% distinct_adj_altdef
thf(fact_6274_distinct__adj__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] : ( distinct_adj @ A @ ( remdups_adj @ A @ Xs ) ) ).
% distinct_adj_remdups_adj
thf(fact_6275_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_6276_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_6277_distinct__adj__ConsD,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( distinct_adj @ A @ ( cons @ A @ X3 @ Xs ) )
=> ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_6278_distinct__adj__singleton,axiom,
! [B: $tType,X3: B] : ( distinct_adj @ B @ ( cons @ B @ X3 @ ( nil @ B ) ) ) ).
% distinct_adj_singleton
thf(fact_6279_max__rpair__set,axiom,
fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_max_strict @ fun_max_weak ) ).
% max_rpair_set
thf(fact_6280_distinct__adj__mapI,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B] :
( ( distinct_adj @ A @ Xs )
=> ( ( inj_on @ A @ B @ F3 @ ( set2 @ A @ Xs ) )
=> ( distinct_adj @ B @ ( map @ A @ B @ F3 @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_6281_distinct__adj__map__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( set2 @ A @ Xs ) )
=> ( ( distinct_adj @ B @ ( map @ A @ B @ F3 @ Xs ) )
= ( distinct_adj @ A @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_6282_distinct__adj__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( distinct_adj @ A @ ( cons @ A @ X3 @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ( ( X3
!= ( hd @ A @ Xs ) )
& ( distinct_adj @ A @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_6283_smax__emptyI,axiom,
! [Y6: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y6 )
=> ( ( Y6
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ Y6 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_6284_wmsI,axiom,
! [A3: multiset @ ( product_prod @ nat @ nat ),B2: multiset @ ( product_prod @ nat @ nat ),Z7: multiset @ ( product_prod @ nat @ nat )] :
( ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A3 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B2 ) ) @ fun_max_strict )
| ( ( A3
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( B2
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z7 @ A3 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z7 @ B2 ) ) @ ms_weak ) ) ).
% wmsI
thf(fact_6285_smsI,axiom,
! [A3: multiset @ ( product_prod @ nat @ nat ),B2: multiset @ ( product_prod @ nat @ nat ),Z7: multiset @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A3 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B2 ) ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z7 @ A3 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z7 @ B2 ) ) @ ms_strict ) ) ).
% smsI
thf(fact_6286_ms__reduction__pair,axiom,
fun_reduction_pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ms_strict @ ms_weak ) ).
% ms_reduction_pair
thf(fact_6287_ms__weakI1,axiom,
! [Z7: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat ),A3: multiset @ ( product_prod @ nat @ nat ),B2: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z7 @ Z10 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A3 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B2 ) ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z7 @ A3 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ B2 ) ) @ ms_weak ) ) ) ).
% ms_weakI1
thf(fact_6288_ms__strictI,axiom,
! [Z7: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat ),A3: multiset @ ( product_prod @ nat @ nat ),B2: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z7 @ Z10 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A3 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B2 ) ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z7 @ A3 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ B2 ) ) @ ms_strict ) ) ) ).
% ms_strictI
thf(fact_6289_pw__leq__lstep,axiom,
! [X3: product_prod @ nat @ nat,Y: product_prod @ nat @ nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_leq )
=> ( pw_leq @ ( add_mset @ ( product_prod @ nat @ nat ) @ X3 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ).
% pw_leq_lstep
thf(fact_6290_pw__leq__step,axiom,
! [X3: product_prod @ nat @ nat,Y: product_prod @ nat @ nat,X5: multiset @ ( product_prod @ nat @ nat ),Y6: multiset @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y ) @ fun_pair_leq )
=> ( ( pw_leq @ X5 @ Y6 )
=> ( pw_leq @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X3 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X5 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y6 ) ) ) ) ).
% pw_leq_step
thf(fact_6291_pw__leq_Osimps,axiom,
( pw_leq
= ( ^ [A13: multiset @ ( product_prod @ nat @ nat ),A23: multiset @ ( product_prod @ nat @ nat )] :
( ( ( A13
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( A23
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
| ? [X4: product_prod @ nat @ nat,Y5: product_prod @ nat @ nat,X9: multiset @ ( product_prod @ nat @ nat ),Y9: multiset @ ( product_prod @ nat @ nat )] :
( ( A13
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X4 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X9 ) )
& ( A23
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y5 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y9 ) )
& ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X4 @ Y5 ) @ fun_pair_leq )
& ( pw_leq @ X9 @ Y9 ) ) ) ) ) ).
% pw_leq.simps
thf(fact_6292_pw__leq_Ocases,axiom,
! [A1: multiset @ ( product_prod @ nat @ nat ),A22: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ A1 @ A22 )
=> ( ( ( A1
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
=> ( A22
!= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
=> ~ ! [X: product_prod @ nat @ nat,Y2: product_prod @ nat @ nat,X11: multiset @ ( product_prod @ nat @ nat )] :
( ( A1
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X11 ) )
=> ! [Y11: multiset @ ( product_prod @ nat @ nat )] :
( ( A22
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y2 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y11 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y2 ) @ fun_pair_leq )
=> ~ ( pw_leq @ X11 @ Y11 ) ) ) ) ) ) ).
% pw_leq.cases
thf(fact_6293_ms__weakI2,axiom,
! [Z7: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z7 @ Z10 )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z7 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) @ ms_weak ) ) ).
% ms_weakI2
thf(fact_6294_pw__leq__split,axiom,
! [X5: multiset @ ( product_prod @ nat @ nat ),Y6: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ X5 @ Y6 )
=> ? [A14: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat ),Z11: multiset @ ( product_prod @ nat @ nat )] :
( ( X5
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ A14 @ Z11 ) )
& ( Y6
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ B6 @ Z11 ) )
& ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A14 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
| ( ( B6
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( A14
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% pw_leq_split
thf(fact_6295_proj__def,axiom,
! [A: $tType,B: $tType] :
( ( equiv_proj @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A ),X4: B] : ( image @ B @ A @ R5 @ ( insert3 @ B @ X4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% proj_def
thf(fact_6296_AboveS__def,axiom,
! [A: $tType] :
( ( order_AboveS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ A
@ ^ [B4: A] :
( ( member @ A @ B4 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ( B4 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B4 ) @ R5 ) ) ) ) ) ) ) ).
% AboveS_def
thf(fact_6297_AboveS__disjoint,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( order_AboveS @ A @ R2 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% AboveS_disjoint
thf(fact_6298_wo__rel_Osuc__greater,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ B3 @ B2 )
=> ( ( ( bNF_Wellorder_wo_suc @ A @ R2 @ B2 )
!= B3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B2 ) ) @ R2 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_6299_wo__rel_Osuc__AboveS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B2 ) @ ( order_AboveS @ A @ R2 @ B2 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_6300_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( member @ A @ A4 @ ( order_AboveS @ A @ R2 @ B2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B2 ) @ A4 ) @ R2 ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_6301_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A,A4: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A4 @ ( order_AboveS @ A @ R2 @ B2 ) )
=> ( ! [A11: A] :
( ( member @ A @ A11 @ ( order_AboveS @ A @ R2 @ B2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A11 ) @ R2 ) )
=> ( A4
= ( bNF_Wellorder_wo_suc @ A @ R2 @ B2 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_6302_wo__rel_Osuc__inField,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( order_AboveS @ A @ R2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R2 @ B2 ) @ ( field2 @ A @ R2 ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_6303_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A3 )
=> ( ( ( order_AboveS @ A @ R2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ ( bNF_Wellorder_wo_suc @ A @ R2 @ A3 ) ) @ R2 )
=> ( ( B3
!= ( bNF_Wellorder_wo_suc @ A @ R2 @ A3 ) )
=> ( member @ A @ B3 @ A3 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_6304_proj__iff,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ ( insert3 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 )
=> ( ( ( equiv_proj @ A @ A @ R2 @ X3 )
= ( equiv_proj @ A @ A @ R2 @ Y ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_6305_quotient__eqI,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X5: set @ A,Y6: set @ A,X3: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ A @ X3 @ X5 )
=> ( ( member @ A @ Y @ Y6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( X5 = Y6 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_6306_quotient__eq__iff,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X5: set @ A,Y6: set @ A,X3: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ A @ X3 @ X5 )
=> ( ( member @ A @ Y @ Y6 )
=> ( ( X5 = Y6 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_6307_in__quotient__imp__closed,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X5: set @ A,X3: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ A @ X3 @ X5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( member @ A @ Y @ X5 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_6308_in__quotient__imp__non__empty,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X5: set @ A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( X5
!= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_6309_equiv__listrel,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( equiv_equiv @ ( list @ A ) @ ( lists @ A @ A3 ) @ ( listrel @ A @ A @ R2 ) ) ) ).
% equiv_listrel
thf(fact_6310_equiv__class__self,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ A @ A4 @ ( image @ A @ A @ R2 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_self
thf(fact_6311_quotient__disj,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X5: set @ A,Y6: set @ A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( X5 = Y6 )
| ( ( inf_inf @ ( set @ A ) @ X5 @ Y6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% quotient_disj
thf(fact_6312_equiv__class__eq__iff,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
= ( ( ( image @ A @ A @ R2 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert3 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X3 @ A3 )
& ( member @ A @ Y @ A3 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_6313_eq__equiv__class__iff,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ( image @ A @ A @ R2 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert3 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_6314_equiv__class__eq,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( image @ A @ A @ R2 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_6315_eq__equiv__class,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A,A3: set @ A] :
( ( ( image @ A @ A @ R2 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ A @ B3 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_6316_eq__equiv__class__iff2,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ( equiv_quotient @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( equiv_quotient @ A @ ( insert3 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_6317_refines__equiv__class__eq2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A3: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A3 @ R )
=> ( ( equiv_equiv @ A @ A3 @ S )
=> ( ( image @ A @ A @ S @ ( image @ A @ A @ R @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_6318_refines__equiv__class__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A3: set @ A,A4: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( equiv_equiv @ A @ A3 @ R )
=> ( ( equiv_equiv @ A @ A3 @ S )
=> ( ( image @ A @ A @ R @ ( image @ A @ A @ S @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_6319_equiv__class__subset,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_6320_subset__equiv__class,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),B3: A,A4: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B3 @ A3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_6321_equiv__class__nondisjoint,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X3: A,A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_6322_in__quotient__imp__in__rel,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),X5: set @ A,X3: A,Y: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( member @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ X3 @ ( insert3 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_6323_ofilterIncl__def,axiom,
! [A: $tType] :
( ( bNF_We413866401316099525erIncl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A7: set @ A,B7: set @ A] :
( ( order_ofilter @ A @ R5 @ A7 )
& ( A7
!= ( field2 @ A @ R5 ) )
& ( order_ofilter @ A @ R5 @ B7 )
& ( B7
!= ( field2 @ A @ R5 ) )
& ( ord_less @ ( set @ A ) @ A7 @ B7 ) ) ) ) ) ) ).
% ofilterIncl_def
thf(fact_6324_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A18: set @ A,R12: set @ ( product_prod @ A @ A ),A24: set @ B,R23: set @ ( product_prod @ B @ B ),F3: A > B > ( set @ C ),A1: A,A22: B] :
( ( equiv_equiv @ A @ A18 @ R12 )
=> ( ( equiv_equiv @ B @ A24 @ R23 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R12 @ R23 @ F3 )
=> ( ( member @ A @ A1 @ A18 )
=> ( ( member @ B @ A22 @ A24 )
=> ( ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X13: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F3 @ X13 ) @ ( image @ B @ B @ R23 @ ( insert3 @ B @ A22 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R12 @ ( insert3 @ A @ A1 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F3 @ A1 @ A22 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_6325_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B > C] :
( ! [Y12: A,Z1: A,Y23: B,Z22: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z1 ) @ R12 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y23 @ Z22 ) @ R23 )
=> ( ( F3 @ Y12 @ Y23 )
= ( F3 @ Z1 @ Z22 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 ) ) ).
% congruent2I'
thf(fact_6326_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B > C,Y1: A,Z12: A,Y22: B,Z23: B] :
( ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z12 ) @ R12 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z23 ) @ R23 )
=> ( ( F3 @ Y1 @ Y22 )
= ( F3 @ Z12 @ Z23 ) ) ) ) ) ).
% congruent2D
thf(fact_6327_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > A > B] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ! [Y2: A,Z3: A] :
( ( member @ A @ Y2 @ A3 )
=> ( ( member @ A @ Z3 @ A3 )
=> ( ( F3 @ Y2 @ Z3 )
= ( F3 @ Z3 @ Y2 ) ) ) )
=> ( ! [Y2: A,Z3: A,W: A] :
( ( member @ A @ W @ A3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R2 )
=> ( ( F3 @ W @ Y2 )
= ( F3 @ W @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R2 @ R2 @ F3 ) ) ) ) ).
% congruent2_commuteI
thf(fact_6328_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A18: set @ A,R12: set @ ( product_prod @ A @ A ),A24: set @ B,R23: set @ ( product_prod @ B @ B ),F3: A > B > C] :
( ( equiv_equiv @ A @ A18 @ R12 )
=> ( ( equiv_equiv @ B @ A24 @ R23 )
=> ( ! [Y2: A,Z3: A,W: B] :
( ( member @ B @ W @ A24 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R12 )
=> ( ( F3 @ Y2 @ W )
= ( F3 @ Z3 @ W ) ) ) )
=> ( ! [Y2: B,Z3: B,W: A] :
( ( member @ A @ W @ A18 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z3 ) @ R23 )
=> ( ( F3 @ W @ Y2 )
= ( F3 @ W @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 ) ) ) ) ) ).
% congruent2I
thf(fact_6329_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A18: set @ A,R12: set @ ( product_prod @ A @ A ),A24: set @ B,R23: set @ ( product_prod @ B @ B ),F3: A > B > ( set @ C ),A4: B] :
( ( equiv_equiv @ A @ A18 @ R12 )
=> ( ( equiv_equiv @ B @ A24 @ R23 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R12 @ R23 @ F3 )
=> ( ( member @ B @ A4 @ A24 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R12
@ ^ [X13: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F3 @ X13 ) @ ( image @ B @ B @ R23 @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_6330_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > ( set @ B ),A4: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R2 @ F3 )
=> ( ( member @ A @ A4 @ A3 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ ( image @ A @ A @ R2 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F3 @ A4 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_6331_congruentD,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B,Y: A,Z2: A] :
( ( equiv_congruent @ A @ B @ R2 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( ( F3 @ Y )
= ( F3 @ Z2 ) ) ) ) ).
% congruentD
thf(fact_6332_congruentI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B] :
( ! [Y2: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z3 ) @ R2 )
=> ( ( F3 @ Y2 )
= ( F3 @ Z3 ) ) )
=> ( equiv_congruent @ A @ B @ R2 @ F3 ) ) ).
% congruentI
thf(fact_6333_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > ( set @ B ),X5: set @ A,Y6: set @ A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R2 @ F3 )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ X5 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ Y6 ) ) )
=> ( ( member @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y6 @ ( equiv_quotient @ A @ A3 @ R2 ) )
=> ( ! [X: A,Y2: A] :
( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( ( ( F3 @ X )
= ( F3 @ Y2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 ) ) ) )
=> ( X5 = Y6 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_6334_disjnt__equiv__class,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( equiv_equiv @ A @ A3 @ R2 )
=> ( ( disjnt @ A @ ( image @ A @ A @ R2 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_6335_chains__alt__def,axiom,
! [A: $tType] :
( ( chains2 @ A )
= ( ^ [A7: set @ ( set @ A )] : ( collect @ ( set @ ( set @ A ) ) @ ( pred_chain @ ( set @ A ) @ A7 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% chains_alt_def
thf(fact_6336_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_6337_disjnt__insert1,axiom,
! [A: $tType,A4: A,X5: set @ A,Y6: set @ A] :
( ( disjnt @ A @ ( insert3 @ A @ A4 @ X5 ) @ Y6 )
= ( ~ ( member @ A @ A4 @ Y6 )
& ( disjnt @ A @ X5 @ Y6 ) ) ) ).
% disjnt_insert1
thf(fact_6338_disjnt__insert2,axiom,
! [A: $tType,Y6: set @ A,A4: A,X5: set @ A] :
( ( disjnt @ A @ Y6 @ ( insert3 @ A @ A4 @ X5 ) )
= ( ~ ( member @ A @ A4 @ Y6 )
& ( disjnt @ A @ Y6 @ X5 ) ) ) ).
% disjnt_insert2
thf(fact_6339_disjnt__Un1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( disjnt @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( ( disjnt @ A @ A3 @ C3 )
& ( disjnt @ A @ B2 @ C3 ) ) ) ).
% disjnt_Un1
thf(fact_6340_disjnt__Un2,axiom,
! [A: $tType,C3: set @ A,A3: set @ A,B2: set @ A] :
( ( disjnt @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ( disjnt @ A @ C3 @ A3 )
& ( disjnt @ A @ C3 @ B2 ) ) ) ).
% disjnt_Un2
thf(fact_6341_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A3: set @ A,C3: set @ B,B2: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : C3 )
@ ( product_Sigma @ A @ B @ B2
@ ^ [Uu3: A] : C3 ) )
= ( ( C3
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A3 @ B2 ) ) ) ).
% disjnt_Times2_iff
thf(fact_6342_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C3: set @ A,A3: set @ B,B2: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C3
@ ^ [Uu3: A] : A3 )
@ ( product_Sigma @ A @ B @ C3
@ ^ [Uu3: A] : B2 ) )
= ( ( C3
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A3 @ B2 ) ) ) ).
% disjnt_Times1_iff
thf(fact_6343_disjnt__subset1,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A,Z7: set @ A] :
( ( disjnt @ A @ X5 @ Y6 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z7 @ X5 )
=> ( disjnt @ A @ Z7 @ Y6 ) ) ) ).
% disjnt_subset1
thf(fact_6344_disjnt__subset2,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A,Z7: set @ A] :
( ( disjnt @ A @ X5 @ Y6 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z7 @ Y6 )
=> ( disjnt @ A @ X5 @ Z7 ) ) ) ).
% disjnt_subset2
thf(fact_6345_disjnt__sym,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( disjnt @ A @ A3 @ B2 )
=> ( disjnt @ A @ B2 @ A3 ) ) ).
% disjnt_sym
thf(fact_6346_disjnt__iff,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X4: A] :
~ ( ( member @ A @ X4 @ A7 )
& ( member @ A @ X4 @ B7 ) ) ) ) ).
% disjnt_iff
thf(fact_6347_disjnt__insert,axiom,
! [A: $tType,X3: A,N7: set @ A,M5: set @ A] :
( ~ ( member @ A @ X3 @ N7 )
=> ( ( disjnt @ A @ M5 @ N7 )
=> ( disjnt @ A @ ( insert3 @ A @ X3 @ M5 ) @ N7 ) ) ) ).
% disjnt_insert
thf(fact_6348_disjnt__empty2,axiom,
! [A: $tType,A3: set @ A] : ( disjnt @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% disjnt_empty2
thf(fact_6349_disjnt__empty1,axiom,
! [A: $tType,A3: set @ A] : ( disjnt @ A @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% disjnt_empty1
thf(fact_6350_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: set @ A,C3: A > ( set @ B ),B2: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ C3 ) @ ( product_Sigma @ A @ B @ B2 @ C3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) )
=> ( ( C3 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A3 @ B2 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_6351_disjnt__def,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( inf_inf @ ( set @ A ) @ A7 @ B7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjnt_def
thf(fact_6352_disjoint__image__subset,axiom,
! [A: $tType,A17: set @ ( set @ A ),F3: ( set @ A ) > ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A17 )
=> ( ! [X11: set @ A] :
( ( member @ ( set @ A ) @ X11 @ A17 )
=> ( ord_less_eq @ ( set @ A ) @ ( F3 @ X11 ) @ X11 ) )
=> ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ F3 @ A17 ) ) ) ) ).
% disjoint_image_subset
thf(fact_6353_disjnt__ge__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y6: set @ A,X5: set @ A] :
( ( finite_finite2 @ A @ Y6 )
=> ( ! [X: A] :
( ( member @ A @ X @ X5 )
=> ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ Y6 ) @ X ) )
=> ( disjnt @ A @ X5 @ Y6 ) ) ) ) ).
% disjnt_ge_max
thf(fact_6354_total__inv__image,axiom,
! [B: $tType,A: $tType,F3: A > B,R2: set @ ( product_prod @ B @ B )] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ R2 )
=> ( total_on @ A @ ( top_top @ ( set @ A ) ) @ ( inv_image @ B @ A @ R2 @ F3 ) ) ) ) ).
% total_inv_image
thf(fact_6355_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A3: set @ C,B2: C > ( set @ D ),F3: D > B] :
( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X: C] :
( ( member @ C @ X @ A3 )
=> ( ( B2 @ X )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ D @ B @ F3 @ ( B2 @ X4 ) )
@ A3 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F3 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B2 @ A3 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F3 @ ( B2 @ X4 ) ) )
@ A3 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_6356_bdd__above__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_above_empty
thf(fact_6357_bdd__above__insert,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: A,A3: set @ A] :
( ( condit941137186595557371_above @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( condit941137186595557371_above @ A @ A3 ) ) ) ).
% bdd_above_insert
thf(fact_6358_in__inv__image,axiom,
! [A: $tType,B: $tType,X3: A,Y: A,R2: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( inv_image @ B @ A @ R2 @ F3 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_6359_converse__inv__image,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( converse @ A @ A @ ( inv_image @ B @ A @ R @ F3 ) )
= ( inv_image @ B @ A @ ( converse @ B @ B @ R ) @ F3 ) ) ).
% converse_inv_image
thf(fact_6360_trans__inv__image,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),F3: B > A] :
( ( trans @ A @ R2 )
=> ( trans @ B @ ( inv_image @ A @ B @ R2 @ F3 ) ) ) ).
% trans_inv_image
thf(fact_6361_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A,A4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S ) @ A4 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ord_less_eq @ A @ X4 @ A4 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_6362_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ A,A3: set @ A] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ? [X8: A] :
( ( member @ A @ X8 @ A3 )
& ( ord_less_eq @ A @ B5 @ X8 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B2 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_6363_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 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ X5 )
& ( ord_less @ A @ Y @ X4 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_6364_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A3: set @ B,Y: A,I: B] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ Y )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ ( F3 @ I ) @ Y ) ) ) ) ) ).
% cSUP_lessD
thf(fact_6365_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,U: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_6366_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: C > A,B2: set @ C,F3: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G @ B2 ) )
=> ( ! [N5: B] :
( ( member @ B @ N5 @ A3 )
=> ? [X8: C] :
( ( member @ C @ X8 @ B2 )
& ( ord_less_eq @ A @ ( F3 @ N5 ) @ ( G @ X8 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_6367_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_6368_cSup__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A4: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X5 )
=> ( ( complete_Sup_Sup @ A @ ( insert3 @ A @ A4 @ X5 ) )
= ( sup_sup @ A @ A4 @ ( complete_Sup_Sup @ A @ X5 ) ) ) ) ) ) ).
% cSup_insert
thf(fact_6369_cSup__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A4: A] :
( ( condit941137186595557371_above @ A @ X5 )
=> ( ( ( X5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert3 @ A @ A4 @ X5 ) )
= A4 ) )
& ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert3 @ A @ A4 @ X5 ) )
= ( sup_sup @ A @ A4 @ ( complete_Sup_Sup @ A @ X5 ) ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_6370_cSup__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_6371_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ B ),F7: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F7 @ X4 ) @ ( F7 @ Y5 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_6372_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: set @ B,F3: B > A,A4: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ A4 @ ( F3 @ X4 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_6373_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,G: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ A3 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ A3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F3 @ A5 ) @ ( G @ A5 ) )
@ A3 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_6374_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: B > A,B2: set @ B,F3: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B2 )
=> ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_6375_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( condit941137186595557371_above @ A @ A3 )
=> ( ( condit941137186595557371_above @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_6376_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,A4: B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( insert3 @ B @ A4 @ A3 ) ) )
= ( sup_sup @ A @ ( F3 @ A4 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_6377_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ B2 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ B2 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_6378_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
@ ^ [X4: A] :
! [Y5: A] :
( ( member @ A @ Y5 @ S )
=> ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_6379_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A3: C > A,I3: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ A3 @ I3 ) )
=> ( ( I3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A3 @ X4 ) )
@ I3 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A3 @ I3 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_6380_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A3 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_6381_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R4: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A ),F7: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F7 ) @ ( inv_image @ A @ B @ S2 @ F7 ) ) ) ) ).
% rp_inv_image_def
thf(fact_6382_lenlex__def,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R5 ) )
@ ^ [Xs3: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs3 ) @ Xs3 ) ) ) ) ).
% lenlex_def
thf(fact_6383_less__than__iff,axiom,
! [X3: nat,Y: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X3 @ Y ) @ less_than )
= ( ord_less @ nat @ X3 @ Y ) ) ).
% less_than_iff
thf(fact_6384_measures__def,axiom,
! [A: $tType] :
( ( measures @ A )
= ( ^ [Fs2: list @ ( A > nat )] :
( inv_image @ ( list @ nat ) @ A @ ( lex @ nat @ less_than )
@ ^ [A5: A] :
( map @ ( A > nat ) @ nat
@ ^ [F7: A > nat] : ( F7 @ A5 )
@ Fs2 ) ) ) ) ).
% measures_def
thf(fact_6385_mlex__prod__def,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F7: A > nat,R4: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R4 )
@ ^ [X4: A] : ( product_Pair @ nat @ A @ ( F7 @ X4 ) @ X4 ) ) ) ) ).
% mlex_prod_def
thf(fact_6386_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A3: set @ C,B2: C > ( set @ D ),F3: D > B] :
( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X: C] :
( ( member @ C @ X @ A3 )
=> ( ( B2 @ X )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ D @ B @ F3 @ ( B2 @ X4 ) )
@ A3 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F3 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B2 @ A3 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F3 @ ( B2 @ X4 ) ) )
@ A3 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_6387_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A3: C > A,I3: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ A3 @ I3 ) )
=> ( ( I3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A3 @ I3 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A3 @ X4 ) )
@ I3 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_6388_bdd__below__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_below_empty
thf(fact_6389_bdd__below__insert,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A4: A,A3: set @ A] :
( ( condit1013018076250108175_below @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( condit1013018076250108175_below @ A @ A3 ) ) ) ).
% bdd_below_insert
thf(fact_6390_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S: set @ A,A4: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S )
=> ( ( ord_less_eq @ A @ A4 @ ( complete_Inf_Inf @ A @ S ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ord_less_eq @ A @ A4 @ X4 ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_6391_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ A,A3: set @ A] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B2 )
=> ? [X8: A] :
( ( member @ A @ X8 @ A3 )
& ( ord_less_eq @ A @ X8 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_6392_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 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ X5 )
& ( ord_less @ A @ X4 @ Y ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_6393_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A3: set @ B,Y: A,I: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) )
=> ( ( member @ B @ I @ A3 )
=> ( ord_less @ A @ Y @ ( F3 @ I ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_6394_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B2: set @ B,F3: C > A,A3: set @ C,G: B > A] :
( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F3 @ A3 ) )
=> ( ! [M6: B] :
( ( member @ B @ M6 @ B2 )
=> ? [X8: C] :
( ( member @ C @ X8 @ A3 )
& ( ord_less_eq @ A @ ( F3 @ X8 ) @ ( G @ M6 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B2 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_6395_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,U: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X4 ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_6396_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B2 ) @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_6397_cInf__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A4: A] :
( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X5 )
=> ( ( complete_Inf_Inf @ A @ ( insert3 @ A @ A4 @ X5 ) )
= ( inf_inf @ A @ A4 @ ( complete_Inf_Inf @ A @ X5 ) ) ) ) ) ) ).
% cInf_insert
thf(fact_6398_cInf__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X5: set @ A,A4: A] :
( ( condit1013018076250108175_below @ A @ X5 )
=> ( ( ( X5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert3 @ A @ A4 @ X5 ) )
= A4 ) )
& ( ( X5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert3 @ A @ A4 @ X5 ) )
= ( inf_inf @ A @ A4 @ ( complete_Inf_Inf @ A @ X5 ) ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_6399_cInf__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B2 ) ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_6400_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: set @ B,F3: B > A,A4: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ A4 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ ( F3 @ X4 ) @ A4 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_6401_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,G: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ A3 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A5: B] : ( inf_inf @ A @ ( F3 @ A5 ) @ ( G @ A5 ) )
@ A3 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_6402_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: B > A,B2: set @ B,F3: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ B2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B2 )
=> ( ! [X: B] :
( ( member @ B @ X @ B2 )
=> ( ord_less_eq @ A @ ( G @ X ) @ ( F3 @ X ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_6403_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( condit1013018076250108175_below @ A @ B2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B2 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B2 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_6404_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,A4: B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( insert3 @ B @ A4 @ A3 ) ) )
= ( inf_inf @ A @ ( F3 @ A4 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) ) ) ) ) ) ).
% cINF_insert
thf(fact_6405_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ B2 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B2 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6406_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_6407_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
@ ^ [X4: A] :
! [Y5: A] :
( ( member @ A @ Y5 @ S )
=> ( ord_less_eq @ A @ X4 @ Y5 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_6408_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A3 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A3 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_6409_filterlim__INF__INF,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,J4: set @ A,I3: set @ B,F3: D > C,F: B > ( filter @ D ),G5: A > ( filter @ C )] :
( ! [M6: A] :
( ( member @ A @ M6 @ J4 )
=> ? [X8: B] :
( ( member @ B @ X8 @ I3 )
& ( ord_less_eq @ ( filter @ C ) @ ( filtermap @ D @ C @ F3 @ ( F @ X8 ) ) @ ( G5 @ M6 ) ) ) )
=> ( filterlim @ D @ C @ F3 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ G5 @ J4 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ F @ I3 ) ) ) ) ).
% filterlim_INF_INF
thf(fact_6410_bsqr__max2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A1: A,A22: A,B15: A,B23: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( product_Pair @ A @ A @ B15 @ B23 ) ) @ ( bNF_Wellorder_bsqr @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ A1 @ A22 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R2 @ B15 @ B23 ) ) @ R2 ) ) ) ).
% bsqr_max2
thf(fact_6411_filtermap__id_H,axiom,
! [A: $tType] :
( ( filtermap @ A @ A
@ ^ [X4: A] : X4 )
= ( ^ [F4: filter @ A] : F4 ) ) ).
% filtermap_id'
thf(fact_6412_filtermap__id,axiom,
! [A: $tType] :
( ( filtermap @ A @ A @ ( id @ A ) )
= ( id @ ( filter @ A ) ) ) ).
% filtermap_id
thf(fact_6413_filtermap__bot,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( filtermap @ B @ A @ F3 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_bot
thf(fact_6414_filtermap__principal,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( filtermap @ B @ A @ F3 @ ( principal @ B @ A3 ) )
= ( principal @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) ) ).
% filtermap_principal
thf(fact_6415_eventually__filtermap,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: B > A,F: filter @ B] :
( ( eventually @ A @ P @ ( filtermap @ B @ A @ F3 @ F ) )
= ( eventually @ B
@ ^ [X4: B] : ( P @ ( F3 @ X4 ) )
@ F ) ) ).
% eventually_filtermap
thf(fact_6416_filtermap__mono,axiom,
! [B: $tType,A: $tType,F: filter @ A,F2: filter @ A,F3: A > B] :
( ( ord_less_eq @ ( filter @ A ) @ F @ F2 )
=> ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F3 @ F ) @ ( filtermap @ A @ B @ F3 @ F2 ) ) ) ).
% filtermap_mono
thf(fact_6417_filterlim__filtermap,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B,F12: filter @ B,G: C > A,F22: filter @ C] :
( ( filterlim @ A @ B @ F3 @ F12 @ ( filtermap @ C @ A @ G @ F22 ) )
= ( filterlim @ C @ B
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
@ F12
@ F22 ) ) ).
% filterlim_filtermap
thf(fact_6418_filtermap__filtermap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G: C > B,F: filter @ C] :
( ( filtermap @ B @ A @ F3 @ ( filtermap @ C @ B @ G @ F ) )
= ( filtermap @ C @ A
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
@ F ) ) ).
% filtermap_filtermap
thf(fact_6419_filtermap__ident,axiom,
! [A: $tType,F: filter @ A] :
( ( filtermap @ A @ A
@ ^ [X4: A] : X4
@ F )
= F ) ).
% filtermap_ident
thf(fact_6420_filtermap__eq__strong,axiom,
! [B: $tType,A: $tType,F3: A > B,F: filter @ A,G5: filter @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( filtermap @ A @ B @ F3 @ F )
= ( filtermap @ A @ B @ F3 @ G5 ) )
= ( F = G5 ) ) ) ).
% filtermap_eq_strong
thf(fact_6421_filtermap__sup,axiom,
! [A: $tType,B: $tType,F3: B > A,F12: filter @ B,F22: filter @ B] :
( ( filtermap @ B @ A @ F3 @ ( sup_sup @ ( filter @ B ) @ F12 @ F22 ) )
= ( sup_sup @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ F12 ) @ ( filtermap @ B @ A @ F3 @ F22 ) ) ) ).
% filtermap_sup
thf(fact_6422_filtermap__bot__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,F: filter @ B] :
( ( ( filtermap @ B @ A @ F3 @ F )
= ( bot_bot @ ( filter @ A ) ) )
= ( F
= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtermap_bot_iff
thf(fact_6423_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_6424_well__order__on__domain,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_well_order_on @ A @ A3 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ( member @ A @ A4 @ A3 )
& ( member @ A @ B3 @ A3 ) ) ) ) ).
% well_order_on_domain
thf(fact_6425_filterlim__def,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F7: A > B,F26: filter @ B,F15: filter @ A] : ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F7 @ F15 ) @ F26 ) ) ) ).
% filterlim_def
thf(fact_6426_map__filter__on__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ B ) ) ).
% map_filter_on_UNIV
thf(fact_6427_filtercomap__filtermap,axiom,
! [B: $tType,A: $tType,F: filter @ A,F3: A > B] : ( ord_less_eq @ ( filter @ A ) @ F @ ( filtercomap @ A @ B @ F3 @ ( filtermap @ A @ B @ F3 @ F ) ) ) ).
% filtercomap_filtermap
thf(fact_6428_filtermap__filtercomap,axiom,
! [B: $tType,A: $tType,F3: B > A,F: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ ( filtercomap @ B @ A @ F3 @ F ) ) @ F ) ).
% filtermap_filtercomap
thf(fact_6429_filtermap__le__iff__le__filtercomap,axiom,
! [B: $tType,A: $tType,F3: B > A,F: filter @ B,G5: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ F ) @ G5 )
= ( ord_less_eq @ ( filter @ B ) @ F @ ( filtercomap @ B @ A @ F3 @ G5 ) ) ) ).
% filtermap_le_iff_le_filtercomap
thf(fact_6430_filtermap__sequentually__ne__bot,axiom,
! [A: $tType,F3: nat > A] :
( ( filtermap @ nat @ A @ F3 @ ( at_top @ nat ) )
!= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_sequentually_ne_bot
thf(fact_6431_filtermap__inf,axiom,
! [A: $tType,B: $tType,F3: B > A,F12: filter @ B,F22: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ ( inf_inf @ ( filter @ B ) @ F12 @ F22 ) ) @ ( inf_inf @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ F12 ) @ ( filtermap @ B @ A @ F3 @ F22 ) ) ) ).
% filtermap_inf
thf(fact_6432_filtermap__fun__inverse,axiom,
! [B: $tType,A: $tType,G: A > B,F: filter @ B,G5: filter @ A,F3: B > A] :
( ( filterlim @ A @ B @ G @ F @ G5 )
=> ( ( filterlim @ B @ A @ F3 @ G5 @ F )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ ( G @ X4 ) )
= X4 )
@ G5 )
=> ( ( filtermap @ B @ A @ F3 @ F )
= G5 ) ) ) ) ).
% filtermap_fun_inverse
thf(fact_6433_filtermap__SUP,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,F: C > ( filter @ B ),B2: set @ C] :
( ( filtermap @ B @ A @ F3 @ ( complete_Sup_Sup @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F @ B2 ) ) )
= ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B4: C] : ( filtermap @ B @ A @ F3 @ ( F @ B4 ) )
@ B2 ) ) ) ).
% filtermap_SUP
thf(fact_6434_filtermap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtermap @ A @ B )
= ( ^ [F7: A > B,F4: filter @ A] :
( abs_filter @ B
@ ^ [P2: B > $o] :
( eventually @ A
@ ^ [X4: A] : ( P2 @ ( F7 @ X4 ) )
@ F4 ) ) ) ) ).
% filtermap_def
thf(fact_6435_natLeq__on__well__order__on,axiom,
! [N: nat] :
( order_well_order_on @ nat
@ ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_6436_filtermap__mono__strong,axiom,
! [B: $tType,A: $tType,F3: A > B,F: filter @ A,G5: filter @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ A @ B @ F3 @ F ) @ ( filtermap @ A @ B @ F3 @ G5 ) )
= ( ord_less_eq @ ( filter @ A ) @ F @ G5 ) ) ) ).
% filtermap_mono_strong
thf(fact_6437_filtermap__image__finite__subsets__at__top,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ A] :
( ( inj_on @ A @ B @ F3 @ A3 )
=> ( ( filtermap @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F3 ) @ ( finite5375528669736107172at_top @ A @ A3 ) )
= ( finite5375528669736107172at_top @ B @ ( image2 @ A @ B @ F3 @ A3 ) ) ) ) ).
% filtermap_image_finite_subsets_at_top
thf(fact_6438_filtermap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,F: C > ( filter @ B ),B2: set @ C] :
( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F @ B2 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B4: C] : ( filtermap @ B @ A @ F3 @ ( F @ B4 ) )
@ B2 ) ) ) ).
% filtermap_INF
thf(fact_6439_natLeq__on__Well__order,axiom,
! [N: nat] :
( order_well_order_on @ nat
@ ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_6440_Linear__order__Well__order__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
= ( ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R2 ) )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A7 )
& ! [Y5: A] :
( ( member @ A @ Y5 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_6441_ofilter__subset__embedS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A3 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embedS
thf(fact_6442_bsqr__ofilter,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),D2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ ( product_prod @ A @ A ) @ ( bNF_Wellorder_bsqr @ A @ R2 ) @ D2 )
=> ( ( ord_less @ ( set @ ( product_prod @ A @ A ) ) @ D2
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : ( field2 @ A @ R2 ) ) )
=> ( ~ ? [A6: A] :
( ( field2 @ A @ R2 )
= ( order_under @ A @ R2 @ A6 ) )
=> ? [A14: set @ A] :
( ( order_ofilter @ A @ R2 @ A14 )
& ( ord_less @ ( set @ A ) @ A14 @ ( field2 @ A @ R2 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ D2
@ ( product_Sigma @ A @ A @ A14
@ ^ [Uu3: A] : A14 ) ) ) ) ) ) ) ).
% bsqr_ofilter
thf(fact_6443_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R5 ) ) ) ) ).
% under_def
thf(fact_6444_under__incr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( trans @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R2 @ A4 ) @ ( order_under @ A @ R2 @ B3 ) ) ) ) ).
% under_incr
thf(fact_6445_embedS__Field,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R2 @ R7 @ F3 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( field2 @ A @ R2 ) ) @ ( field2 @ B @ R7 ) ) ) ) ).
% embedS_Field
thf(fact_6446_ofilter__subset__embedS__iso,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A3 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( id @ A ) ) )
& ( ( A3 = B2 )
= ( bNF_Wellorder_iso @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) )
@ ( id @ A ) ) ) ) ) ) ) ).
% ofilter_subset_embedS_iso
thf(fact_6447_iso__forward,axiom,
! [A: $tType,B: $tType,X3: A,Y: A,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( ( bNF_Wellorder_iso @ A @ B @ R2 @ R7 @ F3 )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) ) @ R7 ) ) ) ).
% iso_forward
thf(fact_6448_ofilter__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A3 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ ( field2 @ A @ R2 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
@ R2 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% ofilter_ordLess
thf(fact_6449_ofilter__subset__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A3 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B2 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_6450_ordLess__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R2 ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ).
% ordLess_irreflexive
thf(fact_6451_ordLess__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R7 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_transitive
thf(fact_6452_finite__ordLess__infinite,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ~ ( finite_finite2 @ B @ ( field2 @ B @ R7 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ).
% finite_ordLess_infinite
thf(fact_6453_underS__Restr__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R2 @ A4 )
@ ^ [Uu3: A] : ( order_underS @ A @ R2 @ A4 ) ) )
@ R2 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% underS_Restr_ordLess
thf(fact_6454_ofilter__subset__ordLeq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A,B2: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_ofilter @ A @ R2 @ A3 )
=> ( ( order_ofilter @ A @ R2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B2
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLeq
thf(fact_6455_ordLess__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R7 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordLeq_trans
thf(fact_6456_ordLeq__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R7 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLeq_ordLess_trans
thf(fact_6457_ordLess__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordLess_imp_ordLeq
thf(fact_6458_not__ordLess__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% not_ordLess_ordLeq
thf(fact_6459_underS__Field2,axiom,
! [A: $tType,A4: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R2 @ A4 ) @ ( field2 @ A @ R2 ) ) ) ).
% underS_Field2
thf(fact_6460_underS__empty,axiom,
! [A: $tType,A4: A,R2: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( order_underS @ A @ R2 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% underS_empty
thf(fact_6461_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] :
( ( B4 != A5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R5 ) ) ) ) ) ).
% underS_def
thf(fact_6462_underS__I,axiom,
! [A: $tType,I: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( I != J )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( member @ A @ I @ ( order_underS @ A @ R @ J ) ) ) ) ).
% underS_I
thf(fact_6463_underS__E,axiom,
! [A: $tType,I: A,R: set @ ( product_prod @ A @ A ),J: A] :
( ( member @ A @ I @ ( order_underS @ A @ R @ J ) )
=> ( ( I != J )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R ) ) ) ).
% underS_E
thf(fact_6464_ordLeq__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R7 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_transitive
thf(fact_6465_exists__minim__Well__order,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X @ R )
=> ( order_well_order_on @ A @ ( field2 @ A @ X ) @ X ) )
=> ? [X: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X @ R )
& ! [Xa3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa3 @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X @ Xa3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Well_order
thf(fact_6466_ordLeq__total,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLeq_total
thf(fact_6467_ordLeq__reflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_reflexive
thf(fact_6468_ordLeq__Well__order__simp,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
& ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 ) ) ) ).
% ordLeq_Well_order_simp
thf(fact_6469_ordLeq__iff__ordLess__Restr,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R2 @ X4 )
@ ^ [Uu3: A] : ( order_underS @ A @ R2 @ X4 ) ) )
@ R7 )
@ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
thf(fact_6470_underS__Field3,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R2 @ A4 ) @ ( field2 @ A @ R2 ) ) ) ).
% underS_Field3
thf(fact_6471_ordLess__or__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLess_or_ordLeq
thf(fact_6472_not__ordLeq__iff__ordLess,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ).
% not_ordLeq_iff_ordLess
thf(fact_6473_not__ordLess__iff__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R2 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% not_ordLess_iff_ordLeq
thf(fact_6474_underS__incl__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B3 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A4 ) @ ( order_underS @ A @ R2 @ B3 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_6475_underS__incr,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A,B3: A] :
( ( trans @ A @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A4 ) @ ( order_underS @ A @ R2 @ B3 ) ) ) ) ) ).
% underS_incr
thf(fact_6476_Refl__under__underS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( ( order_under @ A @ R2 @ A4 )
= ( sup_sup @ ( set @ A ) @ ( order_underS @ A @ R2 @ A4 ) @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Refl_under_underS
thf(fact_6477_ordLess__iff__ordIso__Restr,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R2 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R2 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R2 @ X4 )
@ ^ [Uu3: A] : ( order_underS @ A @ R2 @ X4 ) ) ) )
@ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
thf(fact_6478_embed__ordLess__ofilterIncl,axiom,
! [B: $tType,A: $tType,C: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C ),F132: A > C,F232: B > C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R12 @ R23 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R23 @ R32 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( ( bNF_Wellorder_embed @ A @ C @ R12 @ R32 @ F132 )
=> ( ( bNF_Wellorder_embed @ B @ C @ R23 @ R32 @ F232 )
=> ( member @ ( product_prod @ ( set @ C ) @ ( set @ C ) ) @ ( product_Pair @ ( set @ C ) @ ( set @ C ) @ ( image2 @ A @ C @ F132 @ ( field2 @ A @ R12 ) ) @ ( image2 @ B @ C @ F232 @ ( field2 @ B @ R23 ) ) ) @ ( bNF_We413866401316099525erIncl @ C @ R32 ) ) ) ) ) ) ).
% embed_ordLess_ofilterIncl
thf(fact_6479_ordIso__iff__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% ordIso_iff_ordLeq
thf(fact_6480_ordIso__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordIso_imp_ordLeq
thf(fact_6481_ordIso__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R7 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordIso_ordLeq_trans
thf(fact_6482_ordLeq__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R7 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_ordIso_trans
thf(fact_6483_ordIso__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R7 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R9 ) @ ( bNF_Wellorder_ordIso @ A @ C ) ) ) ) ).
% ordIso_transitive
thf(fact_6484_ordIso__symmetric,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R2 ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% ordIso_symmetric
thf(fact_6485_ordLess__not__embed,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ? [X_12: B > A] : ( bNF_Wellorder_embed @ B @ A @ R7 @ R2 @ X_12 ) ) ).
% ordLess_not_embed
thf(fact_6486_not__ordLess__ordIso,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% not_ordLess_ordIso
thf(fact_6487_ordIso__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R7 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordIso_ordLess_trans
thf(fact_6488_ordLess__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R7 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R2 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordIso_trans
thf(fact_6489_finite__well__order__on__ordIso,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( order_well_order_on @ A @ A3 @ R2 )
=> ( ( order_well_order_on @ A @ A3 @ R7 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% finite_well_order_on_ordIso
thf(fact_6490_ordIso__reflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R2 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_reflexive
thf(fact_6491_ordLeq__iff__ordLess__or__ordIso,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% ordLeq_iff_ordLess_or_ordIso
thf(fact_6492_internalize__ordLeq,axiom,
! [A: $tType,B: $tType,R7: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R7 @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [P7: set @ ( product_prod @ B @ B )] :
( ( ord_less_eq @ ( set @ B ) @ ( field2 @ B @ P7 ) @ ( field2 @ B @ R2 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R7 @ P7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P7 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_ordLeq
thf(fact_6493_BNF__Wellorder__Constructions_OordLess__Field,axiom,
! [A: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R12 @ R23 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( bNF_Wellorder_embed @ A @ B @ R12 @ R23 @ F3 )
=> ( ( image2 @ A @ B @ F3 @ ( field2 @ A @ R12 ) )
!= ( field2 @ B @ R23 ) ) ) ) ).
% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_6494_internalize__ordLess,axiom,
! [A: $tType,B: $tType,R7: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R7 @ R2 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ? [P7: set @ ( product_prod @ B @ B )] :
( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P7 ) @ ( field2 @ B @ R2 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R7 @ P7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P7 @ R2 ) @ ( bNF_We4044943003108391690rdLess @ B @ B ) ) ) ) ) ).
% internalize_ordLess
thf(fact_6495_ordLess__iff,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
& ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 )
& ~ ? [X9: B > A] : ( bNF_Wellorder_embed @ B @ A @ R7 @ R2 @ X9 ) ) ) ).
% ordLess_iff
thf(fact_6496_embedS__iff,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R2 @ R7 @ F3 )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R2 @ R7 @ F3 )
= ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( field2 @ A @ R2 ) ) @ ( field2 @ B @ R7 ) ) ) ) ) ).
% embedS_iff
thf(fact_6497_dir__image__ordIso,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( inj_on @ A @ B @ F3 @ ( field2 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ ( bNF_We2720479622203943262_image @ A @ B @ R2 @ F3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image_ordIso
thf(fact_6498_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F: filter @ A,X3: B] :
( ( prod_filter @ A @ B @ F @ ( principal @ B @ ( insert3 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A5: A] : ( product_Pair @ A @ B @ A5 @ X3 )
@ F ) ) ).
% prod_filter_principal_singleton2
thf(fact_6499_prod__filter__eq__bot,axiom,
! [A: $tType,B: $tType,A3: filter @ A,B2: filter @ B] :
( ( ( prod_filter @ A @ B @ A3 @ B2 )
= ( bot_bot @ ( filter @ ( product_prod @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( filter @ A ) ) )
| ( B2
= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% prod_filter_eq_bot
thf(fact_6500_eventually__prod__same,axiom,
! [A: $tType,P: ( product_prod @ A @ A ) > $o,F: filter @ A] :
( ( eventually @ ( product_prod @ A @ A ) @ P @ ( prod_filter @ A @ A @ F @ F ) )
= ( ? [Q2: A > $o] :
( ( eventually @ A @ Q2 @ F )
& ! [X4: A,Y5: A] :
( ( Q2 @ X4 )
=> ( ( Q2 @ Y5 )
=> ( P @ ( product_Pair @ A @ A @ X4 @ Y5 ) ) ) ) ) ) ) ).
% eventually_prod_same
thf(fact_6501_eventually__prod__filter,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,F: filter @ A,G5: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ P @ ( prod_filter @ A @ B @ F @ G5 ) )
= ( ? [Pf: A > $o,Pg: B > $o] :
( ( eventually @ A @ Pf @ F )
& ( eventually @ B @ Pg @ G5 )
& ! [X4: A,Y5: B] :
( ( Pf @ X4 )
=> ( ( Pg @ Y5 )
=> ( P @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) ) ) ) ) ) ).
% eventually_prod_filter
thf(fact_6502_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F: filter @ A,G5: filter @ B,H7: filter @ C] :
( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F @ G5 ) @ H7 )
= ( filtermap @ ( product_prod @ A @ ( product_prod @ B @ C ) ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ( product_case_prod @ A @ ( product_prod @ B @ C ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [X4: A] :
( product_case_prod @ B @ C @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [Y5: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F @ ( prod_filter @ B @ C @ G5 @ H7 ) ) ) ) ).
% prod_filter_assoc
thf(fact_6503_prod__filter__mono,axiom,
! [A: $tType,B: $tType,F: filter @ A,F2: filter @ A,G5: filter @ B,G6: filter @ B] :
( ( ord_less_eq @ ( filter @ A ) @ F @ F2 )
=> ( ( ord_less_eq @ ( filter @ B ) @ G5 @ G6 )
=> ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ F @ G5 ) @ ( prod_filter @ A @ B @ F2 @ G6 ) ) ) ) ).
% prod_filter_mono
thf(fact_6504_filtermap__fst__prod__filter,axiom,
! [B: $tType,A: $tType,A3: filter @ A,B2: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( prod_filter @ A @ B @ A3 @ B2 ) ) @ A3 ) ).
% filtermap_fst_prod_filter
thf(fact_6505_filtermap__snd__prod__filter,axiom,
! [B: $tType,A: $tType,A3: filter @ B,B2: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( prod_filter @ B @ A @ A3 @ B2 ) ) @ B2 ) ).
% filtermap_snd_prod_filter
thf(fact_6506_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,A3: filter @ A,B2: filter @ B,C3: filter @ A,D2: filter @ B] :
( ( A3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( B2
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ A3 @ B2 ) @ ( prod_filter @ A @ B @ C3 @ D2 ) )
= ( ( ord_less_eq @ ( filter @ A ) @ A3 @ C3 )
& ( ord_less_eq @ ( filter @ B ) @ B2 @ D2 ) ) ) ) ) ).
% prod_filter_mono_iff
thf(fact_6507_prod__filtermap1,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,F: filter @ C,G5: filter @ B] :
( ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F3 @ F ) @ G5 )
= ( filtermap @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ ( product_apfst @ C @ A @ B @ F3 ) @ ( prod_filter @ C @ B @ F @ G5 ) ) ) ).
% prod_filtermap1
thf(fact_6508_prod__filtermap2,axiom,
! [B: $tType,A: $tType,C: $tType,F: filter @ A,G: C > B,G5: filter @ C] :
( ( prod_filter @ A @ B @ F @ ( filtermap @ C @ B @ G @ G5 ) )
= ( filtermap @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B ) @ ( product_apsnd @ C @ B @ A @ G ) @ ( prod_filter @ A @ C @ F @ G5 ) ) ) ).
% prod_filtermap2
thf(fact_6509_dir__image__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We2720479622203943262_image @ A @ A2 )
= ( ^ [R5: set @ ( product_prod @ A @ A ),F7: A > A2] :
( collect @ ( product_prod @ A2 @ A2 )
@ ^ [Uu3: product_prod @ A2 @ A2] :
? [A5: A,B4: A] :
( ( Uu3
= ( product_Pair @ A2 @ A2 @ ( F7 @ A5 ) @ ( F7 @ B4 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R5 ) ) ) ) ) ).
% dir_image_def
thf(fact_6510_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B,G5: filter @ B,F: filter @ A,G: A > C,H7: filter @ C] :
( ( filterlim @ A @ B @ F3 @ G5 @ F )
=> ( ( filterlim @ A @ C @ G @ H7 @ F )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( prod_filter @ B @ C @ G5 @ H7 )
@ F ) ) ) ).
% filterlim_Pair
thf(fact_6511_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,G: C > B,F: filter @ C] :
( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) )
@ ( filtermap @ C @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ F )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F3 @ F ) @ ( filtermap @ C @ B @ G @ F ) ) ) ).
% filtermap_Pair
thf(fact_6512_principal__prod__principal,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ A3 ) @ ( principal @ B @ B2 ) )
= ( principal @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) ) ) ).
% principal_prod_principal
thf(fact_6513_prod__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F4: filter @ A,G8: filter @ B] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) @ ( filter @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ ( filter @ ( product_prod @ A @ B ) )
@ ^ [P2: A > $o,Q2: B > $o] :
( principal @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y5: B] :
( ( P2 @ X4 )
& ( Q2 @ Y5 ) ) ) ) ) )
@ ( collect @ ( product_prod @ ( A > $o ) @ ( B > $o ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ $o
@ ^ [P2: A > $o,Q2: B > $o] :
( ( eventually @ A @ P2 @ F4 )
& ( eventually @ B @ Q2 @ G8 ) ) ) ) ) ) ) ) ).
% prod_filter_def
thf(fact_6514_le__prod__filterI,axiom,
! [A: $tType,B: $tType,F: filter @ ( product_prod @ A @ B ),A3: filter @ A,B2: filter @ B] :
( ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ F ) @ A3 )
=> ( ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ F ) @ B2 )
=> ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ F @ ( prod_filter @ A @ B @ A3 @ B2 ) ) ) ) ).
% le_prod_filterI
thf(fact_6515_eventually__prod__sequentially,axiom,
! [P: ( product_prod @ nat @ nat ) > $o] :
( ( eventually @ ( product_prod @ nat @ nat ) @ P @ ( prod_filter @ nat @ nat @ ( at_top @ nat ) @ ( at_top @ nat ) ) )
= ( ? [N9: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ N9 @ M )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ N9 @ N2 )
=> ( P @ ( product_Pair @ nat @ nat @ N2 @ M ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_6516_eventually__prodI,axiom,
! [A: $tType,B: $tType,P: A > $o,F: filter @ A,Q: B > $o,G5: filter @ B] :
( ( eventually @ A @ P @ F )
=> ( ( eventually @ B @ Q @ G5 )
=> ( eventually @ ( product_prod @ A @ B )
@ ^ [X4: product_prod @ A @ B] :
( ( P @ ( product_fst @ A @ B @ X4 ) )
& ( Q @ ( product_snd @ A @ B @ X4 ) ) )
@ ( prod_filter @ A @ B @ F @ G5 ) ) ) ) ).
% eventually_prodI
thf(fact_6517_eventually__prod2,axiom,
! [A: $tType,B: $tType,A3: filter @ A,P: B > $o,B2: filter @ B] :
( ( A3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A] : P )
@ ( prod_filter @ A @ B @ A3 @ B2 ) )
= ( eventually @ B @ P @ B2 ) ) ) ).
% eventually_prod2
thf(fact_6518_eventually__prod1,axiom,
! [A: $tType,B: $tType,B2: filter @ A,P: B > $o,A3: filter @ B] :
( ( B2
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X4: B,Y5: A] : ( P @ X4 ) )
@ ( prod_filter @ B @ A @ A3 @ B2 ) )
= ( eventually @ B @ P @ A3 ) ) ) ).
% eventually_prod1
thf(fact_6519_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J4: set @ A,A3: filter @ B,B2: A > ( filter @ C )] :
( ( J4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A3 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B2 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I4: A] : ( prod_filter @ B @ C @ A3 @ ( B2 @ I4 ) )
@ J4 ) ) ) ) ).
% prod_filter_INF2
thf(fact_6520_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I3: set @ A,A3: A > ( filter @ B ),B2: filter @ C] :
( ( I3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A3 @ I3 ) ) @ B2 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I4: A] : ( prod_filter @ B @ C @ ( A3 @ I4 ) @ B2 )
@ I3 ) ) ) ) ).
% prod_filter_INF1
thf(fact_6521_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I3: set @ A,J4: set @ B,A3: A > ( filter @ C ),B2: B > ( filter @ D )] :
( ( I3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ A3 @ I3 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B2 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I4: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J2: B] : ( prod_filter @ C @ D @ ( A3 @ I4 ) @ ( B2 @ J2 ) )
@ J4 ) )
@ I3 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_6522_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X3: A,F: filter @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ F )
= ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ F ) ) ).
% prod_filter_principal_singleton
thf(fact_6523_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_6524_subset__singleton__iff__Uniq,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [A5: A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert3 @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( uniq @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_6525_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_6526_Min_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Min.semilattice_order_set_axioms
thf(fact_6527_pairwise__disjnt__iff,axiom,
! [A: $tType,A17: set @ ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A17 )
= ( ! [X4: A] :
( uniq @ ( set @ A )
@ ^ [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ A17 )
& ( member @ A @ X4 @ X9 ) ) ) ) ) ).
% pairwise_disjnt_iff
thf(fact_6528_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_6529_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( uniq @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs3 )
& ( ( set2 @ A @ Xs3 )
= A3 ) ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_6530_gen__length__def,axiom,
! [A: $tType] :
( ( gen_length @ A )
= ( ^ [N2: nat,Xs3: list @ A] : ( plus_plus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_6531_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] :
( lattic501386751177426532rg_min @ A @ B @ F3
@ ^ [X4: A] : ( member @ A @ X4 @ S )
@ X_1 ) ) ) ) ).
% ex_is_arg_min_if_finite
thf(fact_6532_is__arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic501386751177426532rg_min @ B @ A )
= ( ^ [F7: B > A,P2: B > $o,X4: B] :
( ( P2 @ X4 )
& ~ ? [Y5: B] :
( ( P2 @ Y5 )
& ( ord_less @ A @ ( F7 @ Y5 ) @ ( F7 @ X4 ) ) ) ) ) ) ) ).
% is_arg_min_def
thf(fact_6533_gen__length__code_I1_J,axiom,
! [A: $tType,N: nat] :
( ( gen_length @ A @ N @ ( nil @ A ) )
= N ) ).
% gen_length_code(1)
thf(fact_6534_gen__length__code_I2_J,axiom,
! [B: $tType,N: nat,X3: B,Xs: list @ B] :
( ( gen_length @ B @ N @ ( cons @ B @ X3 @ Xs ) )
= ( gen_length @ B @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_6535_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_6536_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B2: set @ A,I3: set @ B,A3: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X: B] :
( ( member @ B @ X @ I3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A3 @ X ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A3 @ I3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_6537_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr4221423524335903396lImage @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ B ),F7: B > A] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [A13: B,A23: B] :
( ( Uu3
= ( product_Pair @ A @ A @ ( F7 @ A13 ) @ ( F7 @ A23 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A13 @ A23 ) @ R4 ) ) ) ) ) ).
% relImage_def
thf(fact_6538_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : A3 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(4)
thf(fact_6539_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(3)
thf(fact_6540_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(2)
thf(fact_6541_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(1)
thf(fact_6542_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% card_of_Times_infinite
thf(fact_6543_card__of__Times1,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : A3 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_6544_card__of__Times2,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_6545_card__of__Times__same__infinite,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% card_of_Times_same_infinite
thf(fact_6546_card__of__Times__commute,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B2
@ ^ [Uu3: B] : A3 ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% card_of_Times_commute
thf(fact_6547_card__of__refl,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ).
% card_of_refl
thf(fact_6548_card__of__Pow__Func,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > $o ) @ ( bNF_Wellorder_Func @ A @ $o @ A3 @ ( top_top @ ( set @ $o ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( A > $o ) ) ) ).
% card_of_Pow_Func
thf(fact_6549_card__of__Func__Times,axiom,
! [C: $tType,B: $tType,A: $tType,A3: set @ A,B2: set @ B,C3: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( ( product_prod @ A @ B ) > C )
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 )
@ C3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B > C ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A3 @ ( bNF_Wellorder_Func @ B @ C @ B2 @ C3 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) ) ) ).
% card_of_Func_Times
thf(fact_6550_Func__Times__Range,axiom,
! [C: $tType,B: $tType,A: $tType,A3: set @ A,B2: set @ B,C3: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > ( product_prod @ B @ C ) )
@ ( bNF_Wellorder_Func @ A @ ( product_prod @ B @ C ) @ A3
@ ( product_Sigma @ B @ C @ B2
@ ^ [Uu3: B] : C3 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ ( A > B ) @ ( A > C ) )
@ ( product_Sigma @ ( A > B ) @ ( A > C ) @ ( bNF_Wellorder_Func @ A @ B @ A3 @ B2 )
@ ^ [Uu3: A > B] : ( bNF_Wellorder_Func @ A @ C @ A3 @ C3 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% Func_Times_Range
thf(fact_6551_card__of__bool,axiom,
! [A: $tType,A1: A,A22: A] :
( ( A1 != A22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ $o @ ( top_top @ ( set @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( insert3 @ A @ A1 @ ( insert3 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ) ).
% card_of_bool
thf(fact_6552_card__of__Pow,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ A3 ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ).
% card_of_Pow
thf(fact_6553_card__of__UNION__Sigma,axiom,
! [B: $tType,A: $tType,A3: B > ( set @ A ),I3: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A3 @ I3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A ) @ ( product_Sigma @ B @ A @ I3 @ A3 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ).
% card_of_UNION_Sigma
thf(fact_6554_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,B2: set @ A,I3: set @ B,A3: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X: B] :
( ( member @ B @ X @ I3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A3 @ X ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I3 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
thf(fact_6555_card__of__Times3,axiom,
! [A: $tType,A3: set @ A] :
( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ A3
@ ^ [Uu3: A] : A3 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ A ) ) ) ).
% card_of_Times3
thf(fact_6556_card__of__Sigma__mono1,axiom,
! [C: $tType,B: $tType,A: $tType,I3: set @ A,A3: A > ( set @ B ),B2: A > ( set @ C )] :
( ! [X: A] :
( ( member @ A @ X @ I3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A3 @ X ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( B2 @ X ) ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ I3 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C ) @ ( product_Sigma @ A @ C @ I3 @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Sigma_mono1
thf(fact_6557_card__of__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A3: set @ A,B2: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ C @ A3
@ ^ [Uu3: A] : C3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ B2
@ ^ [Uu3: B] : C3 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% card_of_Times_mono1
thf(fact_6558_card__of__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A3: set @ A,B2: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
@ ( product_Sigma @ C @ A @ C3
@ ^ [Uu3: C] : A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ C3
@ ^ [Uu3: C] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% card_of_Times_mono2
thf(fact_6559_infinite__iff__card__of__nat,axiom,
! [A: $tType,A3: set @ A] :
( ( ~ ( finite_finite2 @ A @ A3 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_card_of_nat
thf(fact_6560_card__of__image,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( image2 @ B @ A @ F3 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_image
thf(fact_6561_card__of__empty3,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty3
thf(fact_6562_card__of__empty,axiom,
! [B: $tType,A: $tType,A3: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_empty
thf(fact_6563_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( finite_finite2 @ A @ A3 )
=> ~ ( finite_finite2 @ B @ B2 ) ) ) ).
% card_of_ordLeq_infinite
thf(fact_6564_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( finite_finite2 @ B @ B2 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% card_of_ordLeq_finite
thf(fact_6565_card__of__mono1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_mono1
thf(fact_6566_card__of__empty__ordIso,axiom,
! [B: $tType,A: $tType] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% card_of_empty_ordIso
thf(fact_6567_card__of__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty2
thf(fact_6568_card__of__ordIso__finite,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ A3 )
= ( finite_finite2 @ B @ B2 ) ) ) ).
% card_of_ordIso_finite
thf(fact_6569_card__of__mono2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R7 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_mono2
thf(fact_6570_card__of__cong,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R7 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_cong
thf(fact_6571_card__of__ordLeqI,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B2: set @ B] :
( ( inj_on @ A @ B @ F3 @ A3 )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( member @ B @ ( F3 @ A6 ) @ B2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeqI
thf(fact_6572_card__of__least,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A3 @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_least
thf(fact_6573_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ).
% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_6574_type__copy__set__bd,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,S: A > ( set @ B ),Bd: set @ ( product_prod @ C @ C ),Rep: D > A,X3: D] :
( ! [Y2: A] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( S @ Y2 ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( comp @ A @ ( set @ B ) @ D @ S @ Rep @ X3 ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) ) ).
% type_copy_set_bd
thf(fact_6575_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ B2 )
= A3 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_6576_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B2: set @ A,F3: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image2 @ B @ A @ F3 @ A3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% surj_imp_ordLeq
thf(fact_6577_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A3: set @ A,B3: B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert3 @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_singl_ordLeq
thf(fact_6578_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B2: set @ A,A3: set @ B] :
( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F7: B > A] :
( ( image2 @ B @ A @ F7 @ A3 )
= B2 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_6579_internalize__card__of__ordLeq2,axiom,
! [A: $tType,B: $tType,A3: set @ A,C3: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ C3 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq2
thf(fact_6580_card__of__Field__ordLess,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_Field_ordLess
thf(fact_6581_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,B2: set @ B] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > B ) @ ( bNF_Wellorder_Func @ A @ B @ ( top_top @ ( set @ A ) ) @ B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F7: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F7 @ ( top_top @ ( set @ A ) ) ) @ B2 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% card_of_Func_UNIV
thf(fact_6582_ordLeq__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ C @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : C3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ ( field2 @ B @ R7 )
@ ^ [Uu3: B] : C3 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% ordLeq_Times_mono1
thf(fact_6583_ordLeq__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),A3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
@ ( product_Sigma @ C @ A @ A3
@ ^ [Uu3: C] : ( field2 @ A @ R2 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ A3
@ ^ [Uu3: C] : ( field2 @ B @ R7 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% ordLeq_Times_mono2
thf(fact_6584_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( ? [F7: A > B] :
( ( inj_on @ A @ B @ F7 @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F7 @ A3 ) @ B2 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_ordLeq
thf(fact_6585_internalize__card__of__ordLeq,axiom,
! [A: $tType,B: $tType,A3: set @ A,R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ ( field2 @ B @ R2 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq
thf(fact_6586_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( ~ ? [F7: A > B] :
( ( inj_on @ A @ B @ F7 @ A3 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F7 @ A3 ) @ B2 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% card_of_ordLess
thf(fact_6587_regularCard__def,axiom,
! [A: $tType] :
( ( bNF_Ca7133664381575040944arCard @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [K8: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ K8 @ ( field2 @ A @ R5 ) )
& ( bNF_Ca7293521722713021262ofinal @ A @ K8 @ R5 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ K8 ) @ R5 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% regularCard_def
thf(fact_6588_comp__set__bd__Union__o__collect,axiom,
! [C: $tType,B: $tType,A: $tType,X3: C,X5: set @ ( C > ( set @ ( set @ A ) ) ),Hbd: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
@ ( bNF_Ca6860139660246222851ard_of @ A
@ ( complete_Sup_Sup @ ( set @ A )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ ( C > ( set @ ( set @ A ) ) ) @ ( set @ ( set @ A ) )
@ ^ [F7: C > ( set @ ( set @ A ) )] : ( F7 @ X3 )
@ X5 ) ) ) )
@ Hbd )
@ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( bNF_collect @ C @ ( set @ A ) @ X5 ) @ X3 ) ) @ Hbd ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% comp_set_bd_Union_o_collect
thf(fact_6589_UNIV__bool,axiom,
( ( top_top @ ( set @ $o ) )
= ( insert3 @ $o @ $false @ ( insert3 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% UNIV_bool
thf(fact_6590_card__of__ordIso__subst,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_ordIso_subst
thf(fact_6591_card__of__Times__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ B,B2: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ A3
@ ^ [Uu3: B] : B2 ) )
@ R2 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Times_ordLeq_infinite_Field
thf(fact_6592_ordLeq__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_refl
thf(fact_6593_Card__order__trans,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y: A,Z2: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( X3 != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R2 )
=> ( ( Y != Z2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( ( X3 != Z2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z2 ) @ R2 ) ) ) ) ) ) ) ).
% Card_order_trans
thf(fact_6594_ordIso__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R2 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_refl
thf(fact_6595_card__order__on__ordIso,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A3 @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ A3 @ R7 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ).
% card_order_on_ordIso
thf(fact_6596_infinite__Card__order__limit,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R2 ) )
& ( A4 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ X ) @ R2 ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_6597_dir__image,axiom,
! [B: $tType,A: $tType,F3: A > B,R2: set @ ( product_prod @ A @ A )] :
( ! [X: A,Y2: A] :
( ( ( F3 @ X )
= ( F3 @ Y2 ) )
= ( X = Y2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ ( bNF_We2720479622203943262_image @ A @ B @ R2 @ F3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image
thf(fact_6598_Card__order__ordIso2,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R7 ) @ R7 ) ) ) ).
% Card_order_ordIso2
thf(fact_6599_Card__order__ordIso,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R2 ) @ ( bNF_Wellorder_ordIso @ B @ A ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R7 ) @ R7 ) ) ) ).
% Card_order_ordIso
thf(fact_6600_card__of__unique,axiom,
! [A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A3 @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_unique
thf(fact_6601_card__order__on__def,axiom,
! [A: $tType] :
( ( bNF_Ca8970107618336181345der_on @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A7 @ R5 )
& ! [R10: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A7 @ R10 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R10 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% card_order_on_def
thf(fact_6602_Card__order__iff__ordLeq__card__of,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% Card_order_iff_ordLeq_card_of
thf(fact_6603_ordIso__card__of__imp__Card__order,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) ) ).
% ordIso_card_of_imp_Card_order
thf(fact_6604_Card__order__iff__ordIso__card__of,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% Card_order_iff_ordIso_card_of
thf(fact_6605_card__of__Field__ordIso,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) @ R2 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_Field_ordIso
thf(fact_6606_exists__minim__Card__order,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X @ R )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X ) @ X ) )
=> ? [X: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X @ R )
& ! [Xa3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa3 @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X @ Xa3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Card_order
thf(fact_6607_Card__order__empty,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% Card_order_empty
thf(fact_6608_card__of__ordIso__finite__Field,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
= ( finite_finite2 @ B @ A3 ) ) ) ) ).
% card_of_ordIso_finite_Field
thf(fact_6609_card__of__underS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A4: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A4 @ ( field2 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( order_underS @ A @ R2 @ A4 ) ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% card_of_underS
thf(fact_6610_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),B3: B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert3 @ B @ B3 @ ( bot_bot @ ( set @ B ) ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% Card_order_singl_ordLeq
thf(fact_6611_card__of__Un__ordLeq__infinite__Field,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ B,B2: set @ B] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ) ).
% card_of_Un_ordLeq_infinite_Field
thf(fact_6612_card__of__empty1,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
| ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_empty1
thf(fact_6613_Card__order__Pow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ ( field2 @ A @ R2 ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% Card_order_Pow
thf(fact_6614_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( B2
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ R2
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% Card_order_Times1
thf(fact_6615_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ R2
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ A3
@ ^ [Uu3: B] : ( field2 @ A @ R2 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% Card_order_Times2
thf(fact_6616_Card__order__Times__same__infinite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : ( field2 @ A @ R2 ) ) )
@ R2 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ A ) @ A ) ) ) ) ).
% Card_order_Times_same_infinite
thf(fact_6617_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set @ ( product_prod @ A @ A ),I3: set @ B,A3: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X: B] :
( ( member @ B @ X @ I3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A3 @ X ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A3 @ I3 ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
thf(fact_6618_Card__order__iff__Restr__underS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R2 @ X4 )
@ ^ [Uu3: A] : ( order_underS @ A @ R2 @ X4 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R2 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% Card_order_iff_Restr_underS
thf(fact_6619_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),As5: A > ( set @ B ),B2: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca7133664381575040944arCard @ A @ R2 )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R2 @ As5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As5 @ ( field2 @ A @ R2 ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R2 ) )
& ( ord_less_eq @ ( set @ B ) @ B2 @ ( As5 @ X ) ) ) ) ) ) ) ) ).
% regularCard_UNION
thf(fact_6620_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ B @ P4 )
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R2 )
@ ^ [Uu3: A] : ( field2 @ B @ P4 ) ) )
@ R2 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ ( field2 @ B @ P4 )
@ ^ [Uu3: B] : ( field2 @ A @ R2 ) ) )
@ R2 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% Card_order_Times_infinite
thf(fact_6621_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),I3: set @ B,A3: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X: B] :
( ( member @ B @ X @ I3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A3 @ X ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I3 @ A3 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
thf(fact_6622_ex__toCard__pred,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ? [X_1: A > B] : ( bNF_Gr1419584066657907630d_pred @ A @ B @ A3 @ R2 @ X_1 ) ) ) ).
% ex_toCard_pred
thf(fact_6623_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),As5: ( set @ A ) > ( set @ B ),B2: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ As5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As5 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B2 @ ( As5 @ X ) ) ) ) ) ) ) ) ).
% cardSuc_UNION
thf(fact_6624_cardSuc__ordLeq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R2 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( set @ A ) ) ) ) ).
% cardSuc_ordLeq
thf(fact_6625_cardSuc__greater,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R2 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% cardSuc_greater
thf(fact_6626_cardSuc__least,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ R7 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_least
thf(fact_6627_cardSuc__ordLess__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ R7 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_ordLess_ordLeq
thf(fact_6628_cardSuc__mono__ordLeq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R7 ) ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ B ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% cardSuc_mono_ordLeq
thf(fact_6629_cardSuc__invar__ordIso,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R7 ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( set @ B ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% cardSuc_invar_ordIso
thf(fact_6630_cardSuc__least__aux,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R7 ) @ R7 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R2 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ R7 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ).
% cardSuc_least_aux
thf(fact_6631_cardSuc__ordLeq__ordLess,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R7 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ ( set @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% cardSuc_ordLeq_ordLess
thf(fact_6632_isCardSuc__def,axiom,
! [A: $tType] :
( ( bNF_Ca6246979054910435723ardSuc @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),R10: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R10 ) @ R10 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R5 @ R10 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
& ! [R11: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R11 ) @ R11 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R5 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R10 @ R11 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) ).
% isCardSuc_def
thf(fact_6633_toCard__pred__toCard,axiom,
! [A: $tType,B: $tType,A3: set @ A,R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( bNF_Gr1419584066657907630d_pred @ A @ B @ A3 @ R2 @ ( bNF_Greatest_toCard @ A @ B @ A3 @ R2 ) ) ) ) ).
% toCard_pred_toCard
thf(fact_6634_toCard__inj,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ B @ B ),X3: A,Y: A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( member @ A @ Y @ A3 )
=> ( ( ( bNF_Greatest_toCard @ A @ B @ A3 @ R2 @ X3 )
= ( bNF_Greatest_toCard @ A @ B @ A3 @ R2 @ Y ) )
= ( X3 = Y ) ) ) ) ) ) ).
% toCard_inj
thf(fact_6635_fromCard__toCard,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ B @ B ),B3: A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( member @ A @ B3 @ A3 )
=> ( ( bNF_Gr5436034075474128252omCard @ A @ B @ A3 @ R2 @ ( bNF_Greatest_toCard @ A @ B @ A3 @ R2 @ B3 ) )
= B3 ) ) ) ) ).
% fromCard_toCard
thf(fact_6636_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),As5: ( set @ A ) > ( set @ B ),B2: set @ B] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) @ As5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As5 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R2 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B2 @ ( As5 @ X ) ) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
thf(fact_6637_cinfinite__mono,axiom,
! [A: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R12 @ R23 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca4139267488887388095finite @ A @ R12 )
=> ( bNF_Ca4139267488887388095finite @ B @ R23 ) ) ) ).
% cinfinite_mono
thf(fact_6638_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R2: set @ ( product_prod @ A @ A ),X22: A] :
( ( member @ A @ X1 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ X22 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R2 ) )
& ( X1 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X ) @ R2 )
& ( X22 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X22 @ X ) @ R2 ) ) ) ) ) ).
% Cinfinite_limit2
thf(fact_6639_Cinfinite__limit,axiom,
! [A: $tType,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R2 ) )
& ( X3 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X ) @ R2 ) ) ) ) ).
% Cinfinite_limit
thf(fact_6640_Cinfinite__cong,axiom,
! [A: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R12 @ R23 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R12 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R12 ) @ R12 ) )
=> ( ( bNF_Ca4139267488887388095finite @ B @ R23 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R23 ) @ R23 ) ) ) ) ).
% Cinfinite_cong
thf(fact_6641_Cinfinite__limit__finite,axiom,
! [A: $tType,X5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( field2 @ A @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R2 ) )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ X5 )
=> ( ( Xa3 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X ) @ R2 ) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
thf(fact_6642_Un__Cinfinite__bound,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ B @ B ),B2: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% Un_Cinfinite_bound
thf(fact_6643_UNION__Cinfinite__bound,axiom,
! [A: $tType,B: $tType,C: $tType,I3: set @ A,R2: set @ ( product_prod @ B @ B ),A3: A > ( set @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ I3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ! [X: A] :
( ( member @ A @ X @ I3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A3 @ X ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A3 @ I3 ) ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) ) ) ) ).
% UNION_Cinfinite_bound
thf(fact_6644_Cfinite__ordLess__Cinfinite,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S3 ) @ S3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ S3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ).
% Cfinite_ordLess_Cinfinite
thf(fact_6645_comp__single__set__bd,axiom,
! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set @ ( product_prod @ A @ A ),Fset: B > ( set @ C ),Gset: D > ( set @ B ),Gbd: set @ ( product_prod @ E @ E ),X3: D] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ Fbd ) @ Fbd )
=> ( ! [X: B] : ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( Fset @ X ) ) @ Fbd ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ! [X: D] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( Gset @ X ) ) @ Gbd ) @ ( bNF_Wellorder_ordLeq @ B @ E ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ Fset @ ( Gset @ X3 ) ) ) ) @ ( bNF_Cardinal_cprod @ E @ A @ Gbd @ Fbd ) ) @ ( bNF_Wellorder_ordLeq @ C @ ( product_prod @ E @ A ) ) ) ) ) ) ).
% comp_single_set_bd
thf(fact_6646_cprod__com,axiom,
! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ A @ P24 @ P13 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% cprod_com
thf(fact_6647_Cfinite__cprod__Cinfinite,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S3 ) @ S3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R2 @ S3 ) @ S3 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ B ) ) ) ) ).
% Cfinite_cprod_Cinfinite
thf(fact_6648_cprod__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R23: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_mono
thf(fact_6649_cprod__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ Q4 ) @ ( bNF_Cardinal_cprod @ B @ C @ R12 @ Q4 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_mono1
thf(fact_6650_cprod__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q4 @ P24 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_mono2
thf(fact_6651_cprod__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R23: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_cong
thf(fact_6652_cprod__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ C @ R12 @ P24 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_cong1
thf(fact_6653_cprod__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q4 @ P24 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_cong2
thf(fact_6654_cprod__infinite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ A @ R2 @ R2 ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% cprod_infinite
thf(fact_6655_cprod__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P4: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P4 ) @ P4 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P4 @ Q4 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ B ) ) ) ) ) ) ) ).
% cprod_cinfinite_bound
thf(fact_6656_cprod__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B ),P9: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca4139267488887388095finite @ A @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P4 @ P9 ) @ ( bNF_Cardinal_cprod @ A @ A @ R2 @ R2 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ ( product_prod @ A @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P4 @ P9 ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% cprod_dup
thf(fact_6657_card__of__Csum__Times_H,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),I3: set @ B,A3: B > ( set @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ! [X: B] :
( ( member @ B @ X @ I3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A3 @ X ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
@ ( bNF_Cardinal_Csum @ B @ C @ ( bNF_Ca6860139660246222851ard_of @ B @ I3 )
@ ^ [I4: B] : ( bNF_Ca6860139660246222851ard_of @ C @ ( A3 @ I4 ) ) )
@ ( bNF_Cardinal_cprod @ B @ A @ ( bNF_Ca6860139660246222851ard_of @ B @ I3 ) @ R2 ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ ( product_prod @ B @ A ) ) ) ) ) ).
% card_of_Csum_Times'
thf(fact_6658_card__of__Csum__Times,axiom,
! [C: $tType,B: $tType,A: $tType,I3: set @ A,A3: A > ( set @ B ),B2: set @ C] :
( ! [X: A] :
( ( member @ A @ X @ I3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A3 @ X ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) )
@ ( bNF_Cardinal_Csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ I3 )
@ ^ [I4: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( A3 @ I4 ) ) )
@ ( bNF_Cardinal_cprod @ A @ C @ ( bNF_Ca6860139660246222851ard_of @ A @ I3 ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Csum_Times
thf(fact_6659_cprod__infinite1_H,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P4 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P4 ) @ P4 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R2 @ P4 ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% cprod_infinite1'
thf(fact_6660_ordLeq__cprod2,axiom,
! [A: $tType,B: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P13 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P13 ) @ P13 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ P24 @ ( bNF_Cardinal_cprod @ A @ B @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ) ).
% ordLeq_cprod2
thf(fact_6661_czero__def,axiom,
! [A: $tType] :
( ( bNF_Cardinal_czero @ A )
= ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% czero_def
thf(fact_6662_czero__ordIso,axiom,
! [B: $tType,A: $tType] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_czero @ A ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% czero_ordIso
thf(fact_6663_cinfinite__not__czero,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca4139267488887388095finite @ B @ R2 )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% cinfinite_not_czero
thf(fact_6664_czeroE,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( field2 @ A @ R2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% czeroE
thf(fact_6665_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A3: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_ordIso_czero_iff_empty
thf(fact_6666_czeroI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( field2 @ A @ R2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% czeroI
thf(fact_6667_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% Cnotzero_imp_not_empty
thf(fact_6668_Cnotzero__mono,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),Q4: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q4 ) @ Q4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ Q4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q4 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q4 ) @ Q4 ) ) ) ) ) ).
% Cnotzero_mono
thf(fact_6669_Cinfinite__Cnotzero,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) ) ) ).
% Cinfinite_Cnotzero
thf(fact_6670_Cnotzero__UNIV,axiom,
! [A: $tType] :
( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Cnotzero_UNIV
thf(fact_6671_cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R12 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R12 ) @ R12 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R23 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R23 ) @ R23 ) )
=> ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R12 @ R23 ) ) ) ) ).
% cinfinite_cprod2
thf(fact_6672_Cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R12 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R12 ) @ R12 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R23 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R23 ) @ R23 ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R12 @ R23 ) )
& ( bNF_Ca8970107618336181345der_on @ ( product_prod @ A @ B ) @ ( field2 @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R12 @ R23 ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R12 @ R23 ) ) ) ) ) ).
% Cinfinite_cprod2
thf(fact_6673_cone__ordLeq__Cnotzero,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ R2 ) @ ( bNF_Wellorder_ordLeq @ product_unit @ A ) ) ) ).
% cone_ordLeq_Cnotzero
thf(fact_6674_cexp__mono2__Cnotzero,axiom,
! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P24 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q4 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2_Cnotzero
thf(fact_6675_cexp__cprod,axiom,
! [A: $tType,C: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ C @ C ),R32: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R12 ) @ R12 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) @ ( bNF_Cardinal_cexp @ ( C > A ) @ B @ ( bNF_Cardinal_cexp @ A @ C @ R12 @ R23 ) @ R32 ) @ ( bNF_Cardinal_cexp @ A @ ( product_prod @ C @ B ) @ R12 @ ( bNF_Cardinal_cprod @ C @ B @ R23 @ R32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( B > C > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) ).
% cexp_cprod
thf(fact_6676_cexp__cone,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cexp @ A @ product_unit @ R2 @ bNF_Cardinal_cone ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( product_unit > A ) @ A ) ) ) ).
% cexp_cone
thf(fact_6677_cone__not__czero,axiom,
! [A: $tType] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ product_unit @ A ) ) ).
% cone_not_czero
thf(fact_6678_cprod__cexp,axiom,
! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ B @ B ),S3: set @ ( product_prod @ C @ C ),T2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R2 @ S3 ) @ T2 ) @ ( bNF_Cardinal_cprod @ ( A > B ) @ ( A > C ) @ ( bNF_Cardinal_cexp @ B @ A @ R2 @ T2 ) @ ( bNF_Cardinal_cexp @ C @ A @ S3 @ T2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% cprod_cexp
thf(fact_6679_cexp__cprod__ordLeq,axiom,
! [A: $tType,B: $tType,C: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R12 ) @ R12 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R23 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R23 ) @ R23 ) )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R32 @ ( bNF_Cardinal_czero @ C ) ) @ ( bNF_Wellorder_ordIso @ C @ C ) )
& ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ R32 ) @ R32 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R32 @ R23 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( bNF_Cardinal_cexp @ ( B > A ) @ C @ ( bNF_Cardinal_cexp @ A @ B @ R12 @ R23 ) @ R32 ) @ ( bNF_Cardinal_cexp @ A @ B @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordIso @ ( C > B > A ) @ ( B > A ) ) ) ) ) ) ) ).
% cexp_cprod_ordLeq
thf(fact_6680_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R23: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( ( field2 @ C @ P24 )
= ( bot_bot @ ( set @ C ) ) )
=> ( ( field2 @ D @ R23 )
= ( bot_bot @ ( set @ D ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% cexp_mono'
thf(fact_6681_cexp__mono1,axiom,
! [B: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ Q4 ) @ ( bNF_Cardinal_cexp @ B @ C @ R12 @ Q4 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_mono1
thf(fact_6682_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( ( field2 @ A @ P24 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( field2 @ B @ R23 )
= ( bot_bot @ ( set @ B ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q4 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2'
thf(fact_6683_ordLeq__cexp1,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),Q4: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q4 ) @ Q4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ Q4 @ ( bNF_Cardinal_cexp @ B @ A @ Q4 @ R2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( A > B ) ) ) ) ) ).
% ordLeq_cexp1
thf(fact_6684_cexp__cong2,axiom,
! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q4 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordIso @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_cong2
thf(fact_6685_cexp__cong1,axiom,
! [B: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ Q4 ) @ ( bNF_Cardinal_cexp @ B @ C @ R12 @ Q4 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_cong1
thf(fact_6686_cexp__cong,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R23: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( ( bNF_Ca8970107618336181345der_on @ D @ ( field2 @ D @ R23 ) @ R23 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_cong
thf(fact_6687_cexp__mono,axiom,
! [E: $tType,F6: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R23: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) @ P24 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ C @ E ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F6 @ F6 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F6 @ F6 ) ) @ R23 @ ( bNF_Cardinal_czero @ F6 ) ) @ ( bNF_Wellorder_ordIso @ D @ F6 ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_mono
thf(fact_6688_cexp__mono2,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ ( bNF_Cardinal_czero @ D ) ) @ ( bNF_Wellorder_ordIso @ A @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ R23 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ B @ E ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q4 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ) ).
% cexp_mono2
thf(fact_6689_ordLeq__cexp2,axiom,
! [A: $tType,B: $tType,Q4: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q4 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ R2 @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R2 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( B > A ) ) ) ) ) ).
% ordLeq_cexp2
thf(fact_6690_Cfinite__cexp__Cinfinite,axiom,
! [A: $tType,B: $tType,S3: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ S3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ S3 ) @ S3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ T2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) @ ( bNF_Cardinal_cexp @ A @ B @ S3 @ T2 ) @ ( bNF_Cardinal_cexp @ $o @ B @ bNF_Cardinal_ctwo @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( B > A ) @ ( B > $o ) ) ) ) ) ).
% Cfinite_cexp_Cinfinite
thf(fact_6691_ctwo__Cnotzero,axiom,
( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) @ bNF_Cardinal_ctwo @ ( bNF_Cardinal_czero @ $o ) ) @ ( bNF_Wellorder_ordIso @ $o @ $o ) )
& ( bNF_Ca8970107618336181345der_on @ $o @ ( field2 @ $o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).
% ctwo_Cnotzero
thf(fact_6692_ordLess__ctwo__cexp,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ R2 @ ( bNF_Cardinal_cexp @ $o @ A @ bNF_Cardinal_ctwo @ R2 ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( A > $o ) ) ) ) ).
% ordLess_ctwo_cexp
thf(fact_6693_ctwo__not__czero,axiom,
! [A: $tType] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ).
% ctwo_not_czero
thf(fact_6694_ctwo__ordLess__Cinfinite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R2 ) @ ( bNF_We4044943003108391690rdLess @ $o @ A ) ) ) ).
% ctwo_ordLess_Cinfinite
thf(fact_6695_ctwo__ordLeq__Cinfinite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R2 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) ) ) ).
% ctwo_ordLeq_Cinfinite
thf(fact_6696_cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q4: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q4 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R2 ) ) ) ) ).
% cinfinite_cexp
thf(fact_6697_Cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q4: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q4 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R2 ) )
& ( bNF_Ca8970107618336181345der_on @ ( B > A ) @ ( field2 @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R2 ) ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R2 ) ) ) ) ) ).
% Cinfinite_cexp
thf(fact_6698_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A3: set @ A,B2: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ A1 @ ( insert3 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A3 @ B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_6699_natLeq__ordLeq__cinfinite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ R2 ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% natLeq_ordLeq_cinfinite
thf(fact_6700_Card__order__Plus2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A3 @ ( field2 @ A @ R2 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% Card_order_Plus2
thf(fact_6701_Card__order__Plus1,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),B2: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ R2 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R2 ) @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% Card_order_Plus1
thf(fact_6702_card__of__Plus__Times__bool,axiom,
! [A: $tType,A3: set @ A] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A3 @ A3 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ $o )
@ ( product_Sigma @ A @ $o @ A3
@ ^ [Uu3: A] : ( top_top @ ( set @ $o ) ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ A ) @ ( product_prod @ A @ $o ) ) ) ).
% card_of_Plus_Times_bool
thf(fact_6703_card__of__Times__Plus__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,A3: set @ A,B2: set @ B,C3: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ ( sum_sum @ B @ C ) )
@ ( product_Sigma @ A @ ( sum_sum @ B @ C ) @ A3
@ ^ [Uu3: A] : ( sum_Plus @ B @ C @ B2 @ C3 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) )
@ ( sum_Plus @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 )
@ ( product_Sigma @ A @ C @ A3
@ ^ [Uu3: A] : C3 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Times_Plus_distrib
thf(fact_6704_card__of__Plus__commute,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A3 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B2 @ A3 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_commute
thf(fact_6705_card__of__Plus__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,A3: set @ A,B2: set @ B,C3: set @ C] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_Plus @ ( sum_sum @ A @ B ) @ C @ ( sum_Plus @ A @ B @ A3 @ B2 ) @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_Plus @ A @ ( sum_sum @ B @ C ) @ A3 @ ( sum_Plus @ B @ C @ B2 @ C3 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_assoc
thf(fact_6706_card__of__Un__Plus__ordLeq,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A3 @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% card_of_Un_Plus_ordLeq
thf(fact_6707_card__of__Plus2,axiom,
! [B: $tType,A: $tType,B2: set @ A,A3: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A3 @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus2
thf(fact_6708_card__of__Plus1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A3 @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus1
thf(fact_6709_ctwo__ordLess__natLeq,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ $o @ nat ) ).
% ctwo_ordLess_natLeq
thf(fact_6710_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A3: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus_empty1
thf(fact_6711_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( bot_bot @ ( set @ B ) ) @ A3 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_empty2
thf(fact_6712_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N ) ) ) ).
% natLeq_underS_less
thf(fact_6713_ordLeq__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),P4: set @ ( product_prod @ C @ C ),P9: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P4 @ P9 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R2 ) @ ( field2 @ C @ P4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R7 ) @ ( field2 @ D @ P9 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordLeq_Plus_mono
thf(fact_6714_ordLeq__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R2 ) @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R7 ) @ C3 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordLeq_Plus_mono1
thf(fact_6715_ordLeq__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),A3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A3 @ ( field2 @ A @ R2 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A3 @ ( field2 @ B @ R7 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordLeq_Plus_mono2
thf(fact_6716_ordIso__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),P4: set @ ( product_prod @ C @ C ),P9: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P4 @ P9 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R2 ) @ ( field2 @ C @ P4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R7 ) @ ( field2 @ D @ P9 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordIso_Plus_cong
thf(fact_6717_ordIso__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R2 ) @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R7 ) @ C3 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordIso_Plus_cong1
thf(fact_6718_ordIso__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),A3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A3 @ ( field2 @ A @ R2 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A3 @ ( field2 @ B @ R7 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordIso_Plus_cong2
thf(fact_6719_card__of__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A3: set @ A,B2: set @ B,C3: set @ C,D2: set @ D] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C3 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D2 ) ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A3 @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B2 @ D2 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_mono
thf(fact_6720_card__of__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A3: set @ A,B2: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A3 @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B2 @ C3 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_mono1
thf(fact_6721_card__of__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A3: set @ A,B2: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C3 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C3 @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_mono2
thf(fact_6722_card__of__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A3: set @ A,B2: set @ B,C3: set @ C,D2: set @ D] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C3 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D2 ) ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A3 @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B2 @ D2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_cong
thf(fact_6723_card__of__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,A3: set @ A,B2: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A3 @ C3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B2 @ C3 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_cong1
thf(fact_6724_card__of__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,A3: set @ A,B2: set @ B,C3: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C3 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C3 @ B2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_cong2
thf(fact_6725_card__of__Plus__infinite2,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B2 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ).
% card_of_Plus_infinite2
thf(fact_6726_card__of__Plus__infinite1,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A3 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% card_of_Plus_infinite1
thf(fact_6727_card__of__Plus__infinite,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A3 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B2 @ A3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Plus_infinite
thf(fact_6728_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,C3: set @ A,A3: set @ B,B2: set @ C] :
( ~ ( finite_finite2 @ A @ C3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C3 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C3 ) ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A3 @ B2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C3 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite
thf(fact_6729_finite__iff__ordLess__natLeq,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A7 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% finite_iff_ordLess_natLeq
thf(fact_6730_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
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_6731_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N ) )
@ ^ [Uu3: nat] :
( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_6732_Card__order__Plus__infinite,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R2 ) @ ( field2 @ B @ P4 ) ) ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( field2 @ B @ P4 ) @ ( field2 @ A @ R2 ) ) ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ) ).
% Card_order_Plus_infinite
thf(fact_6733_card__of__nat,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% card_of_nat
thf(fact_6734_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A3: set @ A,B15: B,B23: B,B2: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert3 @ A @ A1 @ ( insert3 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A3 ) )
=> ( ( ( B15 != B23 )
& ( ord_less_eq @ ( set @ B ) @ ( insert3 @ B @ B15 @ ( insert3 @ B @ B23 @ ( bot_bot @ ( set @ B ) ) ) ) @ B2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A3 @ B2 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A3
@ ^ [Uu3: A] : B2 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times
thf(fact_6735_card__of__Field__natLeq,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% card_of_Field_natLeq
thf(fact_6736_card__of__Plus__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ B,B2: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A3 @ B2 ) ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLeq_infinite_Field
thf(fact_6737_card__of__Plus__ordLess__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ B,B2: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A3 ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B2 ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A3 @ B2 ) ) @ R2 ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite_Field
thf(fact_6738_infinite__iff__natLeq__ordLeq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
!= ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_natLeq_ordLeq
thf(fact_6739_Plus__eq__empty__conv,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( ( sum_Plus @ A @ B @ A3 @ B2 )
= ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Plus_eq_empty_conv
thf(fact_6740_csum__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P4: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q4 @ R2 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P4 ) @ P4 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R2 ) @ R2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_csum @ A @ C @ P4 @ Q4 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ B ) ) ) ) ) ) ) ).
% csum_cinfinite_bound
thf(fact_6741_cprod__csum__distrib1,axiom,
! [C: $tType,B: $tType,A: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( bNF_Cardinal_cprod @ A @ B @ R12 @ R23 ) @ ( bNF_Cardinal_cprod @ A @ C @ R12 @ R32 ) ) @ ( bNF_Cardinal_cprod @ A @ ( sum_sum @ B @ C ) @ R12 @ ( bNF_Cardinal_csum @ B @ C @ R23 @ R32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% cprod_csum_distrib1
thf(fact_6742_ordLeq__csum1,axiom,
! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P13 ) @ P13 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ P13 @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% ordLeq_csum1
thf(fact_6743_ordLeq__csum2,axiom,
! [B: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),P13: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ P24 @ ( bNF_Cardinal_csum @ B @ A @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% ordLeq_csum2
thf(fact_6744_csum__com,axiom,
! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ A @ P24 @ P13 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% csum_com
thf(fact_6745_csum__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R23: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_cong
thf(fact_6746_csum__csum,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),R32: set @ ( product_prod @ C @ C ),R42: set @ ( product_prod @ D @ D )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ ( bNF_Cardinal_csum @ A @ B @ R12 @ R23 ) @ ( bNF_Cardinal_csum @ C @ D @ R32 @ R42 ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) @ ( bNF_Cardinal_csum @ A @ C @ R12 @ R32 ) @ ( bNF_Cardinal_csum @ B @ D @ R23 @ R42 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ).
% csum_csum
thf(fact_6747_csum__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B ),P33: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) @ P33 ) @ ( bNF_Cardinal_csum @ A @ ( sum_sum @ B @ C ) @ P13 @ ( bNF_Cardinal_csum @ B @ C @ P24 @ P33 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_assoc
thf(fact_6748_csum__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ Q4 ) @ ( bNF_Cardinal_csum @ B @ C @ R12 @ Q4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_cong1
thf(fact_6749_csum__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q4 @ P24 ) @ ( bNF_Cardinal_csum @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_cong2
thf(fact_6750_csum__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R23: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_mono
thf(fact_6751_csum__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R12 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ Q4 ) @ ( bNF_Cardinal_csum @ B @ C @ R12 @ Q4 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_mono1
thf(fact_6752_csum__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R23 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q4 @ P24 ) @ ( bNF_Cardinal_csum @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_mono2
thf(fact_6753_Un__csum,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) @ ( bNF_Cardinal_csum @ A @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B2 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% Un_csum
thf(fact_6754_cprod__csum__cexp,axiom,
! [B: $tType,A: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R12 @ R23 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ $o @ ( bNF_Cardinal_csum @ A @ B @ R12 @ R23 ) @ bNF_Cardinal_ctwo ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ).
% cprod_csum_cexp
thf(fact_6755_csum__Cfinite__cexp__Cinfinite,axiom,
! [B: $tType,A: $tType,C: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ B @ B ),T2: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ( bNF_Cardinal_cfinite @ B @ S3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S3 ) @ S3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ C @ T2 )
& ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ R2 @ S3 ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ $o ) @ C @ ( bNF_Cardinal_csum @ A @ $o @ R2 @ bNF_Cardinal_ctwo ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) ) ) ).
% csum_Cfinite_cexp_Cinfinite
thf(fact_6756_Plus__csum,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A3 @ B2 ) ) @ ( bNF_Cardinal_csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ).
% Plus_csum
thf(fact_6757_csum__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B ),P9: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca4139267488887388095finite @ A @ R2 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Cardinal_csum @ B @ C @ P4 @ P9 ) @ ( bNF_Cardinal_csum @ A @ A @ R2 @ R2 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ ( sum_sum @ A @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ C @ P4 @ P9 ) @ R2 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% csum_dup
thf(fact_6758_csum__Cnotzero1,axiom,
! [A: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R12 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R12 ) @ R12 ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ R12 @ R23 ) @ ( bNF_Cardinal_czero @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) )
& ( bNF_Ca8970107618336181345der_on @ ( sum_sum @ A @ B ) @ ( field2 @ ( sum_sum @ A @ B ) @ ( bNF_Cardinal_csum @ A @ B @ R12 @ R23 ) ) @ ( bNF_Cardinal_csum @ A @ B @ R12 @ R23 ) ) ) ) ).
% csum_Cnotzero1
thf(fact_6759_csum__absorb1,axiom,
! [B: $tType,A: $tType,R23: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R23 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R23 ) @ R23 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R12 @ R23 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R23 @ R12 ) @ R23 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% csum_absorb1
thf(fact_6760_csum__absorb1_H,axiom,
! [B: $tType,A: $tType,R23: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R23 ) @ R23 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R12 @ R23 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R12 )
| ( bNF_Ca4139267488887388095finite @ A @ R23 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R23 @ R12 ) @ R23 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ) ).
% csum_absorb1'
thf(fact_6761_csum__absorb2_H,axiom,
! [A: $tType,B: $tType,R23: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R23 ) @ R23 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R12 @ R23 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R12 )
| ( bNF_Ca4139267488887388095finite @ A @ R23 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ A @ R12 @ R23 ) @ R23 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% csum_absorb2'
thf(fact_6762_cprod__cexp__csum__cexp__Cinfinite,axiom,
! [C: $tType,B: $tType,A: $tType,T2: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),S3: set @ ( product_prod @ C @ C )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ T2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R2 @ S3 ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ B @ C ) @ A @ ( bNF_Cardinal_csum @ B @ C @ R2 @ S3 ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) ).
% cprod_cexp_csum_cexp_Cinfinite
thf(fact_6763_subset__mset_Osum__strict__mono,axiom,
! [A: $tType,B: $tType,A3: set @ B,F3: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( subset_mset @ A @ ( F3 @ X ) @ ( G @ X ) ) )
=> ( subset_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ A3 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ A3 ) ) ) ) ) ).
% subset_mset.sum_strict_mono
thf(fact_6764_subset__mset_Osum__pos,axiom,
! [A: $tType,B: $tType,I3: set @ B,F3: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ I3 )
=> ( ( I3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ I3 )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F3 @ I5 ) ) )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ I3 ) ) ) ) ) ).
% subset_mset.sum_pos
thf(fact_6765_subset__implies__mult,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,R2: set @ ( product_prod @ A @ A )] :
( ( subset_mset @ A @ A3 @ B2 )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ A3 @ B2 ) @ ( mult @ A @ R2 ) ) ) ).
% subset_implies_mult
thf(fact_6766_mset__subset__size,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ A3 @ B2 )
=> ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ A3 ) @ ( size_size @ ( multiset @ A ) @ B2 ) ) ) ).
% mset_subset_size
thf(fact_6767_subset__mset_Olift__Suc__mono__less,axiom,
! [A: $tType,F3: nat > ( multiset @ A ),N: nat,N6: nat] :
( ! [N5: nat] : ( subset_mset @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
=> ( ( ord_less @ nat @ N @ N6 )
=> ( subset_mset @ A @ ( F3 @ N ) @ ( F3 @ N6 ) ) ) ) ).
% subset_mset.lift_Suc_mono_less
thf(fact_6768_subset__mset_Olift__Suc__mono__less__iff,axiom,
! [A: $tType,F3: nat > ( multiset @ A ),N: nat,M2: nat] :
( ! [N5: nat] : ( subset_mset @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
=> ( ( subset_mset @ A @ ( F3 @ N ) @ ( F3 @ M2 ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ).
% subset_mset.lift_Suc_mono_less_iff
thf(fact_6769_size__psubset,axiom,
! [A: $tType,M5: multiset @ A,M13: multiset @ A] :
( ( subseteq_mset @ A @ M5 @ M13 )
=> ( ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ M5 ) @ ( size_size @ ( multiset @ A ) @ M13 ) )
=> ( subset_mset @ A @ M5 @ M13 ) ) ) ).
% size_psubset
thf(fact_6770_rel__filter_Ocases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,F: filter @ A,G5: filter @ B] :
( ( rel_filter @ A @ B @ R @ F @ G5 )
=> ~ ! [Z11: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) @ Z11 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_fst @ A @ B ) @ Z11 )
= F )
=> ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_snd @ A @ B ) @ Z11 )
!= G5 ) ) ) ) ).
% rel_filter.cases
thf(fact_6771_rel__filter_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rel_filter @ A @ B )
= ( ^ [R4: A > B > $o,F4: filter @ A,G8: filter @ B] :
? [Z9: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) @ Z9 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) @ ( product_fst @ A @ B ) @ Z9 )
= F4 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) @ ( product_snd @ A @ B ) @ Z9 )
= G8 ) ) ) ) ).
% rel_filter.simps
thf(fact_6772_sup__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > ( filter @ A ) ) @ ( ( filter @ B ) > ( filter @ B ) ) @ ( rel_filter @ A @ B @ A3 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) @ ( rel_filter @ A @ B @ A3 ) ) @ ( sup_sup @ ( filter @ A ) ) @ ( sup_sup @ ( filter @ B ) ) ) ).
% sup_filter_parametric
thf(fact_6773_Sup__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( set @ ( filter @ A ) ) @ ( set @ ( filter @ B ) ) @ ( filter @ A ) @ ( filter @ B ) @ ( bNF_rel_set @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) ) @ ( rel_filter @ A @ B @ A3 ) @ ( complete_Sup_Sup @ ( filter @ A ) ) @ ( complete_Sup_Sup @ ( filter @ B ) ) ) ).
% Sup_filter_parametric
thf(fact_6774_rel__filter__eq,axiom,
! [A: $tType] :
( ( rel_filter @ A @ A
@ ^ [Y4: A,Z4: A] : Y4 = Z4 )
= ( ^ [Y4: filter @ A,Z4: filter @ A] : Y4 = Z4 ) ) ).
% rel_filter_eq
thf(fact_6775_eventually__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A3 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( eventually @ A )
@ ( eventually @ B ) ) ).
% eventually_parametric
thf(fact_6776_rel__filter__mono,axiom,
! [B: $tType,A: $tType,A3: A > B > $o,B2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B2 )
=> ( ord_less_eq @ ( ( filter @ A ) > ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A3 ) @ ( rel_filter @ A @ B @ B2 ) ) ) ).
% rel_filter_mono
thf(fact_6777_principal__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( bNF_rel_set @ A @ B @ A3 ) @ ( rel_filter @ A @ B @ A3 ) @ ( principal @ A ) @ ( principal @ B ) ) ).
% principal_parametric
thf(fact_6778_filtermap__parametric,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A3: A > C > $o,B2: B > D > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( filter @ A ) > ( filter @ B ) ) @ ( ( filter @ C ) > ( filter @ D ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A3 @ B2 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ C ) @ ( filter @ B ) @ ( filter @ D ) @ ( rel_filter @ A @ C @ A3 ) @ ( rel_filter @ B @ D @ B2 ) ) @ ( filtermap @ A @ B ) @ ( filtermap @ C @ D ) ) ).
% filtermap_parametric
thf(fact_6779_bot__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] : ( rel_filter @ A @ B @ A3 @ ( bot_bot @ ( filter @ A ) ) @ ( bot_bot @ ( filter @ B ) ) ) ).
% bot_filter_parametric
thf(fact_6780_rel__filter_Ointros,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Z7: filter @ ( product_prod @ A @ B ),F: filter @ A,G5: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) @ Z7 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_fst @ A @ B ) @ Z7 )
= F )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R ) ) @ ( product_snd @ A @ B ) @ Z7 )
= G5 )
=> ( rel_filter @ A @ B @ R @ F @ G5 ) ) ) ) ).
% rel_filter.intros
thf(fact_6781_mod__h__bot__iff_I8_J,axiom,
! [C: $tType,R: C > assn,H2: heap_ext @ product_unit] :
( ( rep_assn @ ( ex_assn @ C @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ? [X4: C] : ( rep_assn @ ( R @ X4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(8)
thf(fact_6782_take__Cons,axiom,
! [A: $tType,N: nat,X3: A,Xs: list @ A] :
( ( take @ A @ N @ ( cons @ A @ X3 @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( nil @ A )
@ ^ [M: nat] : ( cons @ A @ X3 @ ( take @ A @ M @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_6783_ex__assn__const,axiom,
! [A: $tType,C2: assn] :
( ( ex_assn @ A
@ ^ [X4: A] : C2 )
= C2 ) ).
% ex_assn_const
thf(fact_6784_norm__assertion__simps_I17_J,axiom,
! [B: $tType,R: assn,Q: B > assn] :
( ( times_times @ assn @ R @ ( ex_assn @ B @ Q ) )
= ( ex_assn @ B
@ ^ [X4: B] : ( times_times @ assn @ R @ ( Q @ X4 ) ) ) ) ).
% norm_assertion_simps(17)
thf(fact_6785_norm__assertion__simps_I16_J,axiom,
! [A: $tType,Q: A > assn,R: assn] :
( ( times_times @ assn @ ( ex_assn @ A @ Q ) @ R )
= ( ex_assn @ A
@ ^ [X4: A] : ( times_times @ assn @ ( Q @ X4 ) @ R ) ) ) ).
% norm_assertion_simps(16)
thf(fact_6786_mod__ex__dist,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H2 )
= ( ? [X4: A] : ( rep_assn @ ( P @ X4 ) @ H2 ) ) ) ).
% mod_ex_dist
thf(fact_6787_triv__exI,axiom,
! [A: $tType,Q: A > assn,X3: A] : ( entails @ ( Q @ X3 ) @ ( ex_assn @ A @ Q ) ) ).
% triv_exI
thf(fact_6788_norm__assertion__simps_I20_J,axiom,
! [E: $tType,Q: E > assn,P: assn] :
( ( sup_sup @ assn @ ( ex_assn @ E @ Q ) @ P )
= ( ex_assn @ E
@ ^ [X4: E] : ( sup_sup @ assn @ ( Q @ X4 ) @ P ) ) ) ).
% norm_assertion_simps(20)
thf(fact_6789_norm__assertion__simps_I21_J,axiom,
! [F6: $tType,Q: assn,P: F6 > assn] :
( ( sup_sup @ assn @ Q @ ( ex_assn @ F6 @ P ) )
= ( ex_assn @ F6
@ ^ [X4: F6] : ( sup_sup @ assn @ Q @ ( P @ X4 ) ) ) ) ).
% norm_assertion_simps(21)
thf(fact_6790_norm__assertion__simps_I18_J,axiom,
! [C: $tType,Q: C > assn,P: assn] :
( ( inf_inf @ assn @ ( ex_assn @ C @ Q ) @ P )
= ( ex_assn @ C
@ ^ [X4: C] : ( inf_inf @ assn @ ( Q @ X4 ) @ P ) ) ) ).
% norm_assertion_simps(18)
thf(fact_6791_norm__assertion__simps_I19_J,axiom,
! [D: $tType,Q: assn,P: D > assn] :
( ( inf_inf @ assn @ Q @ ( ex_assn @ D @ P ) )
= ( ex_assn @ D
@ ^ [X4: D] : ( inf_inf @ assn @ Q @ ( P @ X4 ) ) ) ) ).
% norm_assertion_simps(19)
thf(fact_6792_ex__distrib__and,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ( ex_assn @ A
@ ^ [X4: A] : ( inf_inf @ assn @ ( P @ X4 ) @ Q ) )
= ( inf_inf @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ex_distrib_and
thf(fact_6793_mod__exE,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H2 )
=> ~ ! [X: A] :
~ ( rep_assn @ ( P @ X ) @ H2 ) ) ).
% mod_exE
thf(fact_6794_mod__exI,axiom,
! [A: $tType,P: A > assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ? [X8: A] : ( rep_assn @ ( P @ X8 ) @ H2 )
=> ( rep_assn @ ( ex_assn @ A @ P ) @ H2 ) ) ).
% mod_exI
thf(fact_6795_ex__one__point__gen,axiom,
! [A: $tType,P: A > assn,V2: A] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X: A] :
( ( rep_assn @ ( P @ X ) @ H )
=> ( X = V2 ) )
=> ( ( ex_assn @ A @ P )
= ( P @ V2 ) ) ) ).
% ex_one_point_gen
thf(fact_6796_ex__distrib__star,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ( ex_assn @ A
@ ^ [X4: A] : ( times_times @ assn @ ( P @ X4 ) @ Q ) )
= ( times_times @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ex_distrib_star
thf(fact_6797_enorm__exI_H,axiom,
! [A: $tType,Z7: A > $o,P: assn,Q: A > assn] :
( ! [X: A] :
( ( Z7 @ X )
=> ( entails @ P @ ( Q @ X ) ) )
=> ( ? [X_12: A] : ( Z7 @ X_12 )
=> ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ) ).
% enorm_exI'
thf(fact_6798_ent__ex__preI,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ! [X: A] : ( entails @ ( P @ X ) @ Q )
=> ( entails @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ent_ex_preI
thf(fact_6799_ent__ex__postI,axiom,
! [A: $tType,P: assn,Q: A > assn,X3: A] :
( ( entails @ P @ ( Q @ X3 ) )
=> ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ).
% ent_ex_postI
thf(fact_6800_ex__join__or,axiom,
! [A: $tType,P: A > assn,Q: A > assn] :
( ( ex_assn @ A
@ ^ [X4: A] : ( sup_sup @ assn @ ( P @ X4 ) @ ( ex_assn @ A @ Q ) ) )
= ( ex_assn @ A
@ ^ [X4: A] : ( sup_sup @ assn @ ( P @ X4 ) @ ( Q @ X4 ) ) ) ) ).
% ex_join_or
thf(fact_6801_ex__distrib__or,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ( ex_assn @ A
@ ^ [X4: A] : ( sup_sup @ assn @ ( P @ X4 ) @ Q ) )
= ( sup_sup @ assn @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ex_distrib_or
thf(fact_6802_drop__Cons,axiom,
! [A: $tType,N: nat,X3: A,Xs: list @ A] :
( ( drop @ A @ N @ ( cons @ A @ X3 @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ X3 @ Xs )
@ ^ [M: nat] : ( drop @ A @ M @ Xs )
@ N ) ) ).
% drop_Cons
thf(fact_6803_list__update_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,I: nat,V2: A] :
( ( list_update @ A @ ( cons @ A @ X3 @ Xs ) @ I @ V2 )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V2 @ Xs )
@ ^ [J2: nat] : ( cons @ A @ X3 @ ( list_update @ A @ Xs @ J2 @ V2 ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_6804_nth__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X3 @ Xs ) @ N )
= ( case_nat @ A @ X3 @ ( nth @ A @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_6805_frequently__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A3 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( frequently @ A )
@ ( frequently @ B ) ) ).
% frequently_parametric
thf(fact_6806_plus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
@ ^ [X4: product_prod @ int @ int,Y5: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) )
@ ( plus_plus @ rat ) ) ).
% plus_rat.transfer
thf(fact_6807_frequently__const,axiom,
! [A: $tType,F: filter @ A,P: $o] :
( ( F
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( frequently @ A
@ ^ [X4: A] : P
@ F )
= P ) ) ).
% frequently_const
thf(fact_6808_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% one_rat.transfer
thf(fact_6809_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% zero_rat.transfer
thf(fact_6810_frequently__bex__finite,axiom,
! [A: $tType,B: $tType,A3: set @ A,P: B > A > $o,F: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( frequently @ B
@ ^ [X4: B] :
? [Y5: A] :
( ( member @ A @ Y5 @ A3 )
& ( P @ X4 @ Y5 ) )
@ F )
=> ? [X: A] :
( ( member @ A @ X @ A3 )
& ( frequently @ B
@ ^ [Y5: B] : ( P @ Y5 @ X )
@ F ) ) ) ) ).
% frequently_bex_finite
thf(fact_6811_frequently__bex__finite__distrib,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: B > A > $o,F: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( frequently @ B
@ ^ [X4: B] :
? [Y5: A] :
( ( member @ A @ Y5 @ A3 )
& ( P @ X4 @ Y5 ) )
@ F )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( frequently @ B
@ ^ [Y5: B] : ( P @ Y5 @ X4 )
@ F ) ) ) ) ) ).
% frequently_bex_finite_distrib
thf(fact_6812_eventually__frequently__const__simps_I2_J,axiom,
! [A: $tType,C3: $o,P: A > $o,F: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] :
( C3
& ( P @ X4 ) )
@ F )
= ( C3
& ( frequently @ A @ P @ F ) ) ) ).
% eventually_frequently_const_simps(2)
thf(fact_6813_eventually__frequently__const__simps_I1_J,axiom,
! [A: $tType,P: A > $o,C3: $o,F: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] :
( ( P @ X4 )
& C3 )
@ F )
= ( ( frequently @ A @ P @ F )
& C3 ) ) ).
% eventually_frequently_const_simps(1)
thf(fact_6814_frequently__ex,axiom,
! [A: $tType,P: A > $o,F: filter @ A] :
( ( frequently @ A @ P @ F )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% frequently_ex
thf(fact_6815_frequently__disj,axiom,
! [A: $tType,P: A > $o,F: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F )
=> ( ( frequently @ A @ Q @ F )
=> ( frequently @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) )
@ F ) ) ) ).
% frequently_disj
thf(fact_6816_frequently__elim1,axiom,
! [A: $tType,P: A > $o,F: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F )
=> ( ! [I5: A] :
( ( P @ I5 )
=> ( Q @ I5 ) )
=> ( frequently @ A @ Q @ F ) ) ) ).
% frequently_elim1
thf(fact_6817_frequently__disj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) )
@ F )
= ( ( frequently @ A @ P @ F )
| ( frequently @ A @ Q @ F ) ) ) ).
% frequently_disj_iff
thf(fact_6818_not__frequently__False,axiom,
! [A: $tType,F: filter @ A] :
~ ( frequently @ A
@ ^ [X4: A] : $false
@ F ) ).
% not_frequently_False
thf(fact_6819_frequentlyE,axiom,
! [A: $tType,P: A > $o,F: filter @ A] :
( ( frequently @ A @ P @ F )
=> ~ ! [X: A] :
~ ( P @ X ) ) ).
% frequentlyE
thf(fact_6820_frequently__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F: filter @ A] :
( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ( frequently @ A @ P @ F )
=> ( frequently @ A @ Q @ F ) ) ) ).
% frequently_mono
thf(fact_6821_frequently__all,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] :
! [X9: B] : ( P @ X4 @ X9 )
@ F )
= ( ! [Y9: A > B] :
( frequently @ A
@ ^ [X4: A] : ( P @ X4 @ ( Y9 @ X4 ) )
@ F ) ) ) ).
% frequently_all
thf(fact_6822_frequently__imp__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F )
= ( ( eventually @ A @ P @ F )
=> ( frequently @ A @ Q @ F ) ) ) ).
% frequently_imp_iff
thf(fact_6823_frequently__rev__mp,axiom,
! [A: $tType,P: A > $o,F: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F )
=> ( frequently @ A @ Q @ F ) ) ) ).
% frequently_rev_mp
thf(fact_6824_not__frequently,axiom,
! [A: $tType,P: A > $o,F: filter @ A] :
( ( ~ ( frequently @ A @ P @ F ) )
= ( eventually @ A
@ ^ [X4: A] :
~ ( P @ X4 )
@ F ) ) ).
% not_frequently
thf(fact_6825_not__eventually,axiom,
! [A: $tType,P: A > $o,F: filter @ A] :
( ( ~ ( eventually @ A @ P @ F ) )
= ( frequently @ A
@ ^ [X4: A] :
~ ( P @ X4 )
@ F ) ) ).
% not_eventually
thf(fact_6826_frequently__def,axiom,
! [A: $tType] :
( ( frequently @ A )
= ( ^ [P2: A > $o,F4: filter @ A] :
~ ( eventually @ A
@ ^ [X4: A] :
~ ( P2 @ X4 )
@ F4 ) ) ) ).
% frequently_def
thf(fact_6827_frequently__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F )
=> ( ( frequently @ A @ P @ F )
=> ( frequently @ A @ Q @ F ) ) ) ).
% frequently_mp
thf(fact_6828_eventually__frequently__const__simps_I5_J,axiom,
! [A: $tType,P: A > $o,C3: $o,F: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> C3 )
@ F )
= ( ( frequently @ A @ P @ F )
=> C3 ) ) ).
% eventually_frequently_const_simps(5)
thf(fact_6829_eventually__frequentlyE,axiom,
! [A: $tType,P: A > $o,F: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ~ ( P @ X4 )
| ( Q @ X4 ) )
@ F )
=> ( ( F
!= ( bot_bot @ ( filter @ A ) ) )
=> ( frequently @ A @ Q @ F ) ) ) ) ).
% eventually_frequentlyE
thf(fact_6830_frequently__const__iff,axiom,
! [A: $tType,P: $o,F: filter @ A] :
( ( frequently @ A
@ ^ [X4: A] : P
@ F )
= ( P
& ( F
!= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% frequently_const_iff
thf(fact_6831_eventually__frequently,axiom,
! [A: $tType,F: filter @ A,P: A > $o] :
( ( F
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F )
=> ( frequently @ A @ P @ F ) ) ) ).
% eventually_frequently
thf(fact_6832_times__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
@ ^ [X4: product_prod @ int @ int,Y5: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y5 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) )
@ ( times_times @ rat ) ) ).
% times_rat.transfer
thf(fact_6833_uminus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X4 ) ) @ ( product_snd @ int @ int @ X4 ) )
@ ( uminus_uminus @ rat ) ) ).
% uminus_rat.transfer
thf(fact_6834_normalize__negative,axiom,
! [Q4: int,P4: int] :
( ( ord_less @ int @ Q4 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P4 ) @ ( uminus_uminus @ int @ Q4 ) ) ) ) ) ).
% normalize_negative
thf(fact_6835_inverse__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X4: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X4 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_fst @ int @ int @ X4 ) ) )
@ ( inverse_inverse @ rat ) ) ).
% inverse_rat.transfer
thf(fact_6836_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_6837_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A4 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_6838_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A4 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_6839_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_6840_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less @ A @ B3 @ A4 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_6841_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_6842_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_6843_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A4 ) @ A4 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_6844_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A4 @ ( inverse_inverse @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_6845_normalize__denom__zero,axiom,
! [P4: int] :
( ( normalize @ ( product_Pair @ int @ int @ P4 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_6846_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less @ A @ B3 @ A4 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_6847_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A4 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_6848_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B3 @ A4 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_6849_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A4 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_6850_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A4 ) @ ( zero_zero @ A ) )
=> ( ( A4
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A4 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_6851_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A4 ) )
=> ( ( A4
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_6852_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ A4 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A4 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_6853_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A4 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_6854_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A4 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_6855_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X3: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X3 )
= ( times_times @ A @ X3 @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_6856_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_6857_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ).
% divide_inverse
thf(fact_6858_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ ( inverse_inverse @ A @ B4 ) @ A5 ) ) ) ) ).
% divide_inverse_commute
thf(fact_6859_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: nat] :
( ( times_times @ A @ ( power_power @ A @ X3 @ M2 ) @ ( inverse_inverse @ A @ X3 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X3 ) @ ( power_power @ A @ X3 @ M2 ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_6860_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X3 @ M2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ N ) @ ( power_power @ A @ X3 @ M2 ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_6861_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: nat,X3: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X3 )
= ( times_times @ A @ X3 @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_6862_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y: A,X3: A] :
( ( ( times_times @ A @ Y @ X3 )
= ( times_times @ A @ X3 @ Y ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X3 )
= ( times_times @ A @ X3 @ ( inverse_inverse @ A @ Y ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_6863_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A] :
( ( ( times_times @ A @ A4 @ B3 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A4 )
= B3 ) ) ) ).
% inverse_unique
thf(fact_6864_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A4 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_6865_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_6866_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B3 ) @ ( inverse_inverse @ A @ A4 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_6867_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_6868_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X3 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
& ( ord_less @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_6869_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A4 ) ) ) ) ) ).
% one_less_inverse
thf(fact_6870_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A4 ) @ ( plus_plus @ A @ A4 @ B3 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_6871_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A4 @ B3 ) @ ( inverse_inverse @ A @ A4 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% inverse_add
thf(fact_6872_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A4 ) @ A4 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_6873_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A4 ) @ ( minus_minus @ A @ B3 @ A4 ) ) @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_6874_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less @ A @ B3 @ A4 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_6875_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A4 ) @ ( inverse_inverse @ A @ B3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A4 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A4 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A4 @ B3 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_6876_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X3 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
& ( ord_less_eq @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_6877_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X3 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X3 ) ) ) ) ).
% inverse_less_1_iff
thf(fact_6878_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A4 ) ) ) ) ) ).
% one_le_inverse
thf(fact_6879_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ? [N5: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N5 ) ) ) @ X3 ) ) ) ).
% reals_Archimedean
thf(fact_6880_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N5 ) ) @ X3 ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_6881_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: nat,N: nat] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( power_power @ A @ X3 @ ( minus_minus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ X3 @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X3 ) @ M2 ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_6882_normalize__denom__pos,axiom,
! [R2: product_prod @ int @ int,P4: int,Q4: int] :
( ( ( normalize @ R2 )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% normalize_denom_pos
thf(fact_6883_normalize__crossproduct,axiom,
! [Q4: int,S3: int,P4: int,R2: int] :
( ( Q4
!= ( zero_zero @ int ) )
=> ( ( S3
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ R2 @ S3 ) ) )
=> ( ( times_times @ int @ P4 @ S3 )
= ( times_times @ int @ R2 @ Q4 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_6884_rat__inverse__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( A5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A5 ) @ B4 ) @ ( abs_abs @ int @ A5 ) ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_inverse_code
thf(fact_6885_normalize__def,axiom,
( normalize
= ( ^ [P7: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P7 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P7 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P7 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P7 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P7 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P7 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_6886_Gcd__insert,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( gcd_Gcd @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( gcd_gcd @ A @ A4 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ).
% Gcd_insert
thf(fact_6887_Gcd__fin_Oinsert,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( gcd_gcd @ A @ A4 @ ( semiring_gcd_Gcd_fin @ A @ A3 ) ) ) ) ).
% Gcd_fin.insert
thf(fact_6888_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A,B3: A] :
( ( gcd_Gcd @ A @ ( insert3 @ A @ A4 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ).
% Gcd_2
thf(fact_6889_quotient__of__number_I3_J,axiom,
! [K3: num] :
( ( quotient_of @ ( numeral_numeral @ rat @ K3 ) )
= ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K3 ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(3)
thf(fact_6890_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_6891_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_6892_quotient__of__number_I5_J,axiom,
! [K3: num] :
( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K3 ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K3 ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(5)
thf(fact_6893_quotient__of__number_I4_J,axiom,
( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(4)
thf(fact_6894_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,K3: A,N: A] :
( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ ( times_times @ A @ K3 @ M2 ) @ N ) )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add_mult
thf(fact_6895_Gcd__int__set__eq__fold,axiom,
! [Xs: list @ int] :
( ( gcd_Gcd @ int @ ( set2 @ int @ Xs ) )
= ( fold @ int @ int @ ( gcd_gcd @ int ) @ Xs @ ( zero_zero @ int ) ) ) ).
% Gcd_int_set_eq_fold
thf(fact_6896_gcd__dvd__prod,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,K3: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A4 @ B3 ) @ ( times_times @ A @ K3 @ B3 ) ) ) ).
% gcd_dvd_prod
thf(fact_6897_gcd__diff2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: A,M2: A] :
( ( gcd_gcd @ A @ ( minus_minus @ A @ N @ M2 ) @ N )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_diff2
thf(fact_6898_gcd__diff1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [M2: A,N: A] :
( ( gcd_gcd @ A @ ( minus_minus @ A @ M2 @ N ) @ N )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_diff1
thf(fact_6899_quotient__of__div,axiom,
! [R2: rat,N: int,D3: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ N @ D3 ) )
=> ( R2
= ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N ) @ ( ring_1_of_int @ rat @ D3 ) ) ) ) ).
% quotient_of_div
thf(fact_6900_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B3 @ A4 ) @ C2 )
= ( gcd_gcd @ A @ B3 @ C2 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_6901_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B3 @ ( times_times @ A @ C2 @ A4 ) )
= ( gcd_gcd @ A @ B3 @ C2 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_6902_quotient__of__denom__pos,axiom,
! [R2: rat,P4: int,Q4: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% quotient_of_denom_pos
thf(fact_6903_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( gcd_gcd @ A @ A4 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Gcd_fin.insert_remove
thf(fact_6904_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A3 )
= ( gcd_gcd @ A @ A4 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Gcd_fin.remove
thf(fact_6905_Gcd__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs: list @ A] :
( ( gcd_Gcd @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% Gcd_set_eq_fold
thf(fact_6906_Gcd__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% Gcd_fin.set_eq_fold
thf(fact_6907_rat__abs__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A5 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_abs_code
thf(fact_6908_rat__sgn__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( sgn_sgn @ rat @ P4 ) )
= ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P4 ) ) ) @ ( one_one @ int ) ) ) ).
% rat_sgn_code
thf(fact_6909_rat__uminus__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A5 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_uminus_code
thf(fact_6910_rat__divide__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D6: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ D6 ) @ ( times_times @ int @ C6 @ B4 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_divide_code
thf(fact_6911_rat__minus__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D6: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A5 @ D6 ) @ ( times_times @ int @ B4 @ C6 ) ) @ ( times_times @ int @ C6 @ D6 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_minus_code
thf(fact_6912_rat__times__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( times_times @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D6: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ B4 ) @ ( times_times @ int @ C6 @ D6 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_times_code
thf(fact_6913_rat__plus__code,axiom,
! [P4: rat,Q4: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P4 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C6: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D6: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A5 @ D6 ) @ ( times_times @ int @ B4 @ C6 ) ) @ ( times_times @ int @ C6 @ D6 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_plus_code
thf(fact_6914_quotient__of__int,axiom,
! [A4: int] :
( ( quotient_of @ ( of_int @ A4 ) )
= ( product_Pair @ int @ int @ A4 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_6915_Frct__code__post_I5_J,axiom,
! [K3: num] :
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K3 ) ) )
= ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K3 ) ) ) ).
% Frct_code_post(5)
thf(fact_6916_gcd__pos__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M2 @ N ) )
= ( ( M2
!= ( zero_zero @ nat ) )
| ( N
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_pos_nat
thf(fact_6917_Gcd__in,axiom,
! [A3: set @ nat] :
( ! [A6: nat,B5: nat] :
( ( member @ nat @ A6 @ A3 )
=> ( ( member @ nat @ B5 @ A3 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A6 @ B5 ) @ A3 ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A3 ) @ A3 ) ) ) ).
% Gcd_in
thf(fact_6918_Gcd__nat__set__eq__fold,axiom,
! [Xs: list @ nat] :
( ( gcd_Gcd @ nat @ ( set2 @ nat @ Xs ) )
= ( fold @ nat @ nat @ ( gcd_gcd @ nat ) @ Xs @ ( zero_zero @ nat ) ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_6919_Frct__code__post_I6_J,axiom,
! [K3: num,L: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K3 ) @ ( numeral_numeral @ int @ L ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K3 ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_6920_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( gcd_gcd @ nat @ M2 @ N )
= ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D6: nat] :
( ( dvd_dvd @ nat @ D6 @ M2 )
& ( dvd_dvd @ nat @ D6 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_6921_Frct__code__post_I4_J,axiom,
! [K3: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K3 ) @ ( one_one @ int ) ) )
= ( numeral_numeral @ rat @ K3 ) ) ).
% Frct_code_post(4)
thf(fact_6922_Frct__code__post_I2_J,axiom,
! [A4: int] :
( ( frct @ ( product_Pair @ int @ int @ A4 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_6923_Frct__code__post_I1_J,axiom,
! [A4: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A4 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_6924_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) )
= ( one_one @ rat ) ) ).
% Frct_code_post(3)
thf(fact_6925_Frct__code__post_I8_J,axiom,
! [A4: int,B3: int] :
( ( frct @ ( product_Pair @ int @ int @ A4 @ ( uminus_uminus @ int @ B3 ) ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A4 @ B3 ) ) ) ) ).
% Frct_code_post(8)
thf(fact_6926_Frct__code__post_I7_J,axiom,
! [A4: int,B3: int] :
( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A4 ) @ B3 ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A4 @ B3 ) ) ) ) ).
% Frct_code_post(7)
thf(fact_6927_gcd__nat_Opelims,axiom,
! [X3: nat,Xa: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y = X3 ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X3 @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X3 @ Xa ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_6928_plus__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ratrel @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel )
@ ^ [X4: product_prod @ int @ int,Y5: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) )
@ ^ [X4: product_prod @ int @ int,Y5: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) ) ) ).
% plus_rat.rsp
thf(fact_6929_one__rat_Orsp,axiom,
ratrel @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ).
% one_rat.rsp
thf(fact_6930_zero__rat_Orsp,axiom,
ratrel @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ).
% zero_rat.rsp
thf(fact_6931_Fract_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
@ ^ [Y4: int,Z4: int] : Y4 = Z4
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ^ [Y4: int,Z4: int] : Y4 = Z4
@ ratrel )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( B4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B4 ) )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( B4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B4 ) ) ) ).
% Fract.rsp
thf(fact_6932_times__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ratrel @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel )
@ ^ [X4: product_prod @ int @ int,Y5: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y5 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) )
@ ^ [X4: product_prod @ int @ int,Y5: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y5 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) ) ) ).
% times_rat.rsp
thf(fact_6933_uminus__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
@ ^ [X4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X4 ) ) @ ( product_snd @ int @ int @ X4 ) )
@ ^ [X4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X4 ) ) @ ( product_snd @ int @ int @ X4 ) ) ) ).
% uminus_rat.rsp
thf(fact_6934_inverse__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
@ ^ [X4: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X4 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_fst @ int @ int @ X4 ) ) )
@ ^ [X4: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X4 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_fst @ int @ int @ X4 ) ) ) ) ).
% inverse_rat.rsp
thf(fact_6935_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod @ int @ int,X3: product_prod @ int @ int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X3 @ X3 )
=> ( ( plus_plus @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X3 ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_6936_inverse__rat_Oabs__eq,axiom,
! [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
=> ( ( inverse_inverse @ rat @ ( abs_Rat @ X3 ) )
= ( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X3 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_6937_one__rat__def,axiom,
( ( one_one @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% one_rat_def
thf(fact_6938_zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% zero_rat_def
thf(fact_6939_uminus__rat_Oabs__eq,axiom,
! [X3: product_prod @ int @ int] :
( ( ratrel @ X3 @ X3 )
=> ( ( uminus_uminus @ rat @ ( abs_Rat @ X3 ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ).
% uminus_rat.abs_eq
thf(fact_6940_times__rat_Oabs__eq,axiom,
! [Xa: product_prod @ int @ int,X3: product_prod @ int @ int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X3 @ X3 )
=> ( ( times_times @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X3 ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_fst @ int @ int @ X3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_6941_inverse__rat__def,axiom,
( ( inverse_inverse @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X4: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X4 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_fst @ int @ int @ X4 ) ) ) ) ) ).
% inverse_rat_def
thf(fact_6942_flip__pred,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ A @ B ),R: B > A > $o] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( conversep @ B @ A @ R ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X4: A,Y5: B] : ( product_Pair @ B @ A @ Y5 @ X4 ) )
@ A3 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R ) ) ) ) ).
% flip_pred
thf(fact_6943_conversep__eq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [Y4: A,Z4: A] : Y4 = Z4 )
= ( ^ [Y4: A,Z4: A] : Y4 = Z4 ) ) ).
% conversep_eq
thf(fact_6944_conversep__iff,axiom,
! [B: $tType,A: $tType] :
( ( conversep @ A @ B )
= ( ^ [R5: A > B > $o,A5: B,B4: A] : ( R5 @ B4 @ A5 ) ) ) ).
% conversep_iff
thf(fact_6945_conversep__inject,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S3: B > A > $o] :
( ( ( conversep @ B @ A @ R2 )
= ( conversep @ B @ A @ S3 ) )
= ( R2 = S3 ) ) ).
% conversep_inject
thf(fact_6946_conversep__conversep,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ( conversep @ B @ A @ ( conversep @ A @ B @ R2 ) )
= R2 ) ).
% conversep_conversep
thf(fact_6947_conversep__noteq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [X4: A,Y5: A] : X4 != Y5 )
= ( ^ [X4: A,Y5: A] : X4 != Y5 ) ) ).
% conversep_noteq
thf(fact_6948_conversep__mono,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S3: B > A > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ ( conversep @ B @ A @ R2 ) @ ( conversep @ B @ A @ S3 ) )
= ( ord_less_eq @ ( B > A > $o ) @ R2 @ S3 ) ) ).
% conversep_mono
thf(fact_6949_conversep__le__swap,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,S3: B > A > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ ( conversep @ B @ A @ S3 ) )
= ( ord_less_eq @ ( B > A > $o ) @ ( conversep @ A @ B @ R2 ) @ S3 ) ) ).
% conversep_le_swap
thf(fact_6950_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( conversep @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R2 ) )
= ( ^ [X4: B,Y5: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y5 ) @ ( converse @ A @ B @ R2 ) ) ) ) ).
% conversep_converse_eq
thf(fact_6951_converse__join,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S3: B > A > $o] :
( ( conversep @ B @ A @ ( sup_sup @ ( B > A > $o ) @ R2 @ S3 ) )
= ( sup_sup @ ( A > B > $o ) @ ( conversep @ B @ A @ R2 ) @ ( conversep @ B @ A @ S3 ) ) ) ).
% converse_join
thf(fact_6952_conversep_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( conversep @ A @ B )
= ( ^ [R5: A > B > $o,A13: B,A23: A] :
? [A5: A,B4: B] :
( ( A13 = B4 )
& ( A23 = A5 )
& ( R5 @ A5 @ B4 ) ) ) ) ).
% conversep.simps
thf(fact_6953_conversepD,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,B3: B,A4: A] :
( ( conversep @ A @ B @ R2 @ B3 @ A4 )
=> ( R2 @ A4 @ B3 ) ) ).
% conversepD
thf(fact_6954_conversepE,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A1: B,A22: A] :
( ( conversep @ A @ B @ R2 @ A1 @ A22 )
=> ( R2 @ A22 @ A1 ) ) ).
% conversepE
thf(fact_6955_conversepI,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,A4: A,B3: B] :
( ( R2 @ A4 @ B3 )
=> ( conversep @ A @ B @ R2 @ B3 @ A4 ) ) ).
% conversepI
thf(fact_6956_list_Orel__conversep,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( list_all2 @ B @ A @ ( conversep @ A @ B @ R ) )
= ( conversep @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ R ) ) ) ).
% list.rel_conversep
thf(fact_6957_list_Orel__flip,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A4: list @ B,B3: list @ A] :
( ( list_all2 @ B @ A @ ( conversep @ A @ B @ R ) @ A4 @ B3 )
= ( list_all2 @ A @ B @ R @ B3 @ A4 ) ) ).
% list.rel_flip
thf(fact_6958_converse__meet,axiom,
! [A: $tType,B: $tType,R2: B > A > $o,S3: B > A > $o] :
( ( conversep @ B @ A @ ( inf_inf @ ( B > A > $o ) @ R2 @ S3 ) )
= ( inf_inf @ ( A > B > $o ) @ ( conversep @ B @ A @ R2 ) @ ( conversep @ B @ A @ S3 ) ) ) ).
% converse_meet
thf(fact_6959_rel__filter__conversep,axiom,
! [A: $tType,B: $tType,A3: B > A > $o] :
( ( rel_filter @ A @ B @ ( conversep @ B @ A @ A3 ) )
= ( conversep @ ( filter @ B ) @ ( filter @ A ) @ ( rel_filter @ B @ A @ A3 ) ) ) ).
% rel_filter_conversep
thf(fact_6960_converse__def,axiom,
! [B: $tType,A: $tType] :
( ( converse @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ( conversep @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ).
% converse_def
thf(fact_6961_uminus__rat__def,axiom,
( ( uminus_uminus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X4 ) ) @ ( product_snd @ int @ int @ X4 ) ) ) ) ).
% uminus_rat_def
thf(fact_6962_plus__rat__def,axiom,
( ( plus_plus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
@ ^ [X4: product_prod @ int @ int,Y5: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) ) ) ) ).
% plus_rat_def
thf(fact_6963_times__rat__def,axiom,
( ( times_times @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
@ ^ [X4: product_prod @ int @ int,Y5: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y5 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y5 ) ) ) ) ) ).
% times_rat_def
thf(fact_6964_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ Xs )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ ( suc @ I4 ) ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
thf(fact_6965_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X3: A,Y: B] :
( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y ) )
= ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% prod_set_simps(1)
thf(fact_6966_transp__ge,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ).
% transp_ge
thf(fact_6967_transp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% transp_le
thf(fact_6968_transp__trans__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( transp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R2 ) )
= ( trans @ A @ R2 ) ) ).
% transp_trans_eq
thf(fact_6969_transp__trans,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R5: A > A > $o] : ( trans @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ).
% transp_trans
thf(fact_6970_transp__empty,axiom,
! [A: $tType] :
( transp @ A
@ ^ [X4: A,Y5: A] : $false ) ).
% transp_empty
thf(fact_6971_transp__singleton,axiom,
! [A: $tType,A4: A] :
( transp @ A
@ ^ [X4: A,Y5: A] :
( ( X4 = A4 )
& ( Y5 = A4 ) ) ) ).
% transp_singleton
thf(fact_6972_transpD,axiom,
! [A: $tType,R2: A > A > $o,X3: A,Y: A,Z2: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X3 @ Z2 ) ) ) ) ).
% transpD
thf(fact_6973_transpE,axiom,
! [A: $tType,R2: A > A > $o,X3: A,Y: A,Z2: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X3 @ Z2 ) ) ) ) ).
% transpE
thf(fact_6974_transpI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X: A,Y2: A,Z3: A] :
( ( R2 @ X @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X @ Z3 ) ) )
=> ( transp @ A @ R2 ) ) ).
% transpI
thf(fact_6975_transp__def,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R5: A > A > $o] :
! [X4: A,Y5: A,Z6: A] :
( ( R5 @ X4 @ Y5 )
=> ( ( R5 @ Y5 @ Z6 )
=> ( R5 @ X4 @ Z6 ) ) ) ) ) ).
% transp_def
thf(fact_6976_transp__equality,axiom,
! [A: $tType] :
( transp @ A
@ ^ [Y4: A,Z4: A] : Y4 = Z4 ) ).
% transp_equality
thf(fact_6977_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_6978_sorted__wrt2,axiom,
! [A: $tType,P: A > A > $o,X3: A,Y: A,Zs: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Zs ) ) )
= ( ( P @ X3 @ Y )
& ( sorted_wrt @ A @ P @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted_wrt2
thf(fact_6979_transp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less @ A ) ) ) ).
% transp_less
thf(fact_6980_transp__gr,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ) ).
% transp_gr
thf(fact_6981_transp__inf,axiom,
! [A: $tType,R2: A > A > $o,S3: A > A > $o] :
( ( transp @ A @ R2 )
=> ( ( transp @ A @ S3 )
=> ( transp @ A @ ( inf_inf @ ( A > A > $o ) @ R2 @ S3 ) ) ) ) ).
% transp_inf
thf(fact_6982_transp__INF,axiom,
! [B: $tType,A: $tType,S: set @ A,R2: A > B > B > $o] :
( ! [X: A] :
( ( member @ A @ X @ S )
=> ( transp @ B @ ( R2 @ X ) ) )
=> ( transp @ B @ ( complete_Inf_Inf @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R2 @ S ) ) ) ) ).
% transp_INF
thf(fact_6983_prod__set__defs_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( basic_fsts @ A @ B )
= ( ^ [P7: product_prod @ A @ B] : ( insert3 @ A @ ( product_fst @ A @ B @ P7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% prod_set_defs(1)
thf(fact_6984_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X3: A,Y: B] :
( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y ) )
= ( insert3 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) ) ).
% prod_set_simps(2)
thf(fact_6985_prod__set__defs_I2_J,axiom,
! [D: $tType,C: $tType] :
( ( basic_snds @ C @ D )
= ( ^ [P7: product_prod @ C @ D] : ( insert3 @ D @ ( product_snd @ C @ D @ P7 ) @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% prod_set_defs(2)
thf(fact_6986_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( single_valuedp @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R2 ) )
= ( single_valued @ A @ B @ R2 ) ) ).
% single_valuedp_single_valued_eq
thf(fact_6987_Eps__Opt__def,axiom,
! [A: $tType] :
( ( eps_Opt @ A )
= ( ^ [P2: A > $o] :
( if @ ( option @ A )
@ ? [X9: A] : ( P2 @ X9 )
@ ( some @ A @ ( fChoice @ A @ P2 ) )
@ ( none @ A ) ) ) ) ).
% Eps_Opt_def
thf(fact_6988_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X3: A,Y: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X10: A,Y8: B] :
( ( X3 = X10 )
& ( Y = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X3 @ Y ) ) ).
% Eps_case_prod_eq
thf(fact_6989_some__insert__self,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( insert3 @ A
@ ( fChoice @ A
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
@ S )
= S ) ) ).
% some_insert_self
thf(fact_6990_single__valuedp__less__eq,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,S3: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ S3 )
=> ( ( single_valuedp @ A @ B @ S3 )
=> ( single_valuedp @ A @ B @ R2 ) ) ) ).
% single_valuedp_less_eq
thf(fact_6991_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P2: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B4 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_6992_inv__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inv_on @ A @ B )
= ( ^ [F7: A > B,A7: set @ A,X4: B] :
( fChoice @ A
@ ^ [Y5: A] :
( ( member @ A @ Y5 @ A7 )
& ( ( F7 @ Y5 )
= X4 ) ) ) ) ) ).
% inv_on_def
thf(fact_6993_single__valuedpD,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X3: A,Y: B,Z2: B] :
( ( single_valuedp @ A @ B @ R2 )
=> ( ( R2 @ X3 @ Y )
=> ( ( R2 @ X3 @ Z2 )
=> ( Y = Z2 ) ) ) ) ).
% single_valuedpD
thf(fact_6994_single__valuedpI,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ! [X: A,Y2: B,Z3: B] :
( ( R2 @ X @ Y2 )
=> ( ( R2 @ X @ Z3 )
=> ( Y2 = Z3 ) ) )
=> ( single_valuedp @ A @ B @ R2 ) ) ).
% single_valuedpI
thf(fact_6995_single__valuedp__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valuedp @ A @ B )
= ( ^ [R5: A > B > $o] :
! [X4: A,Y5: B] :
( ( R5 @ X4 @ Y5 )
=> ! [Z6: B] :
( ( R5 @ X4 @ Z6 )
=> ( Y5 = Z6 ) ) ) ) ) ).
% single_valuedp_def
thf(fact_6996_some__theI,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ? [A15: A,X_12: B] : ( P @ A15 @ X_12 )
=> ( ! [B18: B,B25: B] :
( ? [A6: A] : ( P @ A6 @ B18 )
=> ( ? [A6: A] : ( P @ A6 @ B25 )
=> ( B18 = B25 ) ) )
=> ( P
@ ( fChoice @ A
@ ^ [A5: A] :
? [X9: B] : ( P @ A5 @ X9 ) )
@ ( the @ B
@ ^ [B4: B] :
? [A5: A] : ( P @ A5 @ B4 ) ) ) ) ) ).
% some_theI
thf(fact_6997_some__elem,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
@ S ) ) ).
% some_elem
thf(fact_6998_some__in__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
@ A3 )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_6999_single__valuedp__bot,axiom,
! [B: $tType,A: $tType] : ( single_valuedp @ A @ B @ ( bot_bot @ ( A > B > $o ) ) ) ).
% single_valuedp_bot
thf(fact_7000_single__valuedp__iff__Uniq,axiom,
! [B: $tType,A: $tType] :
( ( single_valuedp @ A @ B )
= ( ^ [R5: A > B > $o] :
! [X4: A] : ( uniq @ B @ ( R5 @ X4 ) ) ) ) ).
% single_valuedp_iff_Uniq
thf(fact_7001_fun__of__rel__def,axiom,
! [A: $tType,B: $tType] :
( ( fun_of_rel @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A ),X4: B] :
( fChoice @ A
@ ^ [Y5: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y5 ) @ R4 ) ) ) ) ).
% fun_of_rel_def
thf(fact_7002_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X3: B,A3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X3 ) @ ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A3 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y ) @ A3 ) ) ).
% pair_in_swap_image
thf(fact_7003_swap__simp,axiom,
! [A: $tType,B: $tType,X3: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X3 @ Y ) )
= ( product_Pair @ A @ B @ Y @ X3 ) ) ).
% swap_simp
thf(fact_7004_prod__filter__commute,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F4: filter @ A,G8: filter @ B] : ( filtermap @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( prod_filter @ B @ A @ G8 @ F4 ) ) ) ) ).
% prod_filter_commute
thf(fact_7005_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P7: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P7 ) @ ( product_fst @ A @ B @ P7 ) ) ) ) ).
% prod.swap_def
thf(fact_7006_acyclic__insert,axiom,
! [A: $tType,Y: A,X3: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ R2 ) )
= ( ( transitive_acyclic @ A @ R2 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% acyclic_insert
thf(fact_7007_successively__transfer,axiom,
! [A: $tType,B: $tType,A3: 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 ) @ A3
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 ) )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A3 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( successively @ A )
@ ( successively @ B ) ) ).
% successively_transfer
thf(fact_7008_acyclic__empty,axiom,
! [A: $tType] : ( transitive_acyclic @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% acyclic_empty
thf(fact_7009_successively__rev,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( successively @ A @ P @ ( rev @ A @ Xs ) )
= ( successively @ A
@ ^ [X4: A,Y5: A] : ( P @ Y5 @ X4 )
@ Xs ) ) ).
% successively_rev
thf(fact_7010_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_7011_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_7012_subset__mset_OacyclicI__order,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ B ),F3: B > ( multiset @ A )] :
( ! [A6: B,B5: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A6 @ B5 ) @ R2 )
=> ( subset_mset @ A @ ( F3 @ B5 ) @ ( F3 @ A6 ) ) )
=> ( transitive_acyclic @ B @ R2 ) ) ).
% subset_mset.acyclicI_order
thf(fact_7013_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [R2: set @ ( product_prod @ B @ B ),F3: B > A] :
( ! [A6: B,B5: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A6 @ B5 ) @ R2 )
=> ( ord_less @ A @ ( F3 @ B5 ) @ ( F3 @ A6 ) ) )
=> ( transitive_acyclic @ B @ R2 ) ) ) ).
% acyclicI_order
thf(fact_7014_cyclicE,axiom,
! [A: $tType,G: set @ ( product_prod @ A @ A )] :
( ~ ( transitive_acyclic @ A @ G )
=> ~ ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ G ) ) ) ).
% cyclicE
thf(fact_7015_acyclic__def,axiom,
! [A: $tType] :
( ( transitive_acyclic @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ).
% acyclic_def
thf(fact_7016_acyclicI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R2 ) )
=> ( transitive_acyclic @ A @ R2 ) ) ).
% acyclicI
thf(fact_7017_successively__altdef,axiom,
! [A: $tType] :
( ( successively @ A )
= ( ^ [P2: A > A > $o] :
( rec_list @ $o @ A @ $true
@ ^ [X4: A,Xs3: list @ A,B4: $o] :
( case_list @ $o @ A @ $true
@ ^ [Y5: A,Xa5: list @ A] :
( ( P2 @ X4 @ Y5 )
& B4 )
@ Xs3 ) ) ) ) ).
% successively_altdef
thf(fact_7018_successively__remdups__adj__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X @ X ) )
=> ( ( successively @ A @ P @ ( remdups_adj @ A @ Xs ) )
= ( successively @ A @ P @ Xs ) ) ) ).
% successively_remdups_adj_iff
thf(fact_7019_distinct__adj__def,axiom,
! [A: $tType] :
( ( distinct_adj @ A )
= ( successively @ A
@ ^ [X4: A,Y5: A] : X4 != Y5 ) ) ).
% distinct_adj_def
thf(fact_7020_acyclic__union_I1_J,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B2 ) )
=> ( transitive_acyclic @ A @ A3 ) ) ).
% acyclic_union(1)
thf(fact_7021_acyclic__union_I2_J,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B2: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B2 ) )
=> ( transitive_acyclic @ A @ B2 ) ) ).
% acyclic_union(2)
thf(fact_7022_successively_Osimps_I3_J,axiom,
! [A: $tType,P: A > A > $o,X3: A,Y: A,Xs: list @ A] :
( ( successively @ A @ P @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
= ( ( P @ X3 @ Y )
& ( successively @ A @ P @ ( cons @ A @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_7023_successively_Oelims_I3_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A] :
( ~ ( successively @ A @ X3 @ Xa )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( ( X3 @ X @ Y2 )
& ( successively @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_7024_successively__cong,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o,Q: A > A > $o,Ys: list @ A] :
( ! [X: A,Y2: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X @ Y2 )
= ( Q @ X @ Y2 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( successively @ A @ P @ Xs )
= ( successively @ A @ Q @ Ys ) ) ) ) ).
% successively_cong
thf(fact_7025_successively__mono,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,Q: A > A > $o] :
( ( successively @ A @ P @ Xs )
=> ( ! [X: A,Y2: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) ) ) )
=> ( successively @ A @ Q @ Xs ) ) ) ).
% successively_mono
thf(fact_7026_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_7027_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_7028_successively_Osimps_I1_J,axiom,
! [A: $tType,P: A > A > $o] : ( successively @ A @ P @ ( nil @ A ) ) ).
% successively.simps(1)
thf(fact_7029_successively_Oelims_I2_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A] :
( ( successively @ A @ X3 @ Xa )
=> ( ( Xa
!= ( nil @ A ) )
=> ( ! [X: A] :
( Xa
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ~ ( ( X3 @ X @ Y2 )
& ( successively @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_7030_successively_Oelims_I1_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( successively @ A @ X3 @ Xa )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ Y )
=> ( ( ? [X: A] :
( Xa
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ Y )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( Y
= ( ~ ( ( X3 @ X @ Y2 )
& ( successively @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_7031_successively_Osimps_I2_J,axiom,
! [A: $tType,P: A > A > $o,X3: A] : ( successively @ A @ P @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% successively.simps(2)
thf(fact_7032_successively__iff__sorted__wrt__strong,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ! [X: A,Y2: A,Z3: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Z3 @ ( set2 @ A @ Xs ) )
=> ( ( P @ X @ Y2 )
=> ( ( P @ Y2 @ Z3 )
=> ( P @ X @ Z3 ) ) ) ) ) )
=> ( ( successively @ A @ P @ Xs )
= ( sorted_wrt @ A @ P @ Xs ) ) ) ).
% successively_iff_sorted_wrt_strong
thf(fact_7033_successively__map,axiom,
! [A: $tType,B: $tType,P: A > A > $o,F3: B > A,Xs: list @ B] :
( ( successively @ A @ P @ ( map @ B @ A @ F3 @ Xs ) )
= ( successively @ B
@ ^ [X4: B,Y5: B] : ( P @ ( F3 @ X4 ) @ ( F3 @ Y5 ) )
@ Xs ) ) ).
% successively_map
thf(fact_7034_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_7035_successively__Cons,axiom,
! [A: $tType,P: A > A > $o,X3: A,Xs: list @ A] :
( ( successively @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ( ( P @ X3 @ ( hd @ A @ Xs ) )
& ( successively @ A @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_7036_acyclic__insert__cyclic,axiom,
! [A: $tType,G: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( transitive_acyclic @ A @ G )
=> ( ~ ( transitive_acyclic @ A @ ( insert3 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ G ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ ( transitive_rtrancl @ A @ G ) ) ) ) ).
% acyclic_insert_cyclic
thf(fact_7037_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_7038_successively_Opelims_I1_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( successively @ A @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) ) ) )
=> ( ! [X: A] :
( ( Xa
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( ( Y
= ( ( X3 @ X @ Y2 )
& ( successively @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% successively.pelims(1)
thf(fact_7039_successively_Opelims_I2_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A] :
( ( successively @ A @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) ) )
=> ( ! [X: A] :
( ( Xa
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) )
=> ~ ( ( X3 @ X @ Y2 )
& ( successively @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.pelims(2)
thf(fact_7040_successively_Opelims_I3_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A] :
( ~ ( successively @ A @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ Xa ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X3 @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) )
=> ( ( X3 @ X @ Y2 )
& ( successively @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ).
% successively.pelims(3)
thf(fact_7041_su__rel__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( su_rel_fun @ A @ B )
= ( ^ [F4: set @ ( product_prod @ A @ B ),F7: A > B] :
( ! [A7: A,B7: B,B19: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B7 ) @ F4 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B19 ) @ F4 )
=> ( B7 = B19 ) ) )
& ! [A7: A,P2: $o] :
( ! [B7: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B7 ) @ F4 )
=> P2 )
=> P2 )
& ! [A7: A] :
( ( F7 @ A7 )
= ( the @ B
@ ^ [B7: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B7 ) @ F4 ) ) ) ) ) ) ).
% su_rel_fun_def
thf(fact_7042_su__rel__fun_Ointro,axiom,
! [B: $tType,A: $tType,F: set @ ( product_prod @ A @ B ),F3: A > B] :
( ! [A14: A,B6: B,B9: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A14 @ B6 ) @ F )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A14 @ B9 ) @ F )
=> ( B6 = B9 ) ) )
=> ( ! [A14: A,P6: $o] :
( ! [B13: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A14 @ B13 ) @ F )
=> P6 )
=> P6 )
=> ( ! [A14: A] :
( ( F3 @ A14 )
= ( the @ B
@ ^ [B7: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A14 @ B7 ) @ F ) ) )
=> ( su_rel_fun @ A @ B @ F @ F3 ) ) ) ) ).
% su_rel_fun.intro
thf(fact_7043_su__rel__fun_Orepr,axiom,
! [B: $tType,A: $tType,F: set @ ( product_prod @ A @ B ),F3: A > B,A3: A,B2: B] :
( ( su_rel_fun @ A @ B @ F @ F3 )
=> ( ( ( F3 @ A3 )
= B2 )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ F ) ) ) ).
% su_rel_fun.repr
thf(fact_7044_su__rel__fun_Orepr1,axiom,
! [B: $tType,A: $tType,F: set @ ( product_prod @ A @ B ),F3: A > B,A3: A] :
( ( su_rel_fun @ A @ B @ F @ F3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ ( F3 @ A3 ) ) @ F ) ) ).
% su_rel_fun.repr1
thf(fact_7045_su__rel__fun_Orepr2,axiom,
! [B: $tType,A: $tType,F: set @ ( product_prod @ A @ B ),F3: A > B,A3: A,B2: B] :
( ( su_rel_fun @ A @ B @ F @ F3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ F )
=> ( B2
= ( F3 @ A3 ) ) ) ) ).
% su_rel_fun.repr2
thf(fact_7046_su__rel__fun_Ounique,axiom,
! [A: $tType,B: $tType,F: set @ ( product_prod @ A @ B ),F3: A > B,A3: A,B2: B,B11: B] :
( ( su_rel_fun @ A @ B @ F @ F3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ F )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B11 ) @ F )
=> ( B2 = B11 ) ) ) ) ).
% su_rel_fun.unique
thf(fact_7047_su__rel__fun_Osurjective,axiom,
! [B: $tType,A: $tType,F: set @ ( product_prod @ A @ B ),F3: A > B,A3: A] :
( ( su_rel_fun @ A @ B @ F @ F3 )
=> ~ ! [B6: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B6 ) @ F ) ) ).
% su_rel_fun.surjective
thf(fact_7048_su__rel__fun_Of__def,axiom,
! [A: $tType,B: $tType,F: set @ ( product_prod @ A @ B ),F3: A > B,A3: A] :
( ( su_rel_fun @ A @ B @ F @ F3 )
=> ( ( F3 @ A3 )
= ( the @ B
@ ^ [B7: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B7 ) @ F ) ) ) ) ).
% su_rel_fun.f_def
thf(fact_7049_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] :
( ( B4 != A5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_7050_map__of__map,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,Xs: list @ ( product_prod @ A @ C )] :
( ( map_of @ A @ B
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [K4: A,V4: C] : ( product_Pair @ A @ B @ K4 @ ( F3 @ V4 ) ) )
@ Xs ) )
= ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F3 ) @ ( map_of @ A @ C @ Xs ) ) ) ).
% map_of_map
thf(fact_7051_ex__assn__def,axiom,
! [A: $tType] :
( ( ex_assn @ A )
= ( ^ [P2: A > assn] :
( abs_assn
@ ^ [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
? [X4: A] : ( rep_assn @ ( P2 @ X4 ) @ H3 ) ) ) ) ).
% ex_assn_def
thf(fact_7052_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr7122648621184425601vImage @ A @ B )
= ( ^ [A7: set @ A,R4: set @ ( product_prod @ B @ B ),F7: A > B] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [A13: A,A23: A] :
( ( Uu3
= ( product_Pair @ A @ A @ A13 @ A23 ) )
& ( member @ A @ A13 @ A7 )
& ( member @ A @ A23 @ A7 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F7 @ A13 ) @ ( F7 @ A23 ) ) @ R4 ) ) ) ) ) ).
% relInvImage_def
thf(fact_7053_Rep__assn__inverse,axiom,
! [X3: assn] :
( ( abs_assn @ ( rep_assn @ X3 ) )
= X3 ) ).
% Rep_assn_inverse
thf(fact_7054_Abs__assn__eqI_I2_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H )
= ( rep_assn @ Pr @ H ) )
=> ( Pr
= ( abs_assn @ P ) ) ) ).
% Abs_assn_eqI(2)
thf(fact_7055_Abs__assn__eqI_I1_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H )
= ( rep_assn @ Pr @ H ) )
=> ( ( abs_assn @ P )
= Pr ) ) ).
% Abs_assn_eqI(1)
thf(fact_7056_pure__assn__def,axiom,
( pure_assn
= ( ^ [B4: $o] : ( abs_assn @ ( pure_assn_raw @ ( heap_ext @ product_unit ) @ nat @ B4 ) ) ) ) ).
% pure_assn_def
thf(fact_7057_bot__assn__def,axiom,
( ( bot_bot @ assn )
= ( abs_assn
@ ^ [Uu3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : $false ) ) ).
% bot_assn_def
thf(fact_7058_one__assn__def,axiom,
( ( one_one @ assn )
= ( abs_assn @ one_assn_raw ) ) ).
% one_assn_def
thf(fact_7059_sup__assn__def,axiom,
( ( sup_sup @ assn )
= ( ^ [P2: assn,Q2: assn] :
( abs_assn
@ ^ [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H3 )
| ( rep_assn @ Q2 @ H3 ) ) ) ) ) ).
% sup_assn_def
thf(fact_7060_inf__assn__def,axiom,
( ( inf_inf @ assn )
= ( ^ [P2: assn,Q2: assn] :
( abs_assn
@ ^ [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P2 @ H3 )
& ( rep_assn @ Q2 @ H3 ) ) ) ) ) ).
% inf_assn_def
thf(fact_7061_times__assn__def,axiom,
( ( times_times @ assn )
= ( ^ [P2: assn,Q2: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q2 ) ) ) ) ) ).
% times_assn_def
thf(fact_7062_wand__assn__def,axiom,
( wand_assn
= ( ^ [P2: assn,Q2: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q2 ) ) ) ) ) ).
% wand_assn_def
thf(fact_7063_sngr__assn__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( sngr_assn @ A )
= ( ^ [R5: ref @ A,X4: A] : ( abs_assn @ ( sngr_assn_raw @ A @ R5 @ X4 ) ) ) ) ) ).
% sngr_assn_def
thf(fact_7064_snga__assn__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( snga_assn @ A )
= ( ^ [R5: array @ A,A5: list @ A] : ( abs_assn @ ( snga_assn_raw @ A @ R5 @ A5 ) ) ) ) ) ).
% snga_assn_def
thf(fact_7065_uminus__assn__def,axiom,
( ( uminus_uminus @ assn )
= ( ^ [P2: assn] :
( abs_assn
@ ^ [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ H3 )
& ~ ( rep_assn @ P2 @ H3 ) ) ) ) ) ).
% uminus_assn_def
thf(fact_7066_in__range__empty,axiom,
! [H2: heap_ext @ product_unit] : ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% in_range_empty
thf(fact_7067_mod__true,axiom,
! [H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( top_top @ assn ) @ H2 )
= ( in_range @ H2 ) ) ).
% mod_true
thf(fact_7068_in__range__dist__union,axiom,
! [H2: heap_ext @ product_unit,As2: set @ nat,As3: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ ( sup_sup @ ( set @ nat ) @ As2 @ As3 ) ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As3 ) ) ) ) ).
% in_range_dist_union
thf(fact_7069_mod__not__dist,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( uminus_uminus @ assn @ P ) @ H2 )
= ( ( in_range @ H2 )
& ~ ( rep_assn @ P @ H2 ) ) ) ).
% mod_not_dist
thf(fact_7070_in__range__subset,axiom,
! [As2: set @ nat,As3: set @ nat,H2: heap_ext @ product_unit] :
( ( ord_less_eq @ ( set @ nat ) @ As2 @ As3 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As3 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) ) ) ) ).
% in_range_subset
thf(fact_7071_models__in__range,axiom,
! [P: assn,H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H2 )
=> ( in_range @ H2 ) ) ).
% models_in_range
thf(fact_7072_top__assn__def,axiom,
( ( top_top @ assn )
= ( abs_assn @ in_range ) ) ).
% top_assn_def
thf(fact_7073_wand__assnI,axiom,
! [H2: heap_ext @ product_unit,As2: set @ nat,Q: assn,R: assn] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ( ! [H5: heap_ext @ product_unit,As7: set @ nat] :
( ( ( inf_inf @ ( set @ nat ) @ As2 @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( relH @ As2 @ H2 @ H5 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As2 ) )
=> ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As7 ) )
=> ( rep_assn @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As2 @ As7 ) ) ) ) ) ) )
=> ( rep_assn @ ( wand_assn @ Q @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) ) ) ) ).
% wand_assnI
thf(fact_7074_wand__raw_Oelims_I3_J,axiom,
! [X3: ( 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 @ X3 @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
& ! [H5: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H @ H5 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) )
& ( X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As7 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As @ As7 ) ) ) ) ) ) ) ).
% wand_raw.elims(3)
thf(fact_7075_relH__dist__union,axiom,
! [As2: set @ nat,As3: set @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( relH @ ( sup_sup @ ( set @ nat ) @ As2 @ As3 ) @ H2 @ H4 )
= ( ( relH @ As2 @ H2 @ H4 )
& ( relH @ As3 @ H2 @ H4 ) ) ) ).
% relH_dist_union
thf(fact_7076_relH__refl,axiom,
! [H2: heap_ext @ product_unit,As2: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ( relH @ As2 @ H2 @ H2 ) ) ).
% relH_refl
thf(fact_7077_relH__in__rangeI_I1_J,axiom,
! [As2: set @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( relH @ As2 @ H2 @ H4 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) ) ) ).
% relH_in_rangeI(1)
thf(fact_7078_relH__in__rangeI_I2_J,axiom,
! [As2: set @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( relH @ As2 @ H2 @ H4 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ).
% relH_in_rangeI(2)
thf(fact_7079_mod__relH,axiom,
! [As2: set @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,P: assn] :
( ( relH @ As2 @ H2 @ H4 )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ).
% mod_relH
thf(fact_7080_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_7081_relH__sym,axiom,
! [As2: set @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( relH @ As2 @ H2 @ H4 )
=> ( relH @ As2 @ H4 @ H2 ) ) ).
% relH_sym
thf(fact_7082_relH__subset,axiom,
! [Bs: set @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,As2: set @ nat] :
( ( relH @ Bs @ H2 @ H4 )
=> ( ( ord_less_eq @ ( set @ nat ) @ As2 @ Bs )
=> ( relH @ As2 @ H2 @ H4 ) ) ) ).
% relH_subset
thf(fact_7083_wand__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As2: set @ nat] :
( ( wand_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
& ! [H8: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As2 @ H2 @ 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_7084_wand__raw_Oelims_I1_J,axiom,
! [X3: ( 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 @ X3 @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( Y
= ( ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
& ! [H8: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H @ H8 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As ) )
& ( X3 @ ( 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_7085_wand__raw_Oelims_I2_J,axiom,
! [X3: ( 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 @ X3 @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
& ! [H9: heap_ext @ product_unit,As9: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As9 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H @ H9 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As ) )
& ( X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As9 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( sup_sup @ ( set @ nat ) @ As @ As9 ) ) ) ) ) ) ) ).
% wand_raw.elims(2)
thf(fact_7086_wand__raw_Opelims_I3_J,axiom,
! [X3: ( 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 @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ 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 ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
& ! [H5: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H @ H5 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) )
& ( X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As7 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As @ As7 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
thf(fact_7087_wand__raw_Opelims_I2_J,axiom,
! [X3: ( 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 @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ 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 ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
& ! [H9: heap_ext @ product_unit,As9: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As9 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H @ H9 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As ) )
& ( X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As9 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( sup_sup @ ( set @ nat ) @ As @ As9 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
thf(fact_7088_wand__raw_Opelims_I1_J,axiom,
! [X3: ( 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 @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ( Y
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
& ! [H8: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H @ H8 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As ) )
& ( X3 @ ( 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 ) ) ) @ X3 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(1)
thf(fact_7089_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_7090_Range__Union,axiom,
! [A: $tType,B: $tType,S: set @ ( set @ ( product_prod @ B @ A ) )] :
( ( range2 @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ B @ A ) ) @ ( set @ A ) @ ( range2 @ B @ A ) @ S ) ) ) ).
% Range_Union
thf(fact_7091_Range__Id__on,axiom,
! [A: $tType,A3: set @ A] :
( ( range2 @ A @ A @ ( id_on @ A @ A3 ) )
= A3 ) ).
% Range_Id_on
thf(fact_7092_Range__empty,axiom,
! [B: $tType,A: $tType] :
( ( range2 @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Range_empty
thf(fact_7093_Range__Id,axiom,
! [A: $tType] :
( ( range2 @ A @ A @ ( id2 @ A ) )
= ( top_top @ ( set @ A ) ) ) ).
% Range_Id
thf(fact_7094_Range__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( range2 @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ A
@ ^ [Y5: A] :
? [X4: B] : ( P @ X4 @ Y5 ) ) ) ).
% Range_Collect_case_prod
thf(fact_7095_Range__insert,axiom,
! [A: $tType,B: $tType,A4: B,B3: A,R2: set @ ( product_prod @ B @ A )] :
( ( range2 @ B @ A @ ( insert3 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A4 @ B3 ) @ R2 ) )
= ( insert3 @ A @ B3 @ ( range2 @ B @ A @ R2 ) ) ) ).
% Range_insert
thf(fact_7096_Range__empty__iff,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( ( range2 @ B @ A @ R2 )
= ( bot_bot @ ( set @ A ) ) )
= ( R2
= ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% Range_empty_iff
thf(fact_7097_finite__Range,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
=> ( finite_finite2 @ B @ ( range2 @ A @ B @ R2 ) ) ) ).
% finite_Range
thf(fact_7098_Range__Un__eq,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ B @ A ),B2: set @ ( product_prod @ B @ A )] :
( ( range2 @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( range2 @ B @ A @ A3 ) @ ( range2 @ B @ A @ B2 ) ) ) ).
% Range_Un_eq
thf(fact_7099_snd__eq__Range,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ R )
= ( range2 @ B @ A @ R ) ) ).
% snd_eq_Range
thf(fact_7100_Range__snd,axiom,
! [A: $tType,B: $tType] :
( ( range2 @ B @ A )
= ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) ) ) ).
% Range_snd
thf(fact_7101_Range_Ocases,axiom,
! [B: $tType,A: $tType,A4: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A4 @ ( range2 @ A @ B @ R2 ) )
=> ~ ! [A6: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ A4 ) @ R2 ) ) ).
% Range.cases
thf(fact_7102_Range_Osimps,axiom,
! [B: $tType,A: $tType,A4: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A4 @ ( range2 @ A @ B @ R2 ) )
= ( ? [A5: A,B4: B] :
( ( A4 = B4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_7103_Range_Ointros,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( member @ B @ B3 @ ( range2 @ A @ B @ R2 ) ) ) ).
% Range.intros
thf(fact_7104_RangeE,axiom,
! [A: $tType,B: $tType,B3: A,R2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B3 @ ( range2 @ B @ A @ R2 ) )
=> ~ ! [A6: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A6 @ B3 ) @ R2 ) ) ).
% RangeE
thf(fact_7105_Range__iff,axiom,
! [A: $tType,B: $tType,A4: A,R2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A4 @ ( range2 @ B @ A @ R2 ) )
= ( ? [Y5: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y5 @ A4 ) @ R2 ) ) ) ).
% Range_iff
thf(fact_7106_Range__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ B ) @ ( range2 @ A @ B @ R2 ) @ ( range2 @ A @ B @ S3 ) ) ) ).
% Range_mono
thf(fact_7107_Range__Int__subset,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ B @ A ),B2: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( range2 @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ A3 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( range2 @ B @ A @ A3 ) @ ( range2 @ B @ A @ B2 ) ) ) ).
% Range_Int_subset
thf(fact_7108_trancl__Image__in__Range,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),V: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ R ) @ V ) @ ( range2 @ A @ A @ R ) ) ).
% trancl_Image_in_Range
thf(fact_7109_Range__Diff__subset,axiom,
! [A: $tType,B: $tType,A3: set @ ( product_prod @ B @ A ),B2: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( range2 @ B @ A @ A3 ) @ ( range2 @ B @ A @ B2 ) ) @ ( range2 @ B @ A @ ( minus_minus @ ( set @ ( product_prod @ B @ A ) ) @ A3 @ B2 ) ) ) ).
% Range_Diff_subset
thf(fact_7110_Range__rel__restrict,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( range2 @ A @ A @ ( rel_restrict @ A @ R @ A3 ) ) @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R ) @ A3 ) ) ).
% Range_rel_restrict
thf(fact_7111_finite__reachable__restrictedI,axiom,
! [A: $tType,Q: set @ A,I3: set @ A,E5: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ Q )
=> ( ( ord_less_eq @ ( set @ A ) @ I3 @ Q )
=> ( ( ord_less_eq @ ( set @ A ) @ ( range2 @ A @ A @ E5 ) @ Q )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E5 ) @ I3 ) ) ) ) ) ).
% finite_reachable_restrictedI
thf(fact_7112_wf__UN,axiom,
! [B: $tType,A: $tType,I3: set @ A,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [I5: A] :
( ( member @ A @ I5 @ I3 )
=> ( wf @ B @ ( R2 @ I5 ) ) )
=> ( ! [I5: A,J3: A] :
( ( member @ A @ I5 @ I3 )
=> ( ( member @ A @ J3 @ I3 )
=> ( ( ( R2 @ I5 )
!= ( R2 @ J3 ) )
=> ( ( inf_inf @ ( set @ B ) @ ( domain @ B @ B @ ( R2 @ I5 ) ) @ ( range2 @ B @ B @ ( R2 @ J3 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( wf @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ I3 ) ) ) ) ) ).
% wf_UN
thf(fact_7113_dom__ran__disj__comp,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R ) @ ( range2 @ A @ A @ R ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( relcomp @ A @ A @ A @ R @ R )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% dom_ran_disj_comp
thf(fact_7114_Domain__Id__on,axiom,
! [A: $tType,A3: set @ A] :
( ( domain @ A @ A @ ( id_on @ A @ A3 ) )
= A3 ) ).
% Domain_Id_on
thf(fact_7115_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_7116_Domain__Id,axiom,
! [A: $tType] :
( ( domain @ A @ A @ ( id2 @ A ) )
= ( top_top @ ( set @ A ) ) ) ).
% Domain_Id
thf(fact_7117_Range__converse,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( range2 @ B @ A @ ( converse @ A @ B @ R2 ) )
= ( domain @ A @ B @ R2 ) ) ).
% Range_converse
thf(fact_7118_Domain__converse,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( domain @ A @ B @ ( converse @ B @ A @ R2 ) )
= ( range2 @ B @ A @ R2 ) ) ).
% Domain_converse
thf(fact_7119_Domain__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( domain @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
= ( collect @ A
@ ^ [X4: A] :
? [X9: B] : ( P @ X4 @ X9 ) ) ) ).
% Domain_Collect_case_prod
thf(fact_7120_Domain__insert,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert3 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 ) )
= ( insert3 @ A @ A4 @ ( domain @ A @ B @ R2 ) ) ) ).
% Domain_insert
thf(fact_7121_Domain__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 )
=> ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R2 ) @ ( domain @ A @ B @ S3 ) ) ) ).
% Domain_mono
thf(fact_7122_Domain__unfold,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ^ [X4: A] :
? [Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ).
% Domain_unfold
thf(fact_7123_Domain__iff,axiom,
! [A: $tType,B: $tType,A4: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R2 ) )
= ( ? [Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ Y5 ) @ R2 ) ) ) ).
% Domain_iff
thf(fact_7124_DomainE,axiom,
! [B: $tType,A: $tType,A4: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B5 ) @ R2 ) ) ).
% DomainE
thf(fact_7125_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A4: A,B3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
=> ( member @ A @ A4 @ ( domain @ A @ B @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_7126_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A4: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R2 ) )
= ( ? [A5: A,B4: B] :
( ( A4 = A5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_7127_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A4: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A4 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B5 ) @ R2 ) ) ).
% Domain.cases
thf(fact_7128_Not__Domain__rtrancl,axiom,
! [A: $tType,X3: A,R: set @ ( product_prod @ A @ A ),Y: A] :
( ~ ( member @ A @ X3 @ ( domain @ A @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ ( transitive_rtrancl @ A @ R ) )
= ( X3 = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_7129_Domain__fst,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) ).
% Domain_fst
thf(fact_7130_fst__eq__Domain,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ R )
= ( domain @ A @ B @ R ) ) ).
% fst_eq_Domain
thf(fact_7131_Domain__Un__eq,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B ),B2: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ ( domain @ A @ B @ A3 ) @ ( domain @ A @ B @ B2 ) ) ) ).
% Domain_Un_eq
thf(fact_7132_finite__Domain,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
=> ( finite_finite2 @ A @ ( domain @ A @ B @ R2 ) ) ) ).
% finite_Domain
thf(fact_7133_Domain__empty__iff,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( ( domain @ A @ B @ R2 )
= ( bot_bot @ ( set @ A ) ) )
= ( R2
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% Domain_empty_iff
thf(fact_7134_Domain__Int__subset,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B ),B2: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ B2 ) ) @ ( inf_inf @ ( set @ A ) @ ( domain @ A @ B @ A3 ) @ ( domain @ A @ B @ B2 ) ) ) ).
% Domain_Int_subset
thf(fact_7135_for__in__RI,axiom,
! [B: $tType,A: $tType,X3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X3 @ ( domain @ A @ B @ R ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ ( fun_of_rel @ A @ B @ R @ X3 ) ) @ R ) ) ).
% for_in_RI
thf(fact_7136_Field__def,axiom,
! [A: $tType] :
( ( field2 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range2 @ A @ A @ R5 ) ) ) ) ).
% Field_def
thf(fact_7137_Domain__Diff__subset,axiom,
! [B: $tType,A: $tType,A3: set @ ( product_prod @ A @ B ),B2: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ B @ A3 ) @ ( domain @ A @ B @ B2 ) ) @ ( domain @ A @ B @ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ B2 ) ) ) ).
% Domain_Diff_subset
thf(fact_7138_Domain__rel__restrict,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ A @ ( rel_restrict @ A @ R @ A3 ) ) @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R ) @ A3 ) ) ).
% Domain_rel_restrict
thf(fact_7139_wf__no__path,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R ) @ ( range2 @ A @ A @ R ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ R ) ) ).
% wf_no_path
thf(fact_7140_wf__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( R
!= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ! [M6: A] :
~ ( member @ A @ M6 @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R ) @ ( range2 @ A @ A @ R ) ) ) ) ) ).
% wf_min
thf(fact_7141_Domain__Union,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( domain @ A @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ A ) @ ( domain @ A @ B ) @ S ) ) ) ).
% Domain_Union
thf(fact_7142_wf__Un,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( ( wf @ A @ S3 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R2 ) @ ( range2 @ A @ A @ S3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ) ).
% wf_Un
thf(fact_7143_wf__Union,axiom,
! [A: $tType,R: set @ ( set @ ( product_prod @ A @ A ) )] :
( ! [X: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X @ R )
=> ( wf @ A @ X ) )
=> ( ! [X: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X @ R )
=> ! [Xa4: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa4 @ R )
=> ( ( X != Xa4 )
=> ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ X ) @ ( range2 @ A @ A @ Xa4 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( wf @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R ) ) ) ) ).
% wf_Union
thf(fact_7144_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 ) ) ) )
=> ~ ! [M6: A] :
~ ( member @ A @ M6 @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R ) @ ( domain @ A @ A @ R ) ) ) ) ) ).
% wf_max
thf(fact_7145_Range__def,axiom,
! [B: $tType,A: $tType] :
( ( range2 @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ B
@ ( rangep @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% Range_def
thf(fact_7146_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( rangep @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R2 ) )
= ( ^ [X4: B] : ( member @ B @ X4 @ ( range2 @ A @ B @ R2 ) ) ) ) ).
% Rangep_Range_eq
thf(fact_7147_RangepE,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,B3: B] :
( ( rangep @ A @ B @ R2 @ B3 )
=> ~ ! [A6: A] :
~ ( R2 @ A6 @ B3 ) ) ).
% RangepE
thf(fact_7148_RangePI,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A4: A,B3: B] :
( ( R2 @ A4 @ B3 )
=> ( rangep @ A @ B @ R2 @ B3 ) ) ).
% RangePI
thf(fact_7149_Rangep_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rangep @ A @ B )
= ( ^ [R5: A > B > $o,A5: B] :
? [B4: A,C6: B] :
( ( A5 = C6 )
& ( R5 @ B4 @ C6 ) ) ) ) ).
% Rangep.simps
thf(fact_7150_Rangep_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A4: B] :
( ( rangep @ A @ B @ R2 @ A4 )
=> ~ ! [A6: A] :
~ ( R2 @ A6 @ A4 ) ) ).
% Rangep.cases
thf(fact_7151_normalize__stable,axiom,
! [Q4: int,P4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q4 )
=> ( ( algebr8660921524188924756oprime @ int @ P4 @ Q4 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q4 ) )
= ( product_Pair @ int @ int @ P4 @ Q4 ) ) ) ) ).
% normalize_stable
thf(fact_7152_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: set @ A,B2: set @ A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq @ Less )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F3 @ B2 ) @ ( lattic1715443433743089157tice_F @ A @ F3 @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_7153_coprime__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( ( algebr8660921524188924756oprime @ A @ A4 @ C2 )
& ( algebr8660921524188924756oprime @ A @ B3 @ C2 ) ) ) ) ).
% coprime_mult_left_iff
thf(fact_7154_coprime__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ ( times_times @ A @ A4 @ B3 ) )
= ( ( algebr8660921524188924756oprime @ A @ C2 @ A4 )
& ( algebr8660921524188924756oprime @ A @ C2 @ B3 ) ) ) ) ).
% coprime_mult_right_iff
thf(fact_7155_coprime__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ A4 )
= ( dvd_dvd @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% coprime_self
thf(fact_7156_coprime__0__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] :
( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A4 )
= ( dvd_dvd @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% coprime_0_left_iff
thf(fact_7157_coprime__0__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ ( zero_zero @ A ) )
= ( dvd_dvd @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% coprime_0_right_iff
thf(fact_7158_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_7159_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_7160_coprime__diff__one__left,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ A4 ) ) ).
% coprime_diff_one_left
thf(fact_7161_coprime__doff__one__right,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ A4 @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% coprime_doff_one_right
thf(fact_7162_normalize__coprime,axiom,
! [R2: product_prod @ int @ int,P4: int,Q4: int] :
( ( ( normalize @ R2 )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( algebr8660921524188924756oprime @ int @ P4 @ Q4 ) ) ).
% normalize_coprime
thf(fact_7163_coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ! [C5: A] :
( ( dvd_dvd @ A @ C5 @ A4 )
=> ( ( dvd_dvd @ A @ C5 @ B3 )
=> ( dvd_dvd @ A @ C5 @ ( one_one @ A ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ).
% coprimeI
thf(fact_7164_coprime__def,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A5: A,B4: A] :
! [C6: A] :
( ( dvd_dvd @ A @ C6 @ A5 )
=> ( ( dvd_dvd @ A @ C6 @ B4 )
=> ( dvd_dvd @ A @ C6 @ ( one_one @ A ) ) ) ) ) ) ) ).
% coprime_def
thf(fact_7165_not__coprimeE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ~ ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ~ ! [C5: A] :
( ( dvd_dvd @ A @ C5 @ A4 )
=> ( ( dvd_dvd @ A @ C5 @ B3 )
=> ( dvd_dvd @ A @ C5 @ ( one_one @ A ) ) ) ) ) ) ).
% not_coprimeE
thf(fact_7166_not__coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( dvd_dvd @ A @ C2 @ A4 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( ~ ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ~ ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ) ).
% not_coprimeI
thf(fact_7167_coprime__absorb__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X3: A,Y: A] :
( ( dvd_dvd @ A @ X3 @ Y )
=> ( ( algebr8660921524188924756oprime @ A @ X3 @ Y )
= ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_left
thf(fact_7168_coprime__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,D3: A,A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
=> ( ! [E2: A] :
( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E2 @ A4 )
=> ( ( dvd_dvd @ A @ E2 @ B3 )
=> ( dvd_dvd @ A @ E2 @ C2 ) ) ) )
=> ( ! [E2: A] :
( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E2 @ A4 )
=> ( ( dvd_dvd @ A @ E2 @ B3 )
=> ( dvd_dvd @ A @ E2 @ D3 ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ) ).
% coprime_imp_coprime
thf(fact_7169_coprime__absorb__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [Y: A,X3: A] :
( ( dvd_dvd @ A @ Y @ X3 )
=> ( ( algebr8660921524188924756oprime @ A @ X3 @ Y )
= ( dvd_dvd @ A @ Y @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_right
thf(fact_7170_coprime__common__divisor,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ C2 @ A4 )
=> ( ( dvd_dvd @ A @ C2 @ B3 )
=> ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% coprime_common_divisor
thf(fact_7171_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ).
% is_unit_left_imp_coprime
thf(fact_7172_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ).
% is_unit_right_imp_coprime
thf(fact_7173_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [A4: A,N: A,M2: A,B3: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ N ) @ M2 )
= ( modulo_modulo @ A @ ( times_times @ A @ B3 @ N ) @ M2 ) )
=> ( ( algebr8660921524188924756oprime @ A @ M2 @ N )
=> ( ( modulo_modulo @ A @ A4 @ M2 )
= ( modulo_modulo @ A @ B3 @ M2 ) ) ) ) ) ).
% mult_mod_cancel_right
thf(fact_7174_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [N: A,A4: A,M2: A,B3: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ N @ A4 ) @ M2 )
= ( modulo_modulo @ A @ ( times_times @ A @ N @ B3 ) @ M2 ) )
=> ( ( algebr8660921524188924756oprime @ A @ M2 @ N )
=> ( ( modulo_modulo @ A @ A4 @ M2 )
= ( modulo_modulo @ A @ B3 @ M2 ) ) ) ) ) ).
% mult_mod_cancel_left
thf(fact_7175_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ A4 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_7176_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A4 ) ) ).
% coprime_1_left
thf(fact_7177_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C2 )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% coprime_dvd_mult_right_iff
thf(fact_7178_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C2 )
=> ( ( dvd_dvd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ) ).
% coprime_dvd_mult_left_iff
thf(fact_7179_divides__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ C2 )
=> ( ( dvd_dvd @ A @ B3 @ C2 )
=> ( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ) ) ).
% divides_mult
thf(fact_7180_coprime__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [B4: A,A5: A] : ( algebr8660921524188924756oprime @ A @ A5 @ B4 ) ) ) ) ).
% coprime_commute
thf(fact_7181_coprime__divisors,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: A,C2: A,B3: A,D3: A] :
( ( dvd_dvd @ A @ A4 @ C2 )
=> ( ( dvd_dvd @ A @ B3 @ D3 )
=> ( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ) ).
% coprime_divisors
thf(fact_7182_prod__list__coprime__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A,A4: A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ X ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ ( groups5270119922927024881d_list @ A @ Xs ) ) ) ) ).
% prod_list_coprime_right
thf(fact_7183_prod__list__coprime__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A,A4: A] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( algebr8660921524188924756oprime @ A @ X @ A4 ) )
=> ( algebr8660921524188924756oprime @ A @ ( groups5270119922927024881d_list @ A @ Xs ) @ A4 ) ) ) ).
% prod_list_coprime_left
thf(fact_7184_quotient__of__coprime,axiom,
! [R2: rat,P4: int,Q4: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( algebr8660921524188924756oprime @ int @ P4 @ Q4 ) ) ).
% quotient_of_coprime
thf(fact_7185_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( algebr8660921524188924756oprime @ A @ B3 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A4 ) @ B3 )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_left_left_cancel
thf(fact_7186_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B3: A,C2: A,A4: A] :
( ( algebr8660921524188924756oprime @ A @ B3 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ A4 @ C2 ) @ B3 )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_left_right_cancel
thf(fact_7187_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C2 )
=> ( ( gcd_gcd @ A @ A4 @ ( times_times @ A @ C2 @ B3 ) )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_right_left_cancel
thf(fact_7188_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ C2 )
=> ( ( gcd_gcd @ A @ A4 @ ( times_times @ A @ B3 @ C2 ) )
= ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ).
% gcd_mult_right_right_cancel
thf(fact_7189_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,A9: A,B8: A] :
( ( ( gcd_gcd @ A @ A4 @ B3 )
!= ( zero_zero @ A ) )
=> ( ( A4
= ( times_times @ A @ A9 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
=> ( ( B3
= ( times_times @ A @ B8 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A9 @ B8 ) ) ) ) ) ).
% gcd_coprime
thf(fact_7190_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( ( gcd_gcd @ A @ A4 @ B3 )
!= ( zero_zero @ A ) )
=> ? [A11: A,B12: A] :
( ( A4
= ( times_times @ A @ A11 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
& ( B3
= ( times_times @ A @ B12 @ ( gcd_gcd @ A @ A4 @ B3 ) ) )
& ( algebr8660921524188924756oprime @ A @ A11 @ B12 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_7191_invertible__coprime,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ C2 ) ) ) ).
% invertible_coprime
thf(fact_7192_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: set @ A,X3: A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( Less_eq @ X3 @ X4 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_7193_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: set @ A,X3: A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A3 )
=> ( Less_eq @ X3 @ A6 ) )
=> ( Less_eq @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A3 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_7194_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: set @ A,X3: A] :
( ( lattic4895041142388067077er_set @ A @ F3 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A3 ) )
=> ! [A15: A] :
( ( member @ A @ A15 @ A3 )
=> ( Less_eq @ X3 @ A15 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_7195_semilattice__set_Oremove,axiom,
! [A: $tType,F3: A > A > A,A3: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ A3 )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ A3 )
= ( F3 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_7196_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F3: A > A > A,A3: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert3 @ A @ X3 @ A3 ) )
= X3 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert3 @ A @ X3 @ A3 ) )
= ( F3 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_7197_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_7198_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_7199_semilattice__set_Osingleton,axiom,
! [A: $tType,F3: A > A > A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ) ).
% semilattice_set.singleton
thf(fact_7200_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F3: A > A > A,H2: A > A,N7: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ! [X: A,Y2: A] :
( ( H2 @ ( F3 @ X @ Y2 ) )
= ( F3 @ ( H2 @ X ) @ ( H2 @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic1715443433743089157tice_F @ A @ F3 @ N7 ) )
= ( lattic1715443433743089157tice_F @ A @ F3 @ ( image2 @ A @ A @ H2 @ N7 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_7201_semilattice__set_Osubset,axiom,
! [A: $tType,F3: A > A > A,A3: set @ A,B2: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( F3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ B2 ) @ ( lattic1715443433743089157tice_F @ A @ F3 @ A3 ) )
= ( lattic1715443433743089157tice_F @ A @ F3 @ A3 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_7202_semilattice__set_Oclosed,axiom,
! [A: $tType,F3: A > A > A,A3: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A,Y2: A] : ( member @ A @ ( F3 @ X @ Y2 ) @ ( insert3 @ A @ X @ ( insert3 @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic1715443433743089157tice_F @ A @ F3 @ A3 ) @ A3 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_7203_semilattice__set_Oinsert,axiom,
! [A: $tType,F3: A > A > A,A3: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert3 @ A @ X3 @ A3 ) )
= ( F3 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A3 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_7204_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F3: A > A > A,A3: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X3 @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert3 @ A @ X3 @ A3 ) )
= ( F3 @ X3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A3 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_7205_semilattice__set_Ounion,axiom,
! [A: $tType,F3: A > A > A,A3: set @ A,B2: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B2 )
=> ( ( B2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( F3 @ ( lattic1715443433743089157tice_F @ A @ F3 @ A3 ) @ ( lattic1715443433743089157tice_F @ A @ F3 @ B2 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_7206_semilattice__set_Oeq__fold,axiom,
! [A: $tType,F3: A > A > A,A3: set @ A,X3: A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( insert3 @ A @ X3 @ A3 ) )
= ( finite_fold @ A @ A @ F3 @ X3 @ A3 ) ) ) ) ).
% semilattice_set.eq_fold
thf(fact_7207_semilattice__set_Oset__eq__fold,axiom,
! [A: $tType,F3: A > A > A,X3: A,Xs: list @ A] :
( ( lattic149705377957585745ce_set @ A @ F3 )
=> ( ( lattic1715443433743089157tice_F @ A @ F3 @ ( set2 @ A @ ( cons @ A @ X3 @ Xs ) ) )
= ( fold @ A @ A @ F3 @ Xs @ X3 ) ) ) ).
% semilattice_set.set_eq_fold
thf(fact_7208_pred__on_Onot__maxchain__Some,axiom,
! [A: $tType,A3: set @ A,P: A > A > $o,C3: set @ A] :
( ( pred_chain @ A @ A3 @ P @ C3 )
=> ( ~ ( pred_maxchain @ A @ A3 @ P @ C3 )
=> ( ( pred_chain @ A @ A3 @ P
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A3 @ P @ D7 )
& ( ord_less @ ( set @ A ) @ C3 @ D7 ) ) ) )
& ( ord_less @ ( set @ A ) @ C3
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A3 @ P @ D7 )
& ( ord_less @ ( set @ A ) @ C3 @ D7 ) ) ) ) ) ) ) ).
% pred_on.not_maxchain_Some
thf(fact_7209_antisymp__antisym__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( antisymp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R2 ) )
= ( antisym @ A @ R2 ) ) ).
% antisymp_antisym_eq
thf(fact_7210_antisymp__less__eq,axiom,
! [A: $tType,R2: A > A > $o,S3: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R2 @ S3 )
=> ( ( antisymp @ A @ S3 )
=> ( antisymp @ A @ R2 ) ) ) ).
% antisymp_less_eq
thf(fact_7211_antisympD,axiom,
! [A: $tType,R2: A > A > $o,A4: A,B3: A] :
( ( antisymp @ A @ R2 )
=> ( ( R2 @ A4 @ B3 )
=> ( ( R2 @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ).
% antisympD
thf(fact_7212_antisympI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X: A,Y2: A] :
( ( R2 @ X @ Y2 )
=> ( ( R2 @ Y2 @ X )
=> ( X = Y2 ) ) )
=> ( antisymp @ A @ R2 ) ) ).
% antisympI
thf(fact_7213_antisymp__def,axiom,
! [A: $tType] :
( ( antisymp @ A )
= ( ^ [R5: A > A > $o] :
! [X4: A,Y5: A] :
( ( R5 @ X4 @ Y5 )
=> ( ( R5 @ Y5 @ X4 )
=> ( X4 = Y5 ) ) ) ) ) ).
% antisymp_def
thf(fact_7214_antisymp__equality,axiom,
! [A: $tType] :
( antisymp @ A
@ ^ [Y4: A,Z4: A] : Y4 = Z4 ) ).
% antisymp_equality
thf(fact_7215_subset_OHausdorff,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
? [X_1: set @ ( set @ A )] : ( pred_maxchain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X_1 ) ).
% subset.Hausdorff
thf(fact_7216_antisym__bot,axiom,
! [A: $tType] : ( antisymp @ A @ ( bot_bot @ ( A > A > $o ) ) ) ).
% antisym_bot
thf(fact_7217_subset_Omaxchain__def,axiom,
! [A: $tType,A3: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
= ( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
& ~ ? [S2: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ S2 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C3 @ S2 ) ) ) ) ).
% subset.maxchain_def
thf(fact_7218_subset_Omaxchain__imp__chain,axiom,
! [A: $tType,A3: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 ) ) ).
% subset.maxchain_imp_chain
thf(fact_7219_subset_Onot__maxchain__Some,axiom,
! [A: $tType,A3: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ~ ( pred_maxchain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) )
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C3 @ D7 ) ) ) )
& ( ord_less @ ( set @ ( set @ A ) ) @ C3
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C3 @ D7 ) ) ) ) ) ) ) ).
% subset.not_maxchain_Some
thf(fact_7220_pred__on_Omaxchain__def,axiom,
! [A: $tType] :
( ( pred_maxchain @ A )
= ( ^ [A7: set @ A,P2: A > A > $o,C4: set @ A] :
( ( pred_chain @ A @ A7 @ P2 @ C4 )
& ~ ? [S2: set @ A] :
( ( pred_chain @ A @ A7 @ P2 @ S2 )
& ( ord_less @ ( set @ A ) @ C4 @ S2 ) ) ) ) ) ).
% pred_on.maxchain_def
thf(fact_7221_subset__maxchain__max,axiom,
! [A: $tType,A3: set @ ( set @ A ),C3: set @ ( set @ A ),X5: set @ A] :
( ( pred_maxchain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( member @ ( set @ A ) @ X5 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) @ X5 )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ C3 )
= X5 ) ) ) ) ).
% subset_maxchain_max
thf(fact_7222_pred__on_Osuc__def,axiom,
! [A: $tType] :
( ( pred_suc @ A )
= ( ^ [A7: set @ A,P2: A > A > $o,C4: set @ A] :
( if @ ( set @ A )
@ ( ~ ( pred_chain @ A @ A7 @ P2 @ C4 )
| ( pred_maxchain @ A @ A7 @ P2 @ C4 ) )
@ C4
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A7 @ P2 @ D7 )
& ( ord_less @ ( set @ A ) @ C4 @ D7 ) ) ) ) ) ) ).
% pred_on.suc_def
thf(fact_7223_Quotient__multiset,axiom,
! [A: $tType] :
( quotient @ ( A > nat ) @ ( multiset @ A )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
@ ( abs_multiset @ A )
@ ( count @ A )
@ ( cr_multiset @ A ) ) ).
% Quotient_multiset
thf(fact_7224_subset_Onot__chain__suc,axiom,
! [A: $tType,A3: set @ ( set @ A ),X5: set @ ( set @ A )] :
( ~ ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 )
=> ( ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 )
= X5 ) ) ).
% subset.not_chain_suc
thf(fact_7225_subset_Omaxchain__suc,axiom,
! [A: $tType,A3: set @ ( set @ A ),X5: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 )
=> ( ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 )
= X5 ) ) ).
% subset.maxchain_suc
thf(fact_7226_subset_Osuc__def,axiom,
! [A: $tType,A3: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( ( ~ ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
| ( pred_maxchain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 ) )
=> ( ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
= C3 ) )
& ( ~ ( ~ ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
| ( pred_maxchain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 ) )
=> ( ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
= ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C3 @ D7 ) ) ) ) ) ) ).
% subset.suc_def
thf(fact_7227_subset_Osuc__not__equals,axiom,
! [A: $tType,A3: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ~ ( pred_maxchain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ C3 )
!= C3 ) ) ) ).
% subset.suc_not_equals
thf(fact_7228_Quotient__filter,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T4: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T4 )
=> ( quotient @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ A @ R ) @ ( filtermap @ A @ B @ Abs ) @ ( filtermap @ B @ A @ Rep ) @ ( rel_filter @ A @ B @ T4 ) ) ) ).
% Quotient_filter
thf(fact_7229_subset_Ochain__suc,axiom,
! [A: $tType,A3: set @ ( set @ A ),X5: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 )
=> ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 ) ) ) ).
% subset.chain_suc
thf(fact_7230_list_OQuotient,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T4: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T4 )
=> ( quotient @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ A @ R ) @ ( map @ A @ B @ Abs ) @ ( map @ B @ A @ Rep ) @ ( list_all2 @ A @ B @ T4 ) ) ) ).
% list.Quotient
thf(fact_7231_subset_Osuc__in__carrier,axiom,
! [A: $tType,X5: set @ ( set @ A ),A3: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 ) @ A3 ) ) ).
% subset.suc_in_carrier
thf(fact_7232_subset_Osuc__subset,axiom,
! [A: $tType,X5: set @ ( set @ A ),A3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ X5 @ ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 ) ) ).
% subset.suc_subset
thf(fact_7233_subset_Osubset__suc,axiom,
! [A: $tType,X5: set @ ( set @ A ),Y6: set @ ( set @ A ),A3: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X5 @ Y6 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ X5 @ ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ Y6 ) ) ) ).
% subset.subset_suc
thf(fact_7234_subset_Ochain__sucD,axiom,
! [A: $tType,A3: set @ ( set @ A ),X5: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 ) @ A3 )
& ( pred_chain @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 ) ) ) ) ).
% subset.chain_sucD
thf(fact_7235_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,A3: set @ B,F3: B > A] :
( ( bind3 @ B @ A @ A3
@ ^ [X4: B] : ( insert3 @ A @ ( F3 @ X4 ) @ ( bot_bot @ ( set @ A ) ) ) )
= ( image2 @ B @ A @ F3 @ A3 ) ) ).
% bind_singleton_conv_image
thf(fact_7236_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_7237_empty__bind,axiom,
! [B: $tType,A: $tType,F3: B > ( set @ A )] :
( ( bind3 @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_bind
thf(fact_7238_Set_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind3 @ A @ B )
= ( ^ [A7: set @ A,F7: A > ( set @ B )] :
( collect @ B
@ ^ [X4: B] :
? [Y5: set @ B] :
( ( member @ ( set @ B ) @ Y5 @ ( image2 @ A @ ( set @ B ) @ F7 @ A7 ) )
& ( member @ B @ X4 @ Y5 ) ) ) ) ) ).
% Set.bind_def
thf(fact_7239_Set_Obind__bind,axiom,
! [C: $tType,B: $tType,A: $tType,A3: set @ A,B2: A > ( set @ C ),C3: C > ( set @ B )] :
( ( bind3 @ C @ B @ ( bind3 @ A @ C @ A3 @ B2 ) @ C3 )
= ( bind3 @ A @ B @ A3
@ ^ [X4: A] : ( bind3 @ C @ B @ ( B2 @ X4 ) @ C3 ) ) ) ).
% Set.bind_bind
thf(fact_7240_nonempty__bind__const,axiom,
! [A: $tType,B: $tType,A3: set @ A,B2: set @ B] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bind3 @ A @ B @ A3
@ ^ [Uu3: A] : B2 )
= B2 ) ) ).
% nonempty_bind_const
thf(fact_7241_bind__const,axiom,
! [B: $tType,A: $tType,A3: set @ B,B2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind3 @ B @ A @ A3
@ ^ [Uu3: B] : B2 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind3 @ B @ A @ A3
@ ^ [Uu3: B] : B2 )
= B2 ) ) ) ).
% bind_const
thf(fact_7242_Diff__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( set @ A ) > ( set @ A ) ) @ ( ( set @ B ) > ( set @ B ) ) @ ( bNF_rel_set @ A @ B @ A3 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A3 ) @ ( bNF_rel_set @ A @ B @ A3 ) ) @ ( minus_minus @ ( set @ A ) ) @ ( minus_minus @ ( set @ B ) ) ) ) ).
% Diff_transfer
thf(fact_7243_bij__betw__nth,axiom,
! [A: $tType,Xs: list @ A,A3: set @ nat,B2: set @ A] :
( ( distinct @ A @ Xs )
=> ( ( A3
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( ( B2
= ( set2 @ A @ Xs ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs ) @ A3 @ B2 ) ) ) ) ).
% bij_betw_nth
thf(fact_7244_iso__iff2,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_iso @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),R10: set @ ( product_prod @ B @ B ),F7: A > B] :
( ( bij_betw @ A @ B @ F7 @ ( field2 @ A @ R5 ) @ ( field2 @ B @ R10 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R5 ) )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ ( field2 @ A @ R5 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F7 @ X4 ) @ ( F7 @ Y5 ) ) @ R10 ) ) ) ) ) ) ) ).
% iso_iff2
thf(fact_7245_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B3: A,A3: set @ A,F3: A > B,A8: set @ B] :
( ~ ( member @ A @ B3 @ A3 )
=> ( ~ ( member @ B @ ( F3 @ B3 ) @ A8 )
=> ( ( bij_betw @ A @ B @ F3 @ A3 @ A8 )
=> ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A8 @ ( insert3 @ B @ ( F3 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_7246_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B3: A,A3: set @ A,F3: A > B,A8: set @ B] :
( ~ ( member @ A @ B3 @ A3 )
=> ( ~ ( member @ B @ ( F3 @ B3 ) @ A8 )
=> ( ( bij_betw @ A @ B @ F3 @ A3 @ A8 )
= ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A8 @ ( insert3 @ B @ ( F3 @ B3 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_7247_bi__unique__rel__filter,axiom,
! [B: $tType,A: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bi_unique @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) ) ) ).
% bi_unique_rel_filter
thf(fact_7248_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) @ A3 )
=> ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_7249_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ A] :
( ( bij_betw @ A @ B @ F3 @ A3 @ ( bot_bot @ ( set @ B ) ) )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_7250_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_7251_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ A,C3: set @ A,B2: set @ B,D2: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) @ ( sup_sup @ ( set @ B ) @ B2 @ D2 ) )
=> ( ( bij_betw @ A @ B @ F3 @ C3 @ D2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ C3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B2 @ D2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F3 @ A3 @ B2 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_7252_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ A,B2: set @ B,C3: set @ A,D2: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A3 @ B2 )
=> ( ( bij_betw @ A @ B @ F3 @ C3 @ D2 )
=> ( ( ( inf_inf @ ( set @ B ) @ B2 @ D2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) @ ( sup_sup @ ( set @ B ) @ B2 @ D2 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_7253_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ R2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ? [F5: B > A,B6: set @ B] : ( bij_betw @ B @ A @ F5 @ B6 @ A3 ) ) ).
% ex_bij_betw
thf(fact_7254_card__of__ordIsoI,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: set @ A,B2: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A3 @ B2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIsoI
thf(fact_7255_card__of__ordIso,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: set @ B] :
( ( ? [F7: A > B] : ( bij_betw @ A @ B @ F7 @ A3 @ B2 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A3 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B2 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIso
thf(fact_7256_Well__order__iso__copy,axiom,
! [B: $tType,A: $tType,A3: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > B,A8: set @ B] :
( ( order_well_order_on @ A @ A3 @ R2 )
=> ( ( bij_betw @ A @ B @ F3 @ A3 @ A8 )
=> ? [R13: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ B @ A8 @ R13 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R2 @ R13 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% Well_order_iso_copy
thf(fact_7257_strict__subset__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( set @ A ) > $o ) @ ( ( set @ B ) > $o ) @ ( bNF_rel_set @ A @ B @ A3 )
@ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ $o @ $o @ ( bNF_rel_set @ A @ B @ A3 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( ord_less @ ( set @ A ) )
@ ( ord_less @ ( set @ B ) ) ) ) ).
% strict_subset_transfer
thf(fact_7258_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ A,C3: set @ B,G: A > B,B2: set @ A,D2: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A3 @ C3 )
=> ( ( bij_betw @ A @ B @ G @ B2 @ D2 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C3 @ D2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ A3 ) @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup @ ( set @ A ) @ A3 @ B2 )
@ ( sup_sup @ ( set @ B ) @ C3 @ D2 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_7259_remdups__adj__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) @ ( remdups_adj @ A ) @ ( remdups_adj @ B ) ) ) ).
% remdups_adj_transfer
thf(fact_7260_remdups__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) @ ( remdups @ A ) @ ( remdups @ B ) ) ) ).
% remdups_transfer
thf(fact_7261_distinct__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A3 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4
@ ( distinct @ A )
@ ( distinct @ B ) ) ) ).
% distinct_transfer
thf(fact_7262_distinct__adj__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ A3 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4
@ ( distinct_adj @ A )
@ ( distinct_adj @ B ) ) ) ).
% distinct_adj_transfer
thf(fact_7263_infinite__imp__bij__betw2,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [H: A > A] : ( bij_betw @ A @ A @ H @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_7264_infinite__imp__bij__betw,axiom,
! [A: $tType,A3: set @ A,A4: A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [H: A > A] : ( bij_betw @ A @ A @ H @ A3 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_7265_le__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A3 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A3 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( ord_less_eq @ ( filter @ A ) )
@ ( ord_less_eq @ ( filter @ B ) ) ) ) ).
% le_filter_parametric
thf(fact_7266_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S4: set @ B,T6: set @ C,H2: B > C,S: set @ B,T4: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ ( H2 @ A6 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( G @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( G @ ( H2 @ X4 ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T4 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_7267_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S4: set @ B,T6: set @ C,H2: B > C,S: set @ B,T4: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ ( H2 @ A6 ) )
= ( one_one @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( G @ B5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( G @ ( H2 @ X4 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T4 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_7268_remove1__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ A3 @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) ) @ ( remove1 @ A ) @ ( remove1 @ B ) ) ) ).
% remove1_transfer
thf(fact_7269_less__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > $o ) @ ( ( filter @ B ) > $o ) @ ( rel_filter @ A @ B @ A3 )
@ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ $o @ $o @ ( rel_filter @ A @ B @ A3 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( ord_less @ ( filter @ A ) )
@ ( ord_less @ ( filter @ B ) ) ) ) ).
% less_filter_parametric
thf(fact_7270_removeAll__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ A3 @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) ) @ ( removeAll @ A ) @ ( removeAll @ B ) ) ) ).
% removeAll_transfer
thf(fact_7271_List_Oinsert__transfer,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ A @ B @ ( ( list @ A ) > ( list @ A ) ) @ ( ( list @ B ) > ( list @ B ) ) @ A3 @ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) ) @ ( insert @ A ) @ ( insert @ B ) ) ) ).
% List.insert_transfer
thf(fact_7272_Inf__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( ( bi_total @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( set @ ( filter @ A ) ) @ ( set @ ( filter @ B ) ) @ ( filter @ A ) @ ( filter @ B ) @ ( bNF_rel_set @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) ) @ ( rel_filter @ A @ B @ A3 ) @ ( complete_Inf_Inf @ ( filter @ A ) ) @ ( complete_Inf_Inf @ ( filter @ B ) ) ) ) ) ).
% Inf_filter_parametric
thf(fact_7273_inf__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( ( bi_total @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ A ) > ( filter @ A ) ) @ ( ( filter @ B ) > ( filter @ B ) ) @ ( rel_filter @ A @ B @ A3 ) @ ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) @ ( rel_filter @ A @ B @ A3 ) ) @ ( inf_inf @ ( filter @ A ) ) @ ( inf_inf @ ( filter @ B ) ) ) ) ) ).
% inf_filter_parametric
thf(fact_7274_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_7275_bi__total__rel__filter,axiom,
! [B: $tType,A: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( bi_total @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) ) ) ).
% bi_total_rel_filter
thf(fact_7276_top__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( rel_filter @ A @ B @ A3 @ ( top_top @ ( filter @ A ) ) @ ( top_top @ ( filter @ B ) ) ) ) ).
% top_filter_parametric
thf(fact_7277_fun__lub__parametric,axiom,
! [A: $tType,C: $tType,E: $tType,F6: $tType,D: $tType,B: $tType,A3: A > B > $o,B2: C > D > $o,C3: E > F6 > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( ( set @ A ) > C ) @ ( ( set @ B ) > D ) @ ( ( set @ ( E > A ) ) > E > C ) @ ( ( set @ ( F6 > B ) ) > F6 > D ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ C @ D @ ( bNF_rel_set @ A @ B @ A3 ) @ B2 ) @ ( bNF_rel_fun @ ( set @ ( E > A ) ) @ ( set @ ( F6 > B ) ) @ ( E > C ) @ ( F6 > D ) @ ( bNF_rel_set @ ( E > A ) @ ( F6 > B ) @ ( bNF_rel_fun @ E @ F6 @ A @ B @ C3 @ A3 ) ) @ ( bNF_rel_fun @ E @ F6 @ C @ D @ C3 @ B2 ) ) @ ( partial_fun_lub @ A @ C @ E ) @ ( partial_fun_lub @ B @ D @ F6 ) ) ) ) ).
% fun_lub_parametric
thf(fact_7278_monotone__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B2: C > D > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( 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 ) @ A3
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 ) )
@ ( bNF_rel_fun @ ( C > C > $o ) @ ( D > D > $o ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o )
@ ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B2
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B2
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A3 @ B2 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 ) )
@ ( comple7038119648293358887notone @ A @ C )
@ ( comple7038119648293358887notone @ B @ D ) ) ) ).
% monotone_parametric
thf(fact_7279_fun__ord__parametric,axiom,
! [C: $tType,D: $tType,A: $tType,B: $tType,F6: $tType,E: $tType,C3: A > B > $o,A3: C > E > $o,B2: D > F6 > $o] :
( ( bi_total @ A @ B @ C3 )
=> ( bNF_rel_fun @ ( C > D > $o ) @ ( E > F6 > $o ) @ ( ( A > C ) > ( A > D ) > $o ) @ ( ( B > E ) > ( B > F6 ) > $o )
@ ( bNF_rel_fun @ C @ E @ ( D > $o ) @ ( F6 > $o ) @ A3
@ ( bNF_rel_fun @ D @ F6 @ $o @ $o @ B2
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 ) )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > E ) @ ( ( A > D ) > $o ) @ ( ( B > F6 ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ E @ C3 @ A3 )
@ ( bNF_rel_fun @ ( A > D ) @ ( B > F6 ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ D @ F6 @ C3 @ B2 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 ) )
@ ( partial_fun_ord @ C @ D @ A )
@ ( partial_fun_ord @ E @ F6 @ B ) ) ) ).
% fun_ord_parametric
thf(fact_7280_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ X3 @ ( power_int @ A @ X3 @ M2 ) ) ) ) ) ).
% power_int_add_1'
thf(fact_7281_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W2: num,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_7282_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,W2: num,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X3 @ ( numeral_numeral @ A @ W2 ) ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X3 @ M2 ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_7283_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_7284_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int,B3: A] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ B3 ) )
= B3 ) ) ).
% power_int_minus_one_mult_self'
thf(fact_7285_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: num,N: num] :
( ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M2 ) ) @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N ) ) )
= ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_7286_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: num,N: num,B3: A] :
( ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ M2 ) ) @ ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ N ) ) @ B3 ) )
= ( times_times @ A @ ( power_int @ A @ X3 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) @ B3 ) ) ) ).
% power_int_add_numeral2
thf(fact_7287_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N7: int,A4: A] :
( ( ord_less @ int @ N @ N7 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ord_less @ A @ ( power_int @ A @ A4 @ N ) @ ( power_int @ A @ A4 @ N7 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_7288_zero__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,N: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X3 @ N ) ) ) ) ).
% zero_less_power_int
thf(fact_7289_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X3 @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X3 @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib
thf(fact_7290_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,N: int] :
( ( times_times @ A @ ( power_int @ A @ X3 @ N ) @ X3 )
= ( times_times @ A @ X3 @ ( power_int @ A @ X3 @ N ) ) ) ) ).
% power_int_commutes
thf(fact_7291_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N7: int,A4: A] :
( ( ord_less @ int @ N @ N7 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ A @ A4 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A4 @ N7 ) @ ( power_int @ A @ A4 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_7292_power__int__strict__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,N: int] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less @ A @ ( power_int @ A @ B3 @ N ) @ ( power_int @ A @ A4 @ N ) ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_7293_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A4 @ N ) ) ) ) ) ).
% one_less_power_int
thf(fact_7294_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: int,N: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M2 @ N )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M2 @ N ) )
= ( times_times @ A @ ( power_int @ A @ X3 @ M2 ) @ ( power_int @ A @ X3 @ N ) ) ) ) ) ).
% power_int_add
thf(fact_7295_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( division_ring @ A )
& ( one @ B )
& ( uminus @ B ) )
=> ! [X3: C,A4: A,N: int] :
( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X3 )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A4 ) @ N )
= ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A4 @ N ) ) ) ) ) ).
% power_int_minus_left_distrib
thf(fact_7296_power__int__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,N: int] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ B3 @ N ) @ ( power_int @ A @ A4 @ N ) ) ) ) ) ) ).
% power_int_antimono
thf(fact_7297_power__int__strict__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A4: A,B3: A,N: int] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( power_int @ A @ A4 @ N ) @ ( power_int @ A @ B3 @ N ) ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_7298_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X3 )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X3 @ M2 ) @ ( power_int @ A @ X3 @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_7299_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X3 )
=> ( ( ord_less @ A @ ( power_int @ A @ X3 @ M2 ) @ ( power_int @ A @ X3 @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_7300_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X3: A,N: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( N
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X3 @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) @ X3 )
= ( power_int @ A @ X3 @ N ) ) ) ) ).
% power_int_minus_mult
thf(fact_7301_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X3: A,M2: int] :
( ( ( X3
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X3 @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X3 @ M2 ) @ X3 ) ) ) ) ).
% power_int_add_1
thf(fact_7302_Fract_Otransfer,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
@ ^ [Y4: int,Z4: int] : Y4 = Z4
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
@ ^ [Y4: int,Z4: int] : Y4 = Z4
@ pcr_rat )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( B4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B4 ) )
@ fract ) ).
% Fract.transfer
thf(fact_7303_subset__mset_Ostrict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: ( multiset @ A ) > B] :
( ( strict_mono @ ( multiset @ A ) @ B @ ( subset_mset @ A ) @ F3 )
= ( ! [X4: multiset @ A,Y5: multiset @ A] :
( ( subset_mset @ A @ X4 @ Y5 )
=> ( ord_less @ B @ ( F3 @ X4 ) @ ( F3 @ Y5 ) ) ) ) ) ) ).
% subset_mset.strict_mono_def
thf(fact_7304_quotient__of__eq,axiom,
! [A4: int,B3: int,P4: int,Q4: int] :
( ( ( quotient_of @ ( fract @ A4 @ B3 ) )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ( fract @ P4 @ Q4 )
= ( fract @ A4 @ B3 ) ) ) ).
% quotient_of_eq
thf(fact_7305_order_Ostrict__mono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( strict_mono @ A @ B )
= ( strict_mono @ A @ B ) ) ) ).
% order.strict_mono.cong
thf(fact_7306_normalize__eq,axiom,
! [A4: int,B3: int,P4: int,Q4: int] :
( ( ( normalize @ ( product_Pair @ int @ int @ A4 @ B3 ) )
= ( product_Pair @ int @ int @ P4 @ Q4 ) )
=> ( ( fract @ P4 @ Q4 )
= ( fract @ A4 @ B3 ) ) ) ).
% normalize_eq
thf(fact_7307_quotient__of__Fract,axiom,
! [A4: int,B3: int] :
( ( quotient_of @ ( fract @ A4 @ B3 ) )
= ( normalize @ ( product_Pair @ int @ int @ A4 @ B3 ) ) ) ).
% quotient_of_Fract
thf(fact_7308_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa5: int,X4: int] :
( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( X4
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ Xa5 @ X4 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_7309_subset__mset_Ostrict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: ( multiset @ A ) > B,X3: multiset @ A,Y: multiset @ A] :
( ( strict_mono @ ( multiset @ A ) @ B @ ( subset_mset @ A ) @ F3 )
=> ( ( subset_mset @ A @ X3 @ Y )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ) ) ).
% subset_mset.strict_monoD
thf(fact_7310_subset__mset_Ostrict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: ( multiset @ A ) > B] :
( ! [X: multiset @ A,Y2: multiset @ A] :
( ( subset_mset @ A @ X @ Y2 )
=> ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) )
=> ( strict_mono @ ( multiset @ A ) @ B @ ( subset_mset @ A ) @ F3 ) ) ) ).
% subset_mset.strict_monoI
thf(fact_7311_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_7312_folding__insort__key_Oaxioms_I2_J,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S: set @ B,F3: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( foldin3648464208017769352axioms @ B @ A @ S @ F3 ) ) ).
% folding_insort_key.axioms(2)
thf(fact_7313_group_Oleft__cancel,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A,A4: A,B3: A,C2: A] :
( ( group @ A @ F3 @ Z2 @ Inverse )
=> ( ( ( F3 @ A4 @ B3 )
= ( F3 @ A4 @ C2 ) )
= ( B3 = C2 ) ) ) ).
% group.left_cancel
thf(fact_7314_group_Oleft__inverse,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A,A4: A] :
( ( group @ A @ F3 @ Z2 @ Inverse )
=> ( ( F3 @ ( Inverse @ A4 ) @ A4 )
= Z2 ) ) ).
% group.left_inverse
thf(fact_7315_group_Oright__cancel,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A,B3: A,A4: A,C2: A] :
( ( group @ A @ F3 @ Z2 @ Inverse )
=> ( ( ( F3 @ B3 @ A4 )
= ( F3 @ C2 @ A4 ) )
= ( B3 = C2 ) ) ) ).
% group.right_cancel
thf(fact_7316_group_Oright__inverse,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A,A4: A] :
( ( group @ A @ F3 @ Z2 @ Inverse )
=> ( ( F3 @ A4 @ ( Inverse @ A4 ) )
= Z2 ) ) ).
% group.right_inverse
thf(fact_7317_group_Oinverse__unique,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A,A4: A,B3: A] :
( ( group @ A @ F3 @ Z2 @ Inverse )
=> ( ( ( F3 @ A4 @ B3 )
= Z2 )
=> ( ( Inverse @ A4 )
= B3 ) ) ) ).
% group.inverse_unique
thf(fact_7318_group_Oinverse__inverse,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A,A4: A] :
( ( group @ A @ F3 @ Z2 @ Inverse )
=> ( ( Inverse @ ( Inverse @ A4 ) )
= A4 ) ) ).
% group.inverse_inverse
thf(fact_7319_group_Oinverse__neutral,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F3 @ Z2 @ Inverse )
=> ( ( Inverse @ Z2 )
= Z2 ) ) ).
% group.inverse_neutral
thf(fact_7320_group_Ogroup__left__neutral,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A,A4: A] :
( ( group @ A @ F3 @ Z2 @ Inverse )
=> ( ( F3 @ Z2 @ A4 )
= A4 ) ) ).
% group.group_left_neutral
thf(fact_7321_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A,A4: A,B3: A] :
( ( group @ A @ F3 @ Z2 @ Inverse )
=> ( ( Inverse @ ( F3 @ A4 @ B3 ) )
= ( F3 @ ( Inverse @ B3 ) @ ( Inverse @ A4 ) ) ) ) ).
% group.inverse_distrib_swap
thf(fact_7322_folding__insort__key__axioms_Ointro,axiom,
! [A: $tType,B: $tType,F3: B > A,S: set @ B] :
( ( inj_on @ B @ A @ F3 @ S )
=> ( foldin3648464208017769352axioms @ B @ A @ S @ F3 ) ) ).
% folding_insort_key_axioms.intro
thf(fact_7323_folding__insort__key__axioms__def,axiom,
! [A: $tType,B: $tType] :
( ( foldin3648464208017769352axioms @ B @ A )
= ( ^ [S2: set @ B,F7: B > A] : ( inj_on @ B @ A @ F7 @ S2 ) ) ) ).
% folding_insort_key_axioms_def
thf(fact_7324_Powp__Pow__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( powp @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= ( ^ [X4: set @ A] : ( member @ ( set @ A ) @ X4 @ ( pow @ A @ A3 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_7325_mergesort__by__rel__split_Opelims,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X3 @ 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 ) @ X3 @ Xa ) )
=> ( ! [Xs1: list @ A,Xs22: list @ A] :
( ( X3
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ 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 ) @ Xs1 @ Xs22 ) @ ( nil @ A ) ) ) ) ) )
=> ( ! [Xs1: list @ A,Xs22: list @ A] :
( ( X3
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X: A] :
( ( Xa
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs1 ) @ 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 ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ) )
=> ~ ! [Xs1: list @ A,Xs22: list @ A] :
( ( X3
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X12: A,X23: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs2 ) ) )
=> ( ( Y
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ Xs1 ) @ ( 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 ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs2 ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
thf(fact_7326_Powp__mono,axiom,
! [A: $tType,A3: A > $o,B2: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A3 @ B2 )
=> ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A3 ) @ ( powp @ A @ B2 ) ) ) ).
% Powp_mono
thf(fact_7327_Powp__def,axiom,
! [A: $tType] :
( ( powp @ A )
= ( ^ [A7: A > $o,B7: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ B7 )
=> ( A7 @ X4 ) ) ) ) ).
% Powp_def
thf(fact_7328_exhaustive__integer_H_Ocases,axiom,
! [X3: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F5: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D4: code_integer,I5: code_integer] :
( X3
!= ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F5 @ ( product_Pair @ code_integer @ code_integer @ D4 @ I5 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_7329_subset_Osuc__Union__closedp_Ointros_I2_J,axiom,
! [A: $tType,A3: set @ ( set @ A ),Xa6: set @ ( set @ ( set @ A ) )] :
( ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) ) @ Xa6 )
=> ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ Xa6 ) ) ) ).
% subset.suc_Union_closedp.intros(2)
thf(fact_7330_subset_Osuc__Union__closedp_OUnion,axiom,
! [A: $tType,A3: set @ ( set @ A ),X5: set @ ( set @ ( set @ A ) )] :
( ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) ) @ X5 )
=> ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X5 ) ) ) ).
% subset.suc_Union_closedp.Union
thf(fact_7331_subset_Osuc__Union__closedp_Osuc,axiom,
! [A: $tType,A3: set @ ( set @ A ),X5: set @ ( set @ A )] :
( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 )
=> ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X5 ) ) ) ).
% subset.suc_Union_closedp.suc
thf(fact_7332_exhaustive__int_H_Ocases,axiom,
! [X3: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F5: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D4: int,I5: int] :
( X3
!= ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F5 @ ( product_Pair @ int @ int @ D4 @ I5 ) ) ) ).
% exhaustive_int'.cases
thf(fact_7333_full__exhaustive__int_H_Ocases,axiom,
! [X3: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F5: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D4: int,I5: int] :
( X3
!= ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F5 @ ( product_Pair @ int @ int @ D4 @ I5 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_7334_subset_Osuc__Union__closedp_Ocases,axiom,
! [A: $tType,A3: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ A4 )
=> ( ! [X11: set @ ( set @ A )] :
( ( A4
= ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X11 ) )
=> ~ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X11 ) )
=> ~ ! [X11: set @ ( set @ ( set @ A ) )] :
( ( A4
= ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X11 ) )
=> ~ ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) ) @ X11 ) ) ) ) ).
% subset.suc_Union_closedp.cases
thf(fact_7335_subset_Osuc__Union__closedp_Osimps,axiom,
! [A: $tType,A3: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ A4 )
= ( ? [X9: set @ ( set @ A )] :
( ( A4
= ( pred_suc @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X9 ) )
& ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) @ X9 ) )
| ? [X9: set @ ( set @ ( set @ A ) )] :
( ( A4
= ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X9 ) )
& ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A3 @ ( ord_less @ ( set @ A ) ) ) @ X9 ) ) ) ) ).
% subset.suc_Union_closedp.simps
thf(fact_7336_full__exhaustive__integer_H_Ocases,axiom,
! [X3: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F5: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D4: code_integer,I5: code_integer] :
( X3
!= ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F5 @ ( product_Pair @ code_integer @ code_integer @ D4 @ I5 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_7337_list__encode_Oelims,axiom,
! [X3: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X3 )
= Y )
=> ( ( ( X3
= ( nil @ nat ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [X: nat,Xs2: list @ nat] :
( ( X3
= ( cons @ nat @ X @ Xs2 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_7338_irreflp__irrefl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irreflp @ A
@ ^ [A5: A,B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R ) )
= ( irrefl @ A @ R ) ) ).
% irreflp_irrefl_eq
thf(fact_7339_list__encode_Osimps_I2_J,axiom,
! [X3: nat,Xs: list @ nat] :
( ( nat_list_encode @ ( cons @ nat @ X3 @ Xs ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_7340_irreflp__def,axiom,
! [A: $tType] :
( ( irreflp @ A )
= ( ^ [R4: A > A > $o] :
! [A5: A] :
~ ( R4 @ A5 @ A5 ) ) ) ).
% irreflp_def
thf(fact_7341_irreflpI,axiom,
! [A: $tType,R: A > A > $o] :
( ! [A6: A] :
~ ( R @ A6 @ A6 )
=> ( irreflp @ A @ R ) ) ).
% irreflpI
thf(fact_7342_irreflp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ) ).
% irreflp_greater
thf(fact_7343_irreflp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A @ ( ord_less @ A ) ) ) ).
% irreflp_less
thf(fact_7344_le__prod__encode__1,axiom,
! [A4: nat,B3: nat] : ( ord_less_eq @ nat @ A4 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A4 @ B3 ) ) ) ).
% le_prod_encode_1
thf(fact_7345_le__prod__encode__2,axiom,
! [B3: nat,A4: nat] : ( ord_less_eq @ nat @ B3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A4 @ B3 ) ) ) ).
% le_prod_encode_2
thf(fact_7346_list__encode_Opelims,axiom,
! [X3: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X3 )
= Y )
=> ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X3 )
=> ( ( ( X3
= ( nil @ nat ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
=> ~ ! [X: nat,Xs2: list @ nat] :
( ( X3
= ( cons @ nat @ X @ Xs2 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_7347_equivp__equiv,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ ( top_top @ ( set @ A ) ) @ A3 )
= ( equiv_equivp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ A3 ) ) ) ).
% equivp_equiv
thf(fact_7348_is__filter__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_total @ A @ B @ A3 )
=> ( ( bi_unique @ A @ B @ A3 )
=> ( bNF_rel_fun @ ( ( A > $o ) > $o ) @ ( ( B > $o ) > $o ) @ $o @ $o
@ ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ $o @ $o
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A3
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4
@ ( is_filter @ A )
@ ( is_filter @ B ) ) ) ) ).
% is_filter_parametric
thf(fact_7349_asymp__asym__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( asymp @ A
@ ^ [A5: A,B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R ) )
= ( asym @ A @ R ) ) ).
% asymp_asym_eq
thf(fact_7350_is__filter__map__filter__on,axiom,
! [A: $tType,B: $tType,F3: B > A,X5: set @ B,F: filter @ B] :
( ( is_filter @ A
@ ^ [P2: A > $o] :
( eventually @ B
@ ^ [X4: B] :
( ( P2 @ ( F3 @ X4 ) )
& ( member @ B @ X4 @ X5 ) )
@ F ) )
= ( eventually @ B
@ ^ [X4: B] : ( member @ B @ X4 @ X5 )
@ F ) ) ).
% is_filter_map_filter_on
thf(fact_7351_asym__inv__image,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),F3: B > A] :
( ( asym @ A @ R )
=> ( asym @ B @ ( inv_image @ A @ B @ R @ F3 ) ) ) ).
% asym_inv_image
thf(fact_7352_asym__lex,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ R )
=> ( asym @ ( list @ A ) @ ( lex @ A @ R ) ) ) ).
% asym_lex
thf(fact_7353_Abs__filter__cases,axiom,
! [A: $tType,X3: filter @ A] :
~ ! [Y2: ( A > $o ) > $o] :
( ( X3
= ( abs_filter @ A @ Y2 ) )
=> ~ ( member @ ( ( A > $o ) > $o ) @ Y2 @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) ) ) ).
% Abs_filter_cases
thf(fact_7354_Abs__filter__induct,axiom,
! [A: $tType,P: ( filter @ A ) > $o,X3: filter @ A] :
( ! [Y2: ( A > $o ) > $o] :
( ( member @ ( ( A > $o ) > $o ) @ Y2 @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ( P @ ( abs_filter @ A @ Y2 ) ) )
=> ( P @ X3 ) ) ).
% Abs_filter_induct
thf(fact_7355_Abs__filter__inject,axiom,
! [A: $tType,X3: ( A > $o ) > $o,Y: ( A > $o ) > $o] :
( ( member @ ( ( A > $o ) > $o ) @ X3 @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ( ( member @ ( ( A > $o ) > $o ) @ Y @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ( ( ( abs_filter @ A @ X3 )
= ( abs_filter @ A @ Y ) )
= ( X3 = Y ) ) ) ) ).
% Abs_filter_inject
thf(fact_7356_asym__lenlex,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ R )
=> ( asym @ ( list @ A ) @ ( lenlex @ A @ R ) ) ) ).
% asym_lenlex
thf(fact_7357_lexord__asym,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ R )
=> ( asym @ ( list @ A ) @ ( lexord @ A @ R ) ) ) ).
% lexord_asym
thf(fact_7358_is__filter__def,axiom,
! [A: $tType] :
( ( is_filter @ A )
= ( ^ [F4: ( A > $o ) > $o] :
( ( F4
@ ^ [X4: A] : $true )
& ! [P2: A > $o] :
( ( F4 @ P2 )
=> ! [Q2: A > $o] :
( ( F4 @ Q2 )
=> ( F4
@ ^ [X4: A] :
( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) )
& ! [P2: A > $o,Q2: A > $o] :
( ! [X4: A] :
( ( P2 @ X4 )
=> ( Q2 @ X4 ) )
=> ( ( F4 @ P2 )
=> ( F4 @ Q2 ) ) ) ) ) ) ).
% is_filter_def
thf(fact_7359_is__filter_Ointro,axiom,
! [A: $tType,F: ( A > $o ) > $o] :
( ( F
@ ^ [X4: A] : $true )
=> ( ! [P6: A > $o] :
( ( F @ P6 )
=> ! [Q5: A > $o] :
( ( F @ Q5 )
=> ( F
@ ^ [X4: A] :
( ( P6 @ X4 )
& ( Q5 @ X4 ) ) ) ) )
=> ( ! [P6: A > $o,Q5: A > $o] :
( ! [X8: A] :
( ( P6 @ X8 )
=> ( Q5 @ X8 ) )
=> ( ( F @ P6 )
=> ( F @ Q5 ) ) )
=> ( is_filter @ A @ F ) ) ) ) ).
% is_filter.intro
thf(fact_7360_is__filter_Omono,axiom,
! [A: $tType,F: ( A > $o ) > $o,P: A > $o,Q: A > $o] :
( ( is_filter @ A @ F )
=> ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ( F @ P )
=> ( F @ Q ) ) ) ) ).
% is_filter.mono
thf(fact_7361_is__filter_Oconj,axiom,
! [A: $tType,F: ( A > $o ) > $o,P: A > $o,Q: A > $o] :
( ( is_filter @ A @ F )
=> ( ( F @ P )
=> ( ( F @ Q )
=> ( F
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ) ).
% is_filter.conj
thf(fact_7362_is__filter_OTrue,axiom,
! [A: $tType,F: ( A > $o ) > $o] :
( ( is_filter @ A @ F )
=> ( F
@ ^ [X4: A] : $true ) ) ).
% is_filter.True
thf(fact_7363_asym_Ocases,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ A4 )
=> ! [A15: A,B10: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A15 @ B10 ) @ A4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B10 @ A15 ) @ A4 ) ) ) ).
% asym.cases
thf(fact_7364_asym_Osimps,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [A5: set @ ( product_prod @ A @ A )] :
? [R4: set @ ( product_prod @ A @ A )] :
( ( A5 = R4 )
& ! [X4: A,Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R4 ) ) ) ) ) ).
% asym.simps
thf(fact_7365_asym_Ointros,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R ) )
=> ( asym @ A @ R ) ) ).
% asym.intros
thf(fact_7366_asymD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X3: A,Y: A] :
( ( asym @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X3 ) @ R ) ) ) ).
% asymD
thf(fact_7367_asym__iff,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X4: A,Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R4 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R4 ) ) ) ) ).
% asym_iff
thf(fact_7368_eventually__Abs__filter,axiom,
! [A: $tType,F: ( A > $o ) > $o,P: A > $o] :
( ( is_filter @ A @ F )
=> ( ( eventually @ A @ P @ ( abs_filter @ A @ F ) )
= ( F @ P ) ) ) ).
% eventually_Abs_filter
thf(fact_7369_lexord__asymmetric,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: list @ A,B3: list @ A] :
( ( asym @ A @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A4 @ B3 ) @ ( lexord @ A @ R ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ B3 @ A4 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_asymmetric
thf(fact_7370_Abs__filter__inverse,axiom,
! [A: $tType,Y: ( A > $o ) > $o] :
( ( member @ ( ( A > $o ) > $o ) @ Y @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ( ( rep_filter @ A @ ( abs_filter @ A @ Y ) )
= Y ) ) ).
% Abs_filter_inverse
thf(fact_7371_Abs__filter__inverse_H,axiom,
! [A: $tType,F: ( A > $o ) > $o] :
( ( is_filter @ A @ F )
=> ( ( rep_filter @ A @ ( abs_filter @ A @ F ) )
= F ) ) ).
% Abs_filter_inverse'
thf(fact_7372_Rep__filter__iff__eventually,axiom,
! [A: $tType] :
( ( rep_filter @ A )
= ( ^ [F4: filter @ A,P2: A > $o] : ( eventually @ A @ P2 @ F4 ) ) ) ).
% Rep_filter_iff_eventually
thf(fact_7373_is__filter__Rep__filter,axiom,
! [A: $tType,F: filter @ A] : ( is_filter @ A @ ( rep_filter @ A @ F ) ) ).
% is_filter_Rep_filter
thf(fact_7374_Rep__filter__induct,axiom,
! [A: $tType,Y: ( A > $o ) > $o,P: ( ( A > $o ) > $o ) > $o] :
( ( member @ ( ( A > $o ) > $o ) @ Y @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ( ! [X: filter @ A] : ( P @ ( rep_filter @ A @ X ) )
=> ( P @ Y ) ) ) ).
% Rep_filter_induct
thf(fact_7375_Rep__filter__cases,axiom,
! [A: $tType,Y: ( A > $o ) > $o] :
( ( member @ ( ( A > $o ) > $o ) @ Y @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) )
=> ~ ! [X: filter @ A] :
( Y
!= ( rep_filter @ A @ X ) ) ) ).
% Rep_filter_cases
thf(fact_7376_Rep__filter,axiom,
! [A: $tType,X3: filter @ A] : ( member @ ( ( A > $o ) > $o ) @ ( rep_filter @ A @ X3 ) @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) ) ).
% Rep_filter
thf(fact_7377_Rep__filter__inverse,axiom,
! [A: $tType,X3: filter @ A] :
( ( abs_filter @ A @ ( rep_filter @ A @ X3 ) )
= X3 ) ).
% Rep_filter_inverse
thf(fact_7378_Rep__filter__inject,axiom,
! [A: $tType,X3: filter @ A,Y: filter @ A] :
( ( ( rep_filter @ A @ X3 )
= ( rep_filter @ A @ Y ) )
= ( X3 = Y ) ) ).
% Rep_filter_inject
thf(fact_7379_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_min @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
= ( F3 @ ( ord_max @ A @ X3 @ Y ) ) ) ) ) ).
% min_of_antimono
thf(fact_7380_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_max @ B @ ( F3 @ X3 ) @ ( F3 @ Y ) )
= ( F3 @ ( ord_min @ A @ X3 @ Y ) ) ) ) ) ).
% max_of_antimono
thf(fact_7381_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F7: A > B] :
! [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
=> ( ord_less_eq @ B @ ( F7 @ Y5 ) @ ( F7 @ X4 ) ) ) ) ) ) ).
% antimono_def
thf(fact_7382_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) )
=> ( order_antimono @ A @ B @ F3 ) ) ) ).
% antimonoI
thf(fact_7383_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X3 ) ) ) ) ) ).
% antimonoE
thf(fact_7384_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X3: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X3 ) ) ) ) ) ).
% antimonoD
thf(fact_7385_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A3 )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ) ).
% in_chain_finite
thf(fact_7386_chain__empty,axiom,
! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ).
% chain_empty
thf(fact_7387_ccpo__Sup__mono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: set @ A,B2: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A3 )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ B2 )
=> ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ B2 )
& ( ord_less_eq @ A @ X @ Xa3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ) ) ).
% ccpo_Sup_mono
thf(fact_7388_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X3: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_7389_cofinite__bot,axiom,
! [A: $tType] :
( ( ( cofinite @ A )
= ( bot_bot @ ( filter @ A ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% cofinite_bot
thf(fact_7390_wfP__SUP,axiom,
! [B: $tType,A: $tType,R2: A > B > B > $o] :
( ! [I5: A] : ( wfP @ B @ ( R2 @ I5 ) )
=> ( ! [I5: A,J3: A] :
( ( ( R2 @ I5 )
!= ( R2 @ J3 ) )
=> ( ( inf_inf @ ( B > $o ) @ ( domainp @ B @ B @ ( R2 @ I5 ) ) @ ( rangep @ B @ B @ ( R2 @ J3 ) ) )
= ( bot_bot @ ( B > $o ) ) ) )
=> ( wfP @ B @ ( complete_Sup_Sup @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% wfP_SUP
thf(fact_7391_Domainp__Domain__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( domainp @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R2 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( domain @ A @ B @ R2 ) ) ) ) ).
% Domainp_Domain_eq
thf(fact_7392_Domain__def,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ( domainp @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% Domain_def
thf(fact_7393_INFM__nat,axiom,
! [P: nat > $o] :
( ( frequently @ nat @ P @ ( cofinite @ nat ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less @ nat @ M @ N2 )
& ( P @ N2 ) ) ) ) ).
% INFM_nat
thf(fact_7394_INFM__nat__inductI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [I5: nat] :
( ( P @ I5 )
=> ? [J7: nat] :
( ( ord_less @ nat @ I5 @ J7 )
& ( P @ J7 )
& ( Q @ J7 ) ) )
=> ( frequently @ nat @ Q @ ( cofinite @ nat ) ) ) ) ).
% INFM_nat_inductI
thf(fact_7395_MOST__nat,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( cofinite @ nat ) )
= ( ? [M: nat] :
! [N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( P @ N2 ) ) ) ) ).
% MOST_nat
thf(fact_7396_frequently__cofinite,axiom,
! [A: $tType,P: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
= ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) ) ) ) ).
% frequently_cofinite
thf(fact_7397_eventually__cofinite,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
= ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
~ ( P @ X4 ) ) ) ) ).
% eventually_cofinite
thf(fact_7398_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_7399_DomainpE,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A4: A] :
( ( domainp @ A @ B @ R2 @ A4 )
=> ~ ! [B5: B] :
~ ( R2 @ A4 @ B5 ) ) ).
% DomainpE
thf(fact_7400_DomainPI,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,A4: A,B3: B] :
( ( R2 @ A4 @ B3 )
=> ( domainp @ A @ B @ R2 @ A4 ) ) ).
% DomainPI
thf(fact_7401_Domainp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [R5: A > B > $o,A5: A] :
? [B4: A,C6: B] :
( ( A5 = B4 )
& ( R5 @ B4 @ C6 ) ) ) ) ).
% Domainp.simps
thf(fact_7402_Domainp_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A4: A] :
( ( domainp @ A @ B @ R2 @ A4 )
=> ~ ! [B5: B] :
~ ( R2 @ A4 @ B5 ) ) ).
% Domainp.cases
thf(fact_7403_cofinite__eq__sequentially,axiom,
( ( cofinite @ nat )
= ( at_top @ nat ) ) ).
% cofinite_eq_sequentially
thf(fact_7404_wfP__less,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( wfP @ A @ ( ord_less @ A ) ) ) ).
% wfP_less
thf(fact_7405_wfP__less__multiset,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( wfP @ A @ ( ord_less @ A ) )
=> ( wfP @ ( multiset @ A ) @ ( ord_less @ ( multiset @ A ) ) ) ) ) ).
% wfP_less_multiset
thf(fact_7406_wfP__wf__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wfP @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R2 ) )
= ( wf @ A @ R2 ) ) ).
% wfP_wf_eq
thf(fact_7407_cofinite__def,axiom,
! [A: $tType] :
( ( cofinite @ A )
= ( abs_filter @ A
@ ^ [P2: A > $o] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
~ ( P2 @ X4 ) ) ) ) ) ).
% cofinite_def
thf(fact_7408_multiset_Odomain,axiom,
! [B: $tType,C: $tType,T4: C > B > $o] :
( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T4 ) )
= ( ^ [X4: C > nat] :
? [Y5: B > nat] :
( ( bNF_rel_fun @ C @ B @ nat @ nat @ T4
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4
@ X4
@ Y5 )
& ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Z6: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( Y5 @ Z6 ) ) ) ) ) ) ) ).
% multiset.domain
thf(fact_7409_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A4: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A4 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_7410_bit__minus__2__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% bit_minus_2_iff
thf(fact_7411_bit__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2239418461657761734s_mask @ A @ M2 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% bit_mask_iff
thf(fact_7412_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A4: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A4 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A4 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_7413_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_7414_multiset_Odomain__eq,axiom,
! [A: $tType] :
( ( domainp @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y4: A,Z4: A] : Y4 = Z4 ) )
= ( ^ [F7: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) ) ) ).
% multiset.domain_eq
thf(fact_7415_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_7416_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ( semiring_1_of_nat @ A @ N2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_7417_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( semiring_1_of_nat @ A @ N2 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_7418_CHAR__eq__posI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
=> ( ( ord_less @ nat @ X @ C2 )
=> ( ( semiring_1_of_nat @ A @ X )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_7419_multiset_Odomain__par__left__total,axiom,
! [B: $tType,C: $tType,T4: C > B > $o,P3: ( C > nat ) > $o] :
( ( left_total @ ( C > nat ) @ ( B > nat )
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T4
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4 ) )
=> ( ( bNF_rel_fun @ ( C > nat ) @ ( B > nat ) @ $o @ $o
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T4
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4
@ P3
@ ^ [F7: B > nat] :
( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
=> ( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T4 ) )
= P3 ) ) ) ).
% multiset.domain_par_left_total
thf(fact_7420_multiset_Odomain__par,axiom,
! [B: $tType,C: $tType,T4: C > B > $o,DT: C > $o,DS: nat > $o,P25: ( C > nat ) > $o] :
( ( ( domainp @ C @ B @ T4 )
= DT )
=> ( ( ( domainp @ nat @ nat
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4 )
= DS )
=> ( ( left_unique @ C @ B @ T4 )
=> ( ( bNF_rel_fun @ ( C > nat ) @ ( B > nat ) @ $o @ $o
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T4
@ ^ [Y4: nat,Z4: nat] : Y4 = Z4 )
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4
@ P25
@ ^ [F7: B > nat] :
( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F7 @ X4 ) ) ) ) )
=> ( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T4 ) )
= ( inf_inf @ ( ( C > nat ) > $o ) @ ( basic_pred_fun @ C @ nat @ DT @ DS ) @ P25 ) ) ) ) ) ) ).
% multiset.domain_par
thf(fact_7421_list_Oleft__unique__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( left_unique @ A @ B @ R )
=> ( left_unique @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ R ) ) ) ).
% list.left_unique_rel
thf(fact_7422_list_Oleft__total__rel,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( left_total @ A @ B @ R )
=> ( left_total @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ R ) ) ) ).
% list.left_total_rel
thf(fact_7423_left__unique__rel__filter,axiom,
! [B: $tType,A: $tType,A3: A > B > $o] :
( ( left_unique @ A @ B @ A3 )
=> ( left_unique @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) ) ) ).
% left_unique_rel_filter
thf(fact_7424_left__total__rel__filter,axiom,
! [B: $tType,A: $tType,A3: A > B > $o] :
( ( left_total @ A @ B @ A3 )
=> ( left_total @ ( filter @ A ) @ ( filter @ B ) @ ( rel_filter @ A @ B @ A3 ) ) ) ).
% left_total_rel_filter
thf(fact_7425_cut__def,axiom,
! [B: $tType,A: $tType] :
( ( cut @ A @ B )
= ( ^ [F7: A > B,R4: set @ ( product_prod @ A @ A ),X4: A,Y5: A] : ( if @ B @ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R4 ) @ ( F7 @ Y5 ) @ ( undefined @ B ) ) ) ) ).
% cut_def
thf(fact_7426_above__def,axiom,
! [A: $tType] :
( ( order_above @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A5: A] :
( collect @ A
@ ^ [B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R5 ) ) ) ) ).
% above_def
thf(fact_7427_cuts__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,R: set @ ( product_prod @ A @ A ),X3: A,G: A > B] :
( ( ( cut @ A @ B @ F3 @ R @ X3 )
= ( cut @ A @ B @ G @ R @ X3 ) )
= ( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ R )
=> ( ( F3 @ Y5 )
= ( G @ Y5 ) ) ) ) ) ).
% cuts_eq
thf(fact_7428_cut__apply,axiom,
! [B: $tType,A: $tType,X3: A,A4: A,R: set @ ( product_prod @ A @ A ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A4 ) @ R )
=> ( ( cut @ A @ B @ F3 @ R @ A4 @ X3 )
= ( F3 @ X3 ) ) ) ).
% cut_apply
thf(fact_7429_type__definition__filter,axiom,
! [A: $tType] : ( type_definition @ ( filter @ A ) @ ( ( A > $o ) > $o ) @ ( rep_filter @ A ) @ ( abs_filter @ A ) @ ( collect @ ( ( A > $o ) > $o ) @ ( is_filter @ A ) ) ) ).
% type_definition_filter
thf(fact_7430_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_7431_all__interval__int__def,axiom,
( all_interval_int
= ( ^ [P2: int > $o,I4: int,J2: int] :
! [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ I4 @ J2 ) )
=> ( P2 @ X4 ) ) ) ) ).
% all_interval_int_def
thf(fact_7432_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_7433_fixp__mono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A,G: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F3 )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ G )
=> ( ! [Z11: A] : ( ord_less_eq @ A @ ( F3 @ Z11 ) @ ( G @ Z11 ) )
=> ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F3 ) @ ( comple115746919287870866o_fixp @ A @ G ) ) ) ) ) ) ).
% fixp_mono
thf(fact_7434_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X3: B,Xs: list @ B] :
( ( map_filter @ B @ A @ F3 @ ( cons @ B @ X3 @ Xs ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F3 @ Xs )
@ ^ [Y5: A] : ( cons @ A @ Y5 @ ( map_filter @ B @ A @ F3 @ Xs ) )
@ ( F3 @ X3 ) ) ) ).
% map_filter_simps(1)
thf(fact_7435_fixp__induct,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o,F3: A > A] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F3 )
=> ( ( P @ ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X: A] :
( ( P @ X )
=> ( P @ ( F3 @ X ) ) )
=> ( P @ ( comple115746919287870866o_fixp @ A @ F3 ) ) ) ) ) ) ) ).
% fixp_induct
thf(fact_7436_arg__max__nat__le,axiom,
! [A: $tType,P: A > $o,X3: A,F3: A > nat,B3: nat] :
( ( P @ X3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less @ nat @ ( F3 @ Y2 ) @ B3 ) )
=> ( ord_less_eq @ nat @ ( F3 @ X3 ) @ ( F3 @ ( lattices_ord_arg_max @ A @ nat @ F3 @ P ) ) ) ) ) ).
% arg_max_nat_le
thf(fact_7437_arg__max__natI,axiom,
! [A: $tType,P: A > $o,K3: A,F3: A > nat,B3: nat] :
( ( P @ K3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less @ nat @ ( F3 @ Y2 ) @ B3 ) )
=> ( P @ ( lattices_ord_arg_max @ A @ nat @ F3 @ P ) ) ) ) ).
% arg_max_natI
thf(fact_7438_arg__maxI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X3: A,F3: A > B,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ~ ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y2 ) ) )
=> ( ! [X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ~ ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( Q @ X ) ) )
=> ( Q @ ( lattices_ord_arg_max @ A @ B @ F3 @ P ) ) ) ) ) ) ).
% arg_maxI
thf(fact_7439_ccpo_OadmissibleD,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,P: A > $o,A3: set @ A] :
( ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P )
=> ( ( comple1602240252501008431_chain @ A @ Ord @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( P @ X ) )
=> ( P @ ( Lub @ A3 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_7440_ccpo_OadmissibleI,axiom,
! [A: $tType,Ord: A > A > $o,P: A > $o,Lub: ( set @ A ) > A] :
( ! [A14: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A14 )
=> ( ( A14
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X8: A] :
( ( member @ A @ X8 @ A14 )
=> ( P @ X8 ) )
=> ( P @ ( Lub @ A14 ) ) ) ) )
=> ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P ) ) ).
% ccpo.admissibleI
thf(fact_7441_ccpo_Oadmissible__def,axiom,
! [A: $tType] :
( ( comple1908693960933563346ssible @ A )
= ( ^ [Lub2: ( set @ A ) > A,Ord2: A > A > $o,P2: A > $o] :
! [A7: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord2 @ A7 )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( P2 @ X4 ) )
=> ( P2 @ ( Lub2 @ A7 ) ) ) ) ) ) ) ).
% ccpo.admissible_def
thf(fact_7442_arg__max__nat__lemma,axiom,
! [A: $tType,P: A > $o,K3: A,F3: A > nat,B3: nat] :
( ( P @ K3 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less @ nat @ ( F3 @ Y2 ) @ B3 ) )
=> ( ( P @ ( lattices_ord_arg_max @ A @ nat @ F3 @ P ) )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ ( F3 @ Y3 ) @ ( F3 @ ( lattices_ord_arg_max @ A @ nat @ F3 @ P ) ) ) ) ) ) ) ).
% arg_max_nat_lemma
thf(fact_7443_heap__step__admissible,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: E > A > C > B > D > $o,X3: E] :
( comple1908693960933563346ssible @ ( A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) @ ( partial_fun_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) ) @ ( partial_fun_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) )
@ ^ [F7: A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) )] :
! [H3: A,H8: C,R5: B,N2: D] :
( ( ( F7 @ H3 )
= ( some @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ R5 @ ( product_Pair @ C @ D @ H8 @ N2 ) ) ) )
=> ( P @ X3 @ H3 @ H8 @ R5 @ N2 ) ) ) ).
% heap_step_admissible
thf(fact_7444_mult__one__div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ A4 @ ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ B3 ) ) )
= ( divide_divide @ A @ A4 @ ( unit_f5069060285200089521factor @ A @ B3 ) ) ) ) ).
% mult_one_div_unit_factor
thf(fact_7445_unit__factor__idem,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( unit_f5069060285200089521factor @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) )
= ( unit_f5069060285200089521factor @ A @ A4 ) ) ) ).
% unit_factor_idem
thf(fact_7446_unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( ( unit_f5069060285200089521factor @ A @ A4 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% unit_factor_eq_0_iff
thf(fact_7447_unit__factor__0,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% unit_factor_0
thf(fact_7448_unit__factor__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% unit_factor_1
thf(fact_7449_inv__unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A4 ) )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% inv_unit_factor_eq_0_iff
thf(fact_7450_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ A4 @ ( unit_f5069060285200089521factor @ A @ B3 ) ) ) ) ) ).
% unit_factor_mult_unit_left
thf(fact_7451_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B3 @ A4 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B3 ) @ A4 ) ) ) ) ).
% unit_factor_mult_unit_right
thf(fact_7452_gcd__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K3: A,A4: A,B3: A] :
( ( times_times @ A @ K3 @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ K3 @ A4 ) @ ( times_times @ A @ K3 @ B3 ) ) @ ( unit_f5069060285200089521factor @ A @ K3 ) ) ) ) ).
% gcd_mult_distrib
thf(fact_7453_mult__gcd__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A4 @ B3 ) @ C2 )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% mult_gcd_right
thf(fact_7454_mult__gcd__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_gcd_left
thf(fact_7455_unit__factor__self,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] : ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) @ A4 ) ) ).
% unit_factor_self
thf(fact_7456_unit__factor__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A4: A,B3: A] :
( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) @ ( unit_f5069060285200089521factor @ A @ B3 ) ) ) ) ).
% unit_factor_mult
thf(fact_7457_is__unit__unit__factor,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ A4 )
= A4 ) ) ) ).
% is_unit_unit_factor
thf(fact_7458_unit__factor__dvd,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) @ B3 ) ) ) ).
% unit_factor_dvd
thf(fact_7459_dvd__unit__factor__div,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( unit_f5069060285200089521factor @ A @ ( divide_divide @ A @ A4 @ B3 ) )
= ( divide_divide @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) @ ( unit_f5069060285200089521factor @ A @ B3 ) ) ) ) ) ).
% dvd_unit_factor_div
thf(fact_7460_unit__factor__is__unit,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) @ ( one_one @ A ) ) ) ) ).
% unit_factor_is_unit
thf(fact_7461_coprime__crossproduct_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [B3: A,D3: A,A4: A,C2: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ B3 )
= ( unit_f5069060285200089521factor @ A @ D3 ) )
=> ( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
=> ( ( ( times_times @ A @ A4 @ D3 )
= ( times_times @ A @ B3 @ C2 ) )
= ( ( A4 = C2 )
& ( B3 = D3 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
thf(fact_7462_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
@ ^ [A5: A] : ( dvd_dvd @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% Lcm_no_units
thf(fact_7463_is__pure__assn__pure,axiom,
! [P: $o] : ( is_pure_assn @ ( pure_assn @ P ) ) ).
% is_pure_assn_pure
thf(fact_7464_is__pure__assn__starI,axiom,
! [A4: assn,B3: assn] :
( ( is_pure_assn @ A4 )
=> ( ( is_pure_assn @ B3 )
=> ( is_pure_assn @ ( times_times @ assn @ A4 @ B3 ) ) ) ) ).
% is_pure_assn_starI
thf(fact_7465_Lcm__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_empty
thf(fact_7466_Lcm__nat__empty,axiom,
( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( one_one @ nat ) ) ).
% Lcm_nat_empty
thf(fact_7467_is__pure__assn__basic__simps_I2_J,axiom,
is_pure_assn @ ( one_one @ assn ) ).
% is_pure_assn_basic_simps(2)
thf(fact_7468_is__pure__assn__def,axiom,
( is_pure_assn
= ( ^ [A5: assn] :
? [P2: $o] :
( A5
= ( pure_assn @ P2 ) ) ) ) ).
% is_pure_assn_def
thf(fact_7469_is__pure__assnE,axiom,
! [A4: assn] :
( ( is_pure_assn @ A4 )
=> ~ ! [P6: $o] :
( A4
!= ( pure_assn @ P6 ) ) ) ).
% is_pure_assnE
thf(fact_7470_is__pure__assn__basic__simps_I1_J,axiom,
is_pure_assn @ ( bot_bot @ assn ) ).
% is_pure_assn_basic_simps(1)
thf(fact_7471_Lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,B5: A] :
( ( member @ A @ A6 @ A3 )
=> ( ( member @ A @ B5 @ A3 )
=> ( ( A6 != B5 )
=> ( algebr8660921524188924756oprime @ A @ A6 @ B5 ) ) ) )
=> ( ( gcd_Lcm @ A @ A3 )
= ( normal6383669964737779283malize @ A
@ ( groups7121269368397514597t_prod @ A @ A
@ ^ [X4: A] : X4
@ A3 ) ) ) ) ) ) ) ).
% Lcm_coprime
thf(fact_7472_Lcm__eq__Max__nat,axiom,
! [M5: set @ nat] :
( ( finite_finite2 @ nat @ M5 )
=> ( ( M5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ! [M6: nat,N5: nat] :
( ( member @ nat @ M6 @ M5 )
=> ( ( member @ nat @ N5 @ M5 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M6 @ N5 ) @ M5 ) ) )
=> ( ( gcd_Lcm @ nat @ M5 )
= ( lattic643756798349783984er_Max @ nat @ M5 ) ) ) ) ) ) ).
% Lcm_eq_Max_nat
thf(fact_7473_normalize__idem,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( normal6383669964737779283malize @ A @ ( normal6383669964737779283malize @ A @ A4 ) )
= ( normal6383669964737779283malize @ A @ A4 ) ) ) ).
% normalize_idem
thf(fact_7474_normalize__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= ( zero_zero @ A ) )
= ( A4
= ( zero_zero @ A ) ) ) ) ).
% normalize_eq_0_iff
thf(fact_7475_normalize__0,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% normalize_0
thf(fact_7476_normalize__mult__normalize__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ ( normal6383669964737779283malize @ A @ B3 ) ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% normalize_mult_normalize_right
thf(fact_7477_normalize__mult__normalize__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ B3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% normalize_mult_normalize_left
thf(fact_7478_normalize__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% normalize_1
thf(fact_7479_normalize__dvd__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ B3 )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% normalize_dvd_iff
thf(fact_7480_dvd__normalize__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( normal6383669964737779283malize @ A @ B3 ) )
= ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% dvd_normalize_iff
thf(fact_7481_coprime__normalize__left__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ B3 )
= ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ).
% coprime_normalize_left_iff
thf(fact_7482_coprime__normalize__right__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ ( normal6383669964737779283malize @ A @ B3 ) )
= ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ).
% coprime_normalize_right_iff
thf(fact_7483_Lcm__insert,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( gcd_Lcm @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( gcd_lcm @ A @ A4 @ ( gcd_Lcm @ A @ A3 ) ) ) ) ).
% Lcm_insert
thf(fact_7484_unit__factor__mult__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ A4 ) )
= A4 ) ) ).
% unit_factor_mult_normalize
thf(fact_7485_normalize__mult__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( unit_f5069060285200089521factor @ A @ A4 ) )
= A4 ) ) ).
% normalize_mult_unit_factor
thf(fact_7486_div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ A4 @ ( unit_f5069060285200089521factor @ A @ A4 ) )
= ( normal6383669964737779283malize @ A @ A4 ) ) ) ).
% div_unit_factor
thf(fact_7487_div__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ A4 @ ( normal6383669964737779283malize @ A @ A4 ) )
= ( unit_f5069060285200089521factor @ A @ A4 ) ) ) ).
% div_normalize
thf(fact_7488_normalize__mult__unit__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( normal6383669964737779283malize @ A @ A4 ) ) ) ) ).
% normalize_mult_unit_right
thf(fact_7489_normalize__mult__unit__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( normal6383669964737779283malize @ A @ B3 ) ) ) ) ).
% normalize_mult_unit_left
thf(fact_7490_unit__factor__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ).
% unit_factor_normalize
thf(fact_7491_normalize__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ).
% normalize_unit_factor
thf(fact_7492_lcm__mult__gcd,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A4 @ B3 ) @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ B3 ) ) ) ) ).
% lcm_mult_gcd
thf(fact_7493_gcd__mult__lcm,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A4 @ B3 ) @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ B3 ) ) ) ) ).
% gcd_mult_lcm
thf(fact_7494_Lcm__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A,B3: A] :
( ( gcd_Lcm @ A @ ( insert3 @ A @ A4 @ ( insert3 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_lcm @ A @ A4 @ B3 ) ) ) ).
% Lcm_2
thf(fact_7495_Lcm__singleton,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A] :
( ( gcd_Lcm @ A @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( normal6383669964737779283malize @ A @ A4 ) ) ) ).
% Lcm_singleton
thf(fact_7496_normalize__div,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ A4 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A4 ) ) ) ) ).
% normalize_div
thf(fact_7497_Gcd__singleton,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: A] :
( ( gcd_Gcd @ A @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) )
= ( normal6383669964737779283malize @ A @ A4 ) ) ) ).
% Gcd_singleton
thf(fact_7498_Lcm__nat__insert,axiom,
! [N: nat,M5: set @ nat] :
( ( gcd_Lcm @ nat @ ( insert3 @ nat @ N @ M5 ) )
= ( gcd_lcm @ nat @ N @ ( gcd_Lcm @ nat @ M5 ) ) ) ).
% Lcm_nat_insert
thf(fact_7499_Lcm__in__lcm__closed__set__nat,axiom,
! [M5: set @ nat] :
( ( finite_finite2 @ nat @ M5 )
=> ( ( M5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ! [M6: nat,N5: nat] :
( ( member @ nat @ M6 @ M5 )
=> ( ( member @ nat @ N5 @ M5 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M6 @ N5 ) @ M5 ) ) )
=> ( member @ nat @ ( gcd_Lcm @ nat @ M5 ) @ M5 ) ) ) ) ).
% Lcm_in_lcm_closed_set_nat
thf(fact_7500_coprime__crossproduct,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,D3: A,B3: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ D3 )
=> ( ( algebr8660921524188924756oprime @ A @ B3 @ C2 )
=> ( ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ C2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ B3 ) @ ( normal6383669964737779283malize @ A @ D3 ) ) )
= ( ( ( normal6383669964737779283malize @ A @ A4 )
= ( normal6383669964737779283malize @ A @ B3 ) )
& ( ( normal6383669964737779283malize @ A @ C2 )
= ( normal6383669964737779283malize @ A @ D3 ) ) ) ) ) ) ) ).
% coprime_crossproduct
thf(fact_7501_lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( algebr8660921524188924756oprime @ A @ A4 @ B3 )
=> ( ( gcd_lcm @ A @ A4 @ B3 )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ) ).
% lcm_coprime
thf(fact_7502_gcd__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A] :
( ( A4
!= ( zero_zero @ A ) )
=> ( ( B3
!= ( zero_zero @ A ) )
=> ( ( gcd_gcd @ A @ A4 @ B3 )
= ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A4 @ B3 ) @ ( gcd_lcm @ A @ A4 @ B3 ) ) ) ) ) ) ) ).
% gcd_lcm
thf(fact_7503_gcd__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ).
% gcd_mult_distrib'
thf(fact_7504_lcm__gcd__prod,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A4 @ B3 ) @ ( gcd_gcd @ A @ A4 @ B3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) ) ) ) ).
% lcm_gcd_prod
thf(fact_7505_gcd__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_gcd @ A @ B3 @ A4 ) @ C2 ) ) ) ) ).
% gcd_mult_right
thf(fact_7506_gcd__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A4 @ B3 ) ) ) ) ) ).
% gcd_mult_left
thf(fact_7507_lcm__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_lcm @ A )
= ( ^ [A5: A,B4: A] : ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A5 @ B4 ) @ ( gcd_gcd @ A @ A5 @ B4 ) ) ) ) ) ) ).
% lcm_gcd
thf(fact_7508_lcm__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( times_times @ A @ B3 @ A4 ) @ C2 )
= ( gcd_lcm @ A @ B3 @ C2 ) ) ) ) ).
% lcm_mult_unit1
thf(fact_7509_lcm__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B3 @ ( times_times @ A @ C2 @ A4 ) )
= ( gcd_lcm @ A @ B3 @ C2 ) ) ) ) ).
% lcm_mult_unit2
thf(fact_7510_associated__iff__dvd,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= ( normal6383669964737779283malize @ A @ B3 ) )
= ( ( dvd_dvd @ A @ A4 @ B3 )
& ( dvd_dvd @ A @ B3 @ A4 ) ) ) ) ).
% associated_iff_dvd
thf(fact_7511_associated__eqI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( ( normal6383669964737779283malize @ A @ A4 )
= A4 )
=> ( ( ( normal6383669964737779283malize @ A @ B3 )
= B3 )
=> ( A4 = B3 ) ) ) ) ) ) ).
% associated_eqI
thf(fact_7512_associatedD2,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= ( normal6383669964737779283malize @ A @ B3 ) )
=> ( dvd_dvd @ A @ B3 @ A4 ) ) ) ).
% associatedD2
thf(fact_7513_associatedD1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= ( normal6383669964737779283malize @ A @ B3 ) )
=> ( dvd_dvd @ A @ A4 @ B3 ) ) ) ).
% associatedD1
thf(fact_7514_associatedI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( dvd_dvd @ A @ A4 @ B3 )
=> ( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( normal6383669964737779283malize @ A @ A4 )
= ( normal6383669964737779283malize @ A @ B3 ) ) ) ) ) ).
% associatedI
thf(fact_7515_normalize__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A4: A,B3: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ B3 ) ) ) ) ).
% normalize_mult
thf(fact_7516_lcm__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A4 @ B3 ) ) ) ) ) ).
% lcm_mult_left
thf(fact_7517_lcm__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,C2: A,B3: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_lcm @ A @ B3 @ A4 ) @ C2 ) ) ) ) ).
% lcm_mult_right
thf(fact_7518_lcm__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ).
% lcm_mult_distrib'
thf(fact_7519_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= A4 )
=> ( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
= ( A4
= ( one_one @ A ) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
thf(fact_7520_is__unit__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% is_unit_normalize
thf(fact_7521_normalize__1__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= ( one_one @ A ) )
= ( dvd_dvd @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% normalize_1_iff
thf(fact_7522_associated__unit,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= ( normal6383669964737779283malize @ A @ B3 ) )
=> ( ( dvd_dvd @ A @ A4 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B3 @ ( one_one @ A ) ) ) ) ) ).
% associated_unit
thf(fact_7523_dvd__normalize__div,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [B3: A,A4: A] :
( ( dvd_dvd @ A @ B3 @ A4 )
=> ( ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ A4 @ B3 ) )
= ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A4 ) @ ( normal6383669964737779283malize @ A @ B3 ) ) ) ) ) ).
% dvd_normalize_div
thf(fact_7524_Gcd__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [C2: A,A3: set @ A] :
( ( gcd_Gcd @ A @ ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ A3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Gcd @ A @ A3 ) ) ) ) ) ).
% Gcd_mult
thf(fact_7525_lcm__pos__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_lcm @ nat @ M2 @ N ) ) ) ) ).
% lcm_pos_nat
thf(fact_7526_normalize__idem__imp__unit__factor__eq,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= A4 )
=> ( ( unit_f5069060285200089521factor @ A @ A4 )
= ( zero_neq_one_of_bool @ A
@ ( A4
!= ( zero_zero @ A ) ) ) ) ) ) ).
% normalize_idem_imp_unit_factor_eq
thf(fact_7527_lcm__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K3: A,A4: A,B3: A] :
( ( times_times @ A @ K3 @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ K3 @ A4 ) @ ( times_times @ A @ K3 @ B3 ) ) @ ( unit_f5069060285200089521factor @ A @ K3 ) ) ) ) ).
% lcm_mult_distrib
thf(fact_7528_mult__lcm__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A4 @ B3 ) @ C2 )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ A4 @ C2 ) @ ( times_times @ A @ B3 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% mult_lcm_right
thf(fact_7529_mult__lcm__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A4: A,B3: A] :
( ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A4 @ B3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A4 ) @ ( times_times @ A @ C2 @ B3 ) ) ) ) ) ).
% mult_lcm_left
thf(fact_7530_normalize__unit__factor__eqI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A4: A,B3: A] :
( ( ( normal6383669964737779283malize @ A @ A4 )
= ( normal6383669964737779283malize @ A @ B3 ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ A4 )
= ( unit_f5069060285200089521factor @ A @ B3 ) )
=> ( A4 = B3 ) ) ) ) ).
% normalize_unit_factor_eqI
thf(fact_7531_Lcm__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: set @ A,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( gcd_Lcm @ A @ ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ A3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Lcm @ A @ A3 ) ) ) ) ) ) ).
% Lcm_mult
thf(fact_7532_Gcd__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A,B3: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B3 ) @ A3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B3 @ ( semiring_gcd_Gcd_fin @ A @ A3 ) ) ) ) ) ) ).
% Gcd_fin_mult
thf(fact_7533_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_7534_Lcm__nat__set__eq__fold,axiom,
! [Xs: list @ nat] :
( ( gcd_Lcm @ nat @ ( set2 @ nat @ Xs ) )
= ( fold @ nat @ nat @ ( gcd_lcm @ nat ) @ Xs @ ( one_one @ nat ) ) ) ).
% Lcm_nat_set_eq_fold
thf(fact_7535_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_7536_Lcm__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( gcd_lcm @ A @ A4 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Lcm_fin.insert_remove
thf(fact_7537_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_7538_Lcm__fin_Oinsert,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( insert3 @ A @ A4 @ A3 ) )
= ( gcd_lcm @ A @ A4 @ ( semiring_gcd_Lcm_fin @ A @ A3 ) ) ) ) ).
% Lcm_fin.insert
thf(fact_7539_lcm__list__dvd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A,B3: A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Xs ) ) @ B3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( dvd_dvd @ A @ X4 @ B3 ) ) ) ) ) ).
% lcm_list_dvd_iff
thf(fact_7540_lcm__list__least,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Bs: list @ A,A4: A] :
( ! [B5: A] :
( ( member @ A @ B5 @ ( set2 @ A @ Bs ) )
=> ( dvd_dvd @ A @ B5 @ A4 ) )
=> ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Bs ) ) @ A4 ) ) ) ).
% lcm_list_least
thf(fact_7541_Lcm__int__set__eq__fold,axiom,
! [Xs: list @ int] :
( ( gcd_Lcm @ int @ ( set2 @ int @ Xs ) )
= ( fold @ int @ int @ ( gcd_lcm @ int ) @ Xs @ ( one_one @ int ) ) ) ).
% Lcm_int_set_eq_fold
thf(fact_7542_Lcm__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: set @ A,B3: A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B3 ) @ A3 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B3 @ ( semiring_gcd_Lcm_fin @ A @ A3 ) ) ) ) ) ) ).
% Lcm_fin_mult
thf(fact_7543_Lcm__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: A,A3: set @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A3 )
= ( gcd_lcm @ A @ A4 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert3 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Lcm_fin.remove
thf(fact_7544_properI,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ! [As: set @ nat,H: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) )
=> ( ! [As: set @ nat,H: heap_ext @ product_unit,H5: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ( relH @ As @ H @ H5 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) ) ) ) )
=> ( proper @ P ) ) ) ).
% properI
thf(fact_7545_properD2,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As2: set @ nat,H4: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ( ( relH @ As2 @ H2 @ H4 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ) ) ).
% properD2
thf(fact_7546_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
@ ^ [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H3 )
& ( Q @ H3 ) ) ) ) ) ).
% bool_assn_proper(4)
thf(fact_7547_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
@ ^ [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H3 )
| ( Q @ H3 ) ) ) ) ) ).
% bool_assn_proper(3)
thf(fact_7548_bool__assn__proper_I2_J,axiom,
( proper
@ ^ [Uu3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : $false ) ).
% bool_assn_proper(2)
thf(fact_7549_bool__assn__proper_I1_J,axiom,
proper @ in_range ).
% bool_assn_proper(1)
thf(fact_7550_ex__assn__proper,axiom,
! [A: $tType,P: A > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ! [X: A] : ( proper @ ( P @ X ) )
=> ( proper
@ ^ [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
? [X4: A] : ( P @ X4 @ H3 ) ) ) ).
% ex_assn_proper
thf(fact_7551_snga__assn__proper,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: array @ A,X3: list @ A] : ( proper @ ( snga_assn_raw @ A @ R2 @ X3 ) ) ) ).
% snga_assn_proper
thf(fact_7552_one__assn__proper,axiom,
proper @ one_assn_raw ).
% one_assn_proper
thf(fact_7553_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_7554_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_7555_sngr__assn__proper,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,X3: A] : ( proper @ ( sngr_assn_raw @ A @ R2 @ X3 ) ) ) ).
% sngr_assn_proper
thf(fact_7556_bool__assn__proper_I5_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( proper @ P )
=> ( proper
@ ^ [H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ H3 )
& ~ ( P @ H3 ) ) ) ) ).
% bool_assn_proper(5)
thf(fact_7557_pure__assn__proper,axiom,
! [B3: $o] : ( proper @ ( pure_assn_raw @ ( heap_ext @ product_unit ) @ nat @ B3 ) ) ).
% pure_assn_proper
thf(fact_7558_Abs__assn__inverse,axiom,
! [Y: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( ( rep_assn @ ( abs_assn @ Y ) )
= Y ) ) ).
% Abs_assn_inverse
thf(fact_7559_Abs__assn__cases,axiom,
! [X3: assn] :
~ ! [Y2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( X3
= ( abs_assn @ Y2 ) )
=> ~ ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y2 @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) ) ) ).
% Abs_assn_cases
thf(fact_7560_Abs__assn__induct,axiom,
! [P: assn > $o,X3: assn] :
( ! [Y2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y2 @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( P @ ( abs_assn @ Y2 ) ) )
=> ( P @ X3 ) ) ).
% Abs_assn_induct
thf(fact_7561_Abs__assn__inject,axiom,
! [X3: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Y: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ X3 @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( ( ( abs_assn @ X3 )
= ( abs_assn @ Y ) )
= ( X3 = Y ) ) ) ) ).
% Abs_assn_inject
thf(fact_7562_Rep__assn__induct,axiom,
! [Y: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,P: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ( ! [X: assn] : ( P @ ( rep_assn @ X ) )
=> ( P @ Y ) ) ) ).
% Rep_assn_induct
thf(fact_7563_Rep__assn__cases,axiom,
! [Y: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ Y @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) )
=> ~ ! [X: assn] :
( Y
!= ( rep_assn @ X ) ) ) ).
% Rep_assn_cases
thf(fact_7564_Rep__assn,axiom,
! [X3: assn] : ( member @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( rep_assn @ X3 ) @ ( collect @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ proper ) ) ).
% Rep_assn
thf(fact_7565_properD1,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As2: set @ nat] :
( ( proper @ P )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) ) ) ) ).
% properD1
thf(fact_7566_proper__iff,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,As2: set @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( relH @ As2 @ H2 @ H4 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
= ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ) ) ).
% proper_iff
thf(fact_7567_proper__def,axiom,
( proper
= ( ^ [P2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
! [H3: heap_ext @ product_unit,H8: heap_ext @ product_unit,As4: set @ nat] :
( ( ( P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As4 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As4 ) ) )
& ( ( ( P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As4 ) )
& ( relH @ As4 @ H3 @ H8 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As4 ) ) )
=> ( P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As4 ) ) ) ) ) ) ).
% proper_def
thf(fact_7568_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_7569_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_7570_all__interval__nat__def,axiom,
( all_interval_nat
= ( ^ [P2: nat > $o,I4: nat,J2: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ I4 @ J2 ) )
=> ( P2 @ X4 ) ) ) ) ).
% all_interval_nat_def
thf(fact_7571_of__rat__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( one_one @ A ) )
= ( ord_less @ rat @ R2 @ ( one_one @ rat ) ) ) ) ).
% of_rat_less_1_iff
thf(fact_7572_one__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less @ rat @ ( one_one @ rat ) @ R2 ) ) ) ).
% one_less_of_rat_iff
thf(fact_7573_of__rat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( zero_zero @ A ) )
= ( ord_less @ rat @ R2 @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_less_0_iff
thf(fact_7574_zero__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R2 ) )
= ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 ) ) ) ).
% zero_less_of_rat_iff
thf(fact_7575_of__rat__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: rat,B3: rat] :
( ( field_char_0_of_rat @ A @ ( times_times @ rat @ A4 @ B3 ) )
= ( times_times @ A @ ( field_char_0_of_rat @ A @ A4 ) @ ( field_char_0_of_rat @ A @ B3 ) ) ) ) ).
% of_rat_mult
thf(fact_7576_of__rat__diff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A4: rat,B3: rat] :
( ( field_char_0_of_rat @ A @ ( minus_minus @ rat @ A4 @ B3 ) )
= ( minus_minus @ A @ ( field_char_0_of_rat @ A @ A4 ) @ ( field_char_0_of_rat @ A @ B3 ) ) ) ) ).
% of_rat_diff
thf(fact_7577_of__rat__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R2: rat,S3: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R2 ) @ ( field_char_0_of_rat @ A @ S3 ) )
= ( ord_less @ rat @ R2 @ S3 ) ) ) ).
% of_rat_less
thf(fact_7578_listsp__mono,axiom,
! [A: $tType,A3: A > $o,B2: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A3 @ B2 )
=> ( ord_less_eq @ ( ( list @ A ) > $o ) @ ( listsp @ A @ A3 ) @ ( listsp @ A @ B2 ) ) ) ).
% listsp_mono
thf(fact_7579_partial__function__definitions_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,Leq: A > A > $o,Lub: ( set @ A ) > A,U2: C > B > A,F: C > C,C3: ( B > A ) > C,F3: C,P: ( B > A ) > $o] :
( ( partia7178651479351089652itions @ A @ Leq @ Lub )
=> ( ! [X: B] :
( comple7038119648293358887notone @ ( B > A ) @ A @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ Leq
@ ^ [F7: B > A] : ( U2 @ ( F @ ( C3 @ F7 ) ) @ X ) )
=> ( ( F3
= ( C3
@ ( comple187402453842119260l_fixp @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub ) @ ( partial_fun_ord @ A @ A @ B @ Leq )
@ ^ [F7: B > A] : ( U2 @ ( F @ ( C3 @ F7 ) ) ) ) ) )
=> ( ! [F5: B > A] :
( ( U2 @ ( C3 @ F5 ) )
= F5 )
=> ( ( comple1908693960933563346ssible @ ( B > A ) @ ( partial_fun_lub @ A @ A @ B @ Lub ) @ ( partial_fun_ord @ A @ A @ B @ Leq ) @ P )
=> ( ( P
@ ^ [Uu3: B] : ( Lub @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [F5: C] :
( ( P @ ( U2 @ F5 ) )
=> ( P @ ( U2 @ ( F @ F5 ) ) ) )
=> ( P @ ( U2 @ F3 ) ) ) ) ) ) ) ) ) ).
% partial_function_definitions.fixp_induct_uc
thf(fact_7580_listsp__conj__eq,axiom,
! [A: $tType,A3: A > $o,B2: A > $o] :
( ( listsp @ A
@ ^ [X4: A] :
( ( A3 @ X4 )
& ( B2 @ X4 ) ) )
= ( ^ [X4: list @ A] :
( ( listsp @ A @ A3 @ X4 )
& ( listsp @ A @ B2 @ X4 ) ) ) ) ).
% listsp_conj_eq
thf(fact_7581_listsp__simps_I1_J,axiom,
! [A: $tType,A3: A > $o] : ( listsp @ A @ A3 @ ( nil @ A ) ) ).
% listsp_simps(1)
thf(fact_7582_in__listspI,axiom,
! [A: $tType,Xs: list @ A,A3: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( A3 @ X ) )
=> ( listsp @ A @ A3 @ Xs ) ) ).
% in_listspI
thf(fact_7583_append__in__listsp__conv,axiom,
! [A: $tType,A3: A > $o,Xs: list @ A,Ys: list @ A] :
( ( listsp @ A @ A3 @ ( append @ A @ Xs @ Ys ) )
= ( ( listsp @ A @ A3 @ Xs )
& ( listsp @ A @ A3 @ Ys ) ) ) ).
% append_in_listsp_conv
thf(fact_7584_listsp__inf__eq,axiom,
! [A: $tType,A3: A > $o,B2: A > $o] :
( ( listsp @ A @ ( inf_inf @ ( A > $o ) @ A3 @ B2 ) )
= ( inf_inf @ ( ( list @ A ) > $o ) @ ( listsp @ A @ A3 ) @ ( listsp @ A @ B2 ) ) ) ).
% listsp_inf_eq
thf(fact_7585_listspE,axiom,
! [A: $tType,A3: A > $o,X3: A,L: list @ A] :
( ( listsp @ A @ A3 @ ( cons @ A @ X3 @ L ) )
=> ~ ( ( A3 @ X3 )
=> ~ ( listsp @ A @ A3 @ L ) ) ) ).
% listspE
thf(fact_7586_listsp_OCons,axiom,
! [A: $tType,A3: A > $o,A4: A,L: list @ A] :
( ( A3 @ A4 )
=> ( ( listsp @ A @ A3 @ L )
=> ( listsp @ A @ A3 @ ( cons @ A @ A4 @ L ) ) ) ) ).
% listsp.Cons
thf(fact_7587_listsp__simps_I2_J,axiom,
! [A: $tType,A3: A > $o,X3: A,Xs: list @ A] :
( ( listsp @ A @ A3 @ ( cons @ A @ X3 @ Xs ) )
= ( ( A3 @ X3 )
& ( listsp @ A @ A3 @ Xs ) ) ) ).
% listsp_simps(2)
thf(fact_7588_in__listspD,axiom,
! [A: $tType,A3: A > $o,Xs: list @ A] :
( ( listsp @ A @ A3 @ Xs )
=> ! [X8: A] :
( ( member @ A @ X8 @ ( set2 @ A @ Xs ) )
=> ( A3 @ X8 ) ) ) ).
% in_listspD
thf(fact_7589_in__listsp__conv__set,axiom,
! [A: $tType] :
( ( listsp @ A )
= ( ^ [A7: A > $o,Xs3: list @ A] :
! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs3 ) )
=> ( A7 @ X4 ) ) ) ) ).
% in_listsp_conv_set
thf(fact_7590_listsp_ONil,axiom,
! [A: $tType,A3: A > $o] : ( listsp @ A @ A3 @ ( nil @ A ) ) ).
% listsp.Nil
thf(fact_7591_listsp_Ocases,axiom,
! [A: $tType,A3: A > $o,A4: list @ A] :
( ( listsp @ A @ A3 @ A4 )
=> ( ( A4
!= ( nil @ A ) )
=> ~ ! [A6: A,L3: list @ A] :
( ( A4
= ( cons @ A @ A6 @ L3 ) )
=> ( ( A3 @ A6 )
=> ~ ( listsp @ A @ A3 @ L3 ) ) ) ) ) ).
% listsp.cases
thf(fact_7592_listsp_Osimps,axiom,
! [A: $tType] :
( ( listsp @ A )
= ( ^ [A7: A > $o,A5: list @ A] :
( ( A5
= ( nil @ A ) )
| ? [B4: A,L4: list @ A] :
( ( A5
= ( cons @ A @ B4 @ L4 ) )
& ( A7 @ B4 )
& ( listsp @ A @ A7 @ L4 ) ) ) ) ) ).
% listsp.simps
thf(fact_7593_listsp__lists__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( listsp @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= ( ^ [X4: list @ A] : ( member @ ( list @ A ) @ X4 @ ( lists @ A @ A3 ) ) ) ) ).
% listsp_lists_eq
thf(fact_7594_lists__def,axiom,
! [A: $tType] :
( ( lists @ A )
= ( ^ [A7: set @ A] :
( collect @ ( list @ A )
@ ( listsp @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A7 ) ) ) ) ) ).
% lists_def
thf(fact_7595_listsp__infI,axiom,
! [A: $tType,A3: A > $o,L: list @ A,B2: A > $o] :
( ( listsp @ A @ A3 @ L )
=> ( ( listsp @ A @ B2 @ L )
=> ( listsp @ A @ ( inf_inf @ ( A > $o ) @ A3 @ B2 ) @ L ) ) ) ).
% listsp_infI
thf(fact_7596_Gr__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr @ A @ B )
= ( ^ [A7: set @ A,F7: A > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A5: A] :
( ( Uu3
= ( product_Pair @ A @ B @ A5 @ ( F7 @ A5 ) ) )
& ( member @ A @ A5 @ A7 ) ) ) ) ) ).
% Gr_def
thf(fact_7597_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_7598_GrD2,axiom,
! [A: $tType,B: $tType,X3: A,Fx: B,A3: set @ A,F3: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A3 @ F3 ) )
=> ( ( F3 @ X3 )
= Fx ) ) ).
% GrD2
thf(fact_7599_GrD1,axiom,
! [B: $tType,A: $tType,X3: A,Fx: B,A3: set @ A,F3: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Fx ) @ ( bNF_Gr @ A @ B @ A3 @ F3 ) )
=> ( member @ A @ X3 @ A3 ) ) ).
% GrD1
thf(fact_7600_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_7601_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_max @ A )
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ) ).
% max.semilattice_order_axioms
thf(fact_7602_semilattice__neutr__order_Oaxioms_I2_J,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F3 @ Z2 @ Less_eq @ Less )
=> ( semilattice_order @ A @ F3 @ Less_eq @ Less ) ) ).
% semilattice_neutr_order.axioms(2)
thf(fact_7603_semilattice__order_Omono,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,C2: A,B3: A,D3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ C2 )
=> ( ( Less_eq @ B3 @ D3 )
=> ( Less_eq @ ( F3 @ A4 @ B3 ) @ ( F3 @ C2 @ D3 ) ) ) ) ) ).
% semilattice_order.mono
thf(fact_7604_semilattice__order_OorderE,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
=> ( A4
= ( F3 @ A4 @ B3 ) ) ) ) ).
% semilattice_order.orderE
thf(fact_7605_semilattice__order_OorderI,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( A4
= ( F3 @ A4 @ B3 ) )
=> ( Less_eq @ A4 @ B3 ) ) ) ).
% semilattice_order.orderI
thf(fact_7606_semilattice__order_Oabsorb1,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
=> ( ( F3 @ A4 @ B3 )
= A4 ) ) ) ).
% semilattice_order.absorb1
thf(fact_7607_semilattice__order_Oabsorb2,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B3: A,A4: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ B3 @ A4 )
=> ( ( F3 @ A4 @ B3 )
= B3 ) ) ) ).
% semilattice_order.absorb2
thf(fact_7608_semilattice__order_Oabsorb3,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
=> ( ( F3 @ A4 @ B3 )
= A4 ) ) ) ).
% semilattice_order.absorb3
thf(fact_7609_semilattice__order_Oabsorb4,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B3: A,A4: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less @ B3 @ A4 )
=> ( ( F3 @ A4 @ B3 )
= B3 ) ) ) ).
% semilattice_order.absorb4
thf(fact_7610_semilattice__order_OboundedE,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C2: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ ( F3 @ B3 @ C2 ) )
=> ~ ( ( Less_eq @ A4 @ B3 )
=> ~ ( Less_eq @ A4 @ C2 ) ) ) ) ).
% semilattice_order.boundedE
thf(fact_7611_semilattice__order_OboundedI,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C2: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
=> ( ( Less_eq @ A4 @ C2 )
=> ( Less_eq @ A4 @ ( F3 @ B3 @ C2 ) ) ) ) ) ).
% semilattice_order.boundedI
thf(fact_7612_semilattice__order_Oorder__iff,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
= ( A4
= ( F3 @ A4 @ B3 ) ) ) ) ).
% semilattice_order.order_iff
thf(fact_7613_semilattice__order_Ocobounded1,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( Less_eq @ ( F3 @ A4 @ B3 ) @ A4 ) ) ).
% semilattice_order.cobounded1
thf(fact_7614_semilattice__order_Ocobounded2,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( Less_eq @ ( F3 @ A4 @ B3 ) @ B3 ) ) ).
% semilattice_order.cobounded2
thf(fact_7615_semilattice__order_Oabsorb__iff1,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
= ( ( F3 @ A4 @ B3 )
= A4 ) ) ) ).
% semilattice_order.absorb_iff1
thf(fact_7616_semilattice__order_Oabsorb__iff2,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B3: A,A4: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ B3 @ A4 )
= ( ( F3 @ A4 @ B3 )
= B3 ) ) ) ).
% semilattice_order.absorb_iff2
thf(fact_7617_semilattice__order_Obounded__iff,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C2: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ ( F3 @ B3 @ C2 ) )
= ( ( Less_eq @ A4 @ B3 )
& ( Less_eq @ A4 @ C2 ) ) ) ) ).
% semilattice_order.bounded_iff
thf(fact_7618_semilattice__order_OcoboundedI1,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,C2: A,B3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ C2 )
=> ( Less_eq @ ( F3 @ A4 @ B3 ) @ C2 ) ) ) ).
% semilattice_order.coboundedI1
thf(fact_7619_semilattice__order_OcoboundedI2,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B3: A,C2: A,A4: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less_eq @ B3 @ C2 )
=> ( Less_eq @ ( F3 @ A4 @ B3 ) @ C2 ) ) ) ).
% semilattice_order.coboundedI2
thf(fact_7620_semilattice__order_Ostrict__boundedE,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A,C2: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less @ A4 @ ( F3 @ B3 @ C2 ) )
=> ~ ( ( Less @ A4 @ B3 )
=> ~ ( Less @ A4 @ C2 ) ) ) ) ).
% semilattice_order.strict_boundedE
thf(fact_7621_semilattice__order_Ostrict__order__iff,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
= ( ( A4
= ( F3 @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% semilattice_order.strict_order_iff
thf(fact_7622_semilattice__order_Ostrict__coboundedI1,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A4: A,C2: A,B3: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less @ A4 @ C2 )
=> ( Less @ ( F3 @ A4 @ B3 ) @ C2 ) ) ) ).
% semilattice_order.strict_coboundedI1
thf(fact_7623_semilattice__order_Ostrict__coboundedI2,axiom,
! [A: $tType,F3: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B3: A,C2: A,A4: A] :
( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( ( Less @ B3 @ C2 )
=> ( Less @ ( F3 @ A4 @ B3 ) @ C2 ) ) ) ).
% semilattice_order.strict_coboundedI2
thf(fact_7624_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_7625_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_7626_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice_order @ A @ ( sup_sup @ A )
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ) ).
% sup.semilattice_order_axioms
thf(fact_7627_comm__monoid__list__set_Oset__conv__list,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,G: B > A,Xs: list @ B] :
( ( groups4802862169904069756st_set @ A @ F3 @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( set2 @ B @ Xs ) )
= ( groups_monoid_F @ A @ F3 @ Z2 @ ( map @ B @ A @ G @ ( remdups @ B @ Xs ) ) ) ) ) ).
% comm_monoid_list_set.set_conv_list
thf(fact_7628_comm__monoid__list__set_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,Xs: list @ B,G: B > A] :
( ( groups4802862169904069756st_set @ A @ F3 @ Z2 )
=> ( ( distinct @ B @ Xs )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( set2 @ B @ Xs ) )
= ( groups_monoid_F @ A @ F3 @ Z2 @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% comm_monoid_list_set.distinct_set_conv_list
thf(fact_7629_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_7630_comm__monoid__set_Otriangle__reindex,axiom,
! [A: $tType,F3: A > A > A,Z2: A,G: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F3 @ Z2 @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J2 ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2
@ ^ [K4: nat] :
( groups_comm_monoid_F @ A @ nat @ F3 @ Z2
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus @ nat @ K4 @ I4 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.triangle_reindex
thf(fact_7631_comm__monoid__set_Ozero__middle,axiom,
! [A: $tType,F3: A > A > A,Z2: A,P4: nat,K3: nat,G: nat > A,H2: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K3 @ P4 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K3 ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K3 ) @ Z2 @ ( H2 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K3 ) @ ( G @ J2 ) @ ( H2 @ J2 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.zero_middle
thf(fact_7632_comm__monoid__set_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,A3: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X4: B] :
( ( G @ X4 )
= Z2 ) ) ) )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ A3 ) ) ) ) ).
% comm_monoid_set.setdiff_irrelevant
thf(fact_7633_comm__monoid__set_Oempty,axiom,
! [B: $tType,A: $tType,F3: A > A > A,Z2: A,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ) ).
% comm_monoid_set.empty
thf(fact_7634_comm__monoid__set_Oinsert__if,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,A3: set @ B,X3: B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X3 @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ A3 ) ) )
& ( ~ ( member @ B @ X3 @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( F3 @ ( G @ X3 ) @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ A3 ) ) ) ) ) ) ) ).
% comm_monoid_set.insert_if
thf(fact_7635_comm__monoid__set_Oinsert,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,A3: set @ B,X3: B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X3 @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( F3 @ ( G @ X3 ) @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ A3 ) ) ) ) ) ) ).
% comm_monoid_set.insert
thf(fact_7636_comm__monoid__set_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > A > A,Z2: A,S4: set @ B,T6: set @ C,S: set @ B,I: C > B,J: B > C,T4: set @ C,G: B > A,H2: C > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) )
=> ( member @ C @ ( J @ A6 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S @ S4 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ A6 )
= Z2 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( H2 @ B5 )
= Z2 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S )
=> ( ( H2 @ ( J @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ S )
= ( groups_comm_monoid_F @ A @ C @ F3 @ Z2 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.reindex_bij_witness_not_neutral
thf(fact_7637_comm__monoid__set_Ohead__if,axiom,
! [A: $tType,F3: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= Z2 ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% comm_monoid_set.head_if
thf(fact_7638_comm__monoid__set_Oop__ivl__Suc,axiom,
! [A: $tType,F3: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= Z2 ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% comm_monoid_set.op_ivl_Suc
thf(fact_7639_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
! [A: $tType,F3: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( F3 @ ( G @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan
thf(fact_7640_comm__monoid__set_Ocl__ivl__Suc,axiom,
! [A: $tType,F3: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= Z2 ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ nat @ F3 @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% comm_monoid_set.cl_ivl_Suc
thf(fact_7641_comm__monoid__set_Oinsert_H,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,I3: set @ B,P4: B > A,I: B] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I3 )
& ( ( P4 @ X4 )
!= Z2 ) ) ) )
=> ( ( ( member @ B @ I @ I3 )
=> ( ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ P4 @ ( insert3 @ B @ I @ I3 ) )
= ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ P4 @ I3 ) ) )
& ( ~ ( member @ B @ I @ I3 )
=> ( ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ P4 @ ( insert3 @ B @ I @ I3 ) )
= ( F3 @ ( P4 @ I ) @ ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ P4 @ I3 ) ) ) ) ) ) ) ).
% comm_monoid_set.insert'
thf(fact_7642_comm__monoid__set_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,S: set @ B,T4: set @ B,G: B > A,H2: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= Z2 ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ G @ T4 )
= ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ H2 @ S ) ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_cong_right'
thf(fact_7643_comm__monoid__set_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,S: set @ B,T4: set @ B,H2: B > A,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ I5 )
= Z2 ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ G @ S )
= ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ H2 @ T4 ) ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_cong_left'
thf(fact_7644_comm__monoid__set_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,S: set @ B,T4: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= Z2 ) )
=> ( ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ G @ T4 )
= ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ G @ S ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_right'
thf(fact_7645_comm__monoid__set_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,S: set @ B,T4: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= Z2 ) )
=> ( ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ G @ S )
= ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ G @ T4 ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_left'
thf(fact_7646_comm__monoid__set_Oempty_H,axiom,
! [B: $tType,A: $tType,F3: A > A > A,Z2: A,P4: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( groups_comm_monoid_G @ A @ B @ F3 @ Z2 @ P4 @ ( bot_bot @ ( set @ B ) ) )
= Z2 ) ) ).
% comm_monoid_set.empty'
thf(fact_7647_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_7648_comm__monoid__set_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ord @ B )
=> ! [F3: A > A > A,Z2: A,A4: B,C2: B,B3: B,D3: B,G: B > A,H2: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( A4 = C2 )
=> ( ( B3 = D3 )
=> ( ! [X: B] :
( ( ord_less_eq @ B @ C2 @ X )
=> ( ( ord_less @ B @ X @ D3 )
=> ( ( G @ X )
= ( H2 @ X ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( set_or7035219750837199246ssThan @ B @ A4 @ B3 ) )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ) ).
% comm_monoid_set.ivl_cong
thf(fact_7649_comm__monoid__set_Osubset__diff,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,B2: set @ B,A3: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ A3 )
= ( F3 @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ B2 ) ) ) ) ) ) ).
% comm_monoid_set.subset_diff
thf(fact_7650_comm__monoid__set_Osame__carrier,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,C3: set @ B,A3: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A3 ) )
=> ( ( G @ A6 )
= Z2 ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C3 @ B2 ) )
=> ( ( H2 @ B5 )
= Z2 ) )
=> ( ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ A3 )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ H2 @ B2 ) )
= ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ C3 )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ H2 @ C3 ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.same_carrier
thf(fact_7651_comm__monoid__set_Osame__carrierI,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,C3: set @ B,A3: set @ B,B2: set @ B,G: B > A,H2: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B2 @ C3 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C3 @ A3 ) )
=> ( ( G @ A6 )
= Z2 ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C3 @ B2 ) )
=> ( ( H2 @ B5 )
= Z2 ) )
=> ( ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ C3 )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ H2 @ C3 ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ A3 )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ H2 @ B2 ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.same_carrierI
thf(fact_7652_comm__monoid__set_Omono__neutral__left,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,T4: set @ B,S: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= Z2 ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ S )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ T4 ) ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_left
thf(fact_7653_comm__monoid__set_Omono__neutral__right,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,T4: set @ B,S: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= Z2 ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ T4 )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ S ) ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_right
thf(fact_7654_comm__monoid__set_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,T4: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ X )
= Z2 ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ S )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ H2 @ T4 ) ) ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_cong_left
thf(fact_7655_comm__monoid__set_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,T4: set @ B,S: set @ B,G: B > A,H2: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T4 )
=> ( ! [X: B] :
( ( member @ B @ X @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( G @ X )
= Z2 ) )
=> ( ! [X: B] :
( ( member @ B @ X @ S )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ T4 )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ H2 @ S ) ) ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_cong_right
thf(fact_7656_comm__monoid__set_OInt__Diff,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,A3: set @ B,G: B > A,B2: set @ B] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ A3 )
= ( F3 @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) ) ) ) ) ).
% comm_monoid_set.Int_Diff
thf(fact_7657_comm__monoid__set_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,T4: set @ B,S: set @ B,H2: B > A,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ ( minus_minus @ ( set @ B ) @ T4 @ S ) )
=> ( ( H2 @ I5 )
= Z2 ) )
=> ( ! [I5: B] :
( ( member @ B @ I5 @ ( minus_minus @ ( set @ B ) @ S @ T4 ) )
=> ( ( G @ I5 )
= Z2 ) )
=> ( ! [X: B] :
( ( member @ B @ X @ ( inf_inf @ ( set @ B ) @ S @ T4 ) )
=> ( ( G @ X )
= ( H2 @ X ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ S )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ H2 @ T4 ) ) ) ) ) ) ) ) ).
% comm_monoid_set.mono_neutral_cong
thf(fact_7658_comm__monoid__set_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > A > A,Z2: A,S4: set @ B,T6: set @ C,H2: B > C,S: set @ B,T4: set @ C,G: C > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ S4 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S @ S4 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ ( H2 @ A6 ) )
= Z2 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T6 )
=> ( ( G @ B5 )
= Z2 ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2
@ ^ [X4: B] : ( G @ ( H2 @ X4 ) )
@ S )
= ( groups_comm_monoid_F @ A @ C @ F3 @ Z2 @ G @ T4 ) ) ) ) ) ) ) ) ).
% comm_monoid_set.reindex_bij_betw_not_neutral
thf(fact_7659_comm__monoid__set_Oremove,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,A3: set @ B,X3: B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X3 @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ A3 )
= ( F3 @ ( G @ X3 ) @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.remove
thf(fact_7660_comm__monoid__set_Oinsert__remove,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,A3: set @ B,G: B > A,X3: B] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( insert3 @ B @ X3 @ A3 ) )
= ( F3 @ ( G @ X3 ) @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert3 @ B @ X3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.insert_remove
thf(fact_7661_comm__monoid__set_Ounion__disjoint,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,A3: set @ B,B2: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B2 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( F3 @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ A3 ) @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ B2 ) ) ) ) ) ) ) ).
% comm_monoid_set.union_disjoint
thf(fact_7662_comm__monoid__set_Ounion__diff2,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,A3: set @ B,B2: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B2 )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B2 ) )
= ( F3 @ ( F3 @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B2 ) ) @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( minus_minus @ ( set @ B ) @ B2 @ A3 ) ) ) @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B2 ) ) ) ) ) ) ) ).
% comm_monoid_set.union_diff2
thf(fact_7663_comm__monoid__set_Odelta__remove,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,S: set @ B,A4: B,B3: B > A,C2: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A4 @ S )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C2 @ K4 ) )
@ S )
= ( F3 @ ( B3 @ A4 ) @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A4 @ S )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2
@ ^ [K4: B] : ( if @ A @ ( K4 = A4 ) @ ( B3 @ K4 ) @ ( C2 @ K4 ) )
@ S )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert3 @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.delta_remove
thf(fact_7664_comm__monoid__set_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > A > A,Z2: A,I3: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ( finite_finite2 @ B @ I3 )
=> ( ! [X: B] :
( ( member @ B @ X @ I3 )
=> ( finite_finite2 @ C @ ( A3 @ X ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ I3 )
=> ! [Xa4: B] :
( ( member @ B @ Xa4 @ I3 )
=> ( ( X != Xa4 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X ) @ ( A3 @ Xa4 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ C @ F3 @ Z2 @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A3 @ I3 ) ) )
= ( groups_comm_monoid_F @ A @ B @ F3 @ Z2
@ ^ [X4: B] : ( groups_comm_monoid_F @ A @ C @ F3 @ Z2 @ G @ ( A3 @ X4 ) )
@ I3 ) ) ) ) ) ) ).
% comm_monoid_set.UNION_disjoint
thf(fact_7665_comm__monoid__set_OUnion__disjoint,axiom,
! [A: $tType,B: $tType,F3: A > A > A,Z2: A,C3: set @ ( set @ B ),G: B > A] :
( ( groups778175481326437816id_set @ A @ F3 @ Z2 )
=> ( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C3 )
=> ( finite_finite2 @ B @ X ) )
=> ( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C3 )
=> ! [Xa4: set @ B] :
( ( member @ ( set @ B ) @ Xa4 @ C3 )
=> ( ( X != Xa4 )
=> ( ( inf_inf @ ( set @ B ) @ X @ Xa4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C3 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups_comm_monoid_F @ A @ ( set @ B ) @ F3 @ Z2 ) @ ( groups_comm_monoid_F @ A @ B @ F3 @ Z2 ) @ G @ C3 ) ) ) ) ) ).
% comm_monoid_set.Union_disjoint
thf(fact_7666_semilattice__neutr__order_Ointro,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_neutr @ A @ F3 @ Z2 )
=> ( ( semilattice_order @ A @ F3 @ Less_eq @ Less )
=> ( semila1105856199041335345_order @ A @ F3 @ Z2 @ Less_eq @ Less ) ) ) ).
% semilattice_neutr_order.intro
thf(fact_7667_semilattice__neutr__order__def,axiom,
! [A: $tType] :
( ( semila1105856199041335345_order @ A )
= ( ^ [F7: A > A > A,Z6: A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semilattice_neutr @ A @ F7 @ Z6 )
& ( semilattice_order @ A @ F7 @ Less_eq2 @ Less2 ) ) ) ) ).
% semilattice_neutr_order_def
thf(fact_7668_semilattice__neutr__order_Oaxioms_I1_J,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F3 @ Z2 @ Less_eq @ Less )
=> ( semilattice_neutr @ A @ F3 @ Z2 ) ) ).
% semilattice_neutr_order.axioms(1)
thf(fact_7669_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_7670_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_7671_subset__eq__mset__impl_Opelims,axiom,
! [A: $tType,X3: list @ A,Xa: list @ A,Y: option @ $o] :
( ( ( subset_eq_mset_impl @ A @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Xa ) )
=> ( ( ( X3
= ( 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 ) ) ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ 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 )
@ ^ [Y5: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y4: A,Z4: A] : Y4 = Z4
@ X )
@ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% subset_eq_mset_impl.pelims
thf(fact_7672_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_7673_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp_eq @ A @ A1 @ A22 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys5: list @ A] :
( A22
= ( cons @ A @ Y2 @ Ys5 ) )
=> ~ ( ord_less @ A @ X @ Y2 ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys5: list @ A] :
( ( A22
= ( cons @ A @ Y2 @ Ys5 ) )
=> ( ~ ( ord_less @ A @ X @ Y2 )
=> ( ~ ( ord_less @ A @ Y2 @ X )
=> ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys5 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_7674_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A13: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A13
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A13
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ord_less @ A @ X4 @ Y5 ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A13
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( ord_less @ A @ X4 @ Y5 )
& ~ ( ord_less @ A @ Y5 @ X4 )
& ( ord_lexordp_eq @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_7675_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_7676_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_7677_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( ord_less @ A @ X3 @ Y )
| ( ~ ( ord_less @ A @ Y @ X3 )
& ( ord_lexordp_eq @ A @ Xs @ Ys ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_7678_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A] :
~ ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ) ).
% lexordp_eq_simps(3)
thf(fact_7679_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq.Nil
thf(fact_7680_lexordp__eq__pref,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [U: list @ A,V2: list @ A] : ( ord_lexordp_eq @ A @ U @ ( append @ A @ U @ V2 ) ) ) ).
% lexordp_eq_pref
thf(fact_7681_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_7682_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_7683_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_7684_lexordp__eq__refl,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% lexordp_eq_refl
thf(fact_7685_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_7686_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_7687_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_7688_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_7689_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_7690_num__of__nat__inverse,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nat_of_num @ ( num_of_nat @ N ) )
= N ) ) ).
% num_of_nat_inverse
thf(fact_7691_numeral__sqr,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K3: num] :
( ( numeral_numeral @ A @ ( sqr @ K3 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K3 ) @ ( numeral_numeral @ A @ K3 ) ) ) ) ).
% numeral_sqr
thf(fact_7692_nat__of__num__pos,axiom,
! [X3: num] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat_of_num @ X3 ) ) ).
% nat_of_num_pos
thf(fact_7693_less__num__def,axiom,
( ( ord_less @ num )
= ( ^ [M: num,N2: num] : ( ord_less @ nat @ ( nat_of_num @ M ) @ ( nat_of_num @ N2 ) ) ) ) ).
% less_num_def
thf(fact_7694_bdd__below_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit622319405099724424ng_bdd @ A
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_7695_natural__decr,axiom,
! [N: code_natural] :
( ( N
!= ( zero_zero @ code_natural ) )
=> ( ord_less @ nat @ ( minus_minus @ nat @ ( code_nat_of_natural @ N ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% natural_decr
thf(fact_7696_less__natural_Orep__eq,axiom,
( ( ord_less @ code_natural )
= ( ^ [X4: code_natural,Xa5: code_natural] : ( ord_less @ nat @ ( code_nat_of_natural @ X4 ) @ ( code_nat_of_natural @ Xa5 ) ) ) ) ).
% less_natural.rep_eq
thf(fact_7697_bdd__above_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit622319405099724424ng_bdd @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% bdd_above.preordering_bdd_axioms
thf(fact_7698_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P2: C > A > $o,Q2: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P2 @ Q2 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_7699_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P2: A > B > $o,Q2: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q2 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_7700_CODE__ABORT__def,axiom,
! [A: $tType] :
( ( cODE_ABORT @ A )
= ( ^ [F7: product_unit > A] : ( F7 @ product_Unity ) ) ) ).
% CODE_ABORT_def
thf(fact_7701_abstract__filter__def,axiom,
! [A: $tType] :
( ( abstract_filter @ A )
= ( ^ [F7: product_unit > ( filter @ A )] : ( F7 @ product_Unity ) ) ) ).
% abstract_filter_def
thf(fact_7702_next_Osimps,axiom,
! [V2: code_natural,W2: code_natural] :
( ( next @ ( product_Pair @ code_natural @ code_natural @ V2 @ W2 ) )
= ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( one_one @ code_natural ) ) ) @ ( one_one @ code_natural ) ) @ ( product_Pair @ code_natural @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% next.simps
thf(fact_7703_select__weight__member,axiom,
! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A ),S3: product_prod @ code_natural @ code_natural] :
( ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs ) ) )
=> ( member @ A @ ( product_fst @ A @ ( product_prod @ code_natural @ code_natural ) @ ( select_weight @ A @ Xs @ S3 ) ) @ ( set2 @ A @ ( map @ ( product_prod @ code_natural @ A ) @ A @ ( product_snd @ code_natural @ A ) @ Xs ) ) ) ) ).
% select_weight_member
thf(fact_7704_full__exhaustive__natural_H_Ocases,axiom,
! [X3: product_prod @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F5: ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D4: code_natural,I5: code_natural] :
( X3
!= ( product_Pair @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F5 @ ( product_Pair @ code_natural @ code_natural @ D4 @ I5 ) ) ) ).
% full_exhaustive_natural'.cases
thf(fact_7705_full__exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( ( quickc3360725361186068524ustive @ B )
& ( cl_HOL_Oequal @ A )
& ( quickc3360725361186068524ustive @ A ) )
=> ! [X3: product_prod @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F5: ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I5: code_natural,D4: code_natural] :
( X3
!= ( product_Pair @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F5 @ ( product_Pair @ code_natural @ code_natural @ I5 @ D4 ) ) ) ) ).
% full_exhaustive_fun'.cases
thf(fact_7706_exhaustive__natural_H_Ocases,axiom,
! [X3: product_prod @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F5: code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D4: code_natural,I5: code_natural] :
( X3
!= ( product_Pair @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F5 @ ( product_Pair @ code_natural @ code_natural @ D4 @ I5 ) ) ) ).
% exhaustive_natural'.cases
thf(fact_7707_exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( ( quickc658316121487927005ustive @ B )
& ( cl_HOL_Oequal @ A )
& ( quickc658316121487927005ustive @ A ) )
=> ! [X3: product_prod @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F5: ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I5: code_natural,D4: code_natural] :
( X3
!= ( product_Pair @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F5 @ ( product_Pair @ code_natural @ code_natural @ I5 @ D4 ) ) ) ) ).
% exhaustive_fun'.cases
thf(fact_7708_iter_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )] :
~ ! [Random: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ),Nrandom: code_natural,Seed: product_prod @ code_natural @ code_natural] :
( X3
!= ( product_Pair @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ Random @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ Nrandom @ Seed ) ) ) ).
% iter.cases
thf(fact_7709_log_Ocases,axiom,
! [X3: product_prod @ code_natural @ code_natural] :
~ ! [B5: code_natural,I5: code_natural] :
( X3
!= ( product_Pair @ code_natural @ code_natural @ B5 @ I5 ) ) ).
% log.cases
thf(fact_7710_select__weight__cons__zero,axiom,
! [A: $tType,X3: A,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( select_weight @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( zero_zero @ code_natural ) @ X3 ) @ Xs ) )
= ( select_weight @ A @ Xs ) ) ).
% select_weight_cons_zero
thf(fact_7711_pick__member,axiom,
! [A: $tType,I: code_natural,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( ord_less @ code_natural @ I @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs ) ) )
=> ( member @ A @ ( pick @ A @ Xs @ I ) @ ( set2 @ A @ ( map @ ( product_prod @ code_natural @ A ) @ A @ ( product_snd @ code_natural @ A ) @ Xs ) ) ) ) ).
% pick_member
thf(fact_7712_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_7713_select,axiom,
! [A: $tType,Xs: list @ A,S3: product_prod @ code_natural @ code_natural] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( product_fst @ A @ ( product_prod @ code_natural @ code_natural ) @ ( select @ A @ Xs @ S3 ) ) @ ( set2 @ A @ Xs ) ) ) ).
% select
thf(fact_7714_split__seed__def,axiom,
( split_seed
= ( ^ [S7: product_prod @ code_natural @ code_natural] :
( product_case_prod @ code_natural @ code_natural @ ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [V4: code_natural,W3: code_natural] :
( product_case_prod @ code_natural @ code_natural @ ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [V5: code_natural,W4: code_natural] : ( product_Pair @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_Pair @ code_natural @ code_natural @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V4 ) @ W4 ) @ ( product_Pair @ code_natural @ code_natural @ V5 @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W3 ) ) )
@ ( product_snd @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( next @ S7 ) ) )
@ S7 ) ) ) ).
% split_seed_def
thf(fact_7715_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F5: code_natural > A,N5: code_natural,M6: code_natural] :
( X3
!= ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F5 @ ( product_Pair @ code_natural @ code_natural @ N5 @ M6 ) ) ) ).
% Lazy_Sequence.iterate_upto.cases
thf(fact_7716_small__lazy_H_Ocases,axiom,
! [X3: product_prod @ int @ int] :
~ ! [D4: int,I5: int] :
( X3
!= ( product_Pair @ int @ int @ D4 @ I5 ) ) ).
% small_lazy'.cases
thf(fact_7717_select__weight__def,axiom,
! [A: $tType] :
( ( select_weight @ A )
= ( ^ [Xs3: list @ ( product_prod @ code_natural @ A )] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) @ ( range @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs3 ) ) )
@ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( pick @ A @ Xs3 @ K4 ) ) ) ) ) ).
% select_weight_def
thf(fact_7718_Predicate_Oiterate__upto_Opinduct,axiom,
! [A: $tType,A0: code_natural > A,A1: code_natural,A22: code_natural,P: ( code_natural > A ) > code_natural > code_natural > $o] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ A0 @ ( product_Pair @ code_natural @ code_natural @ A1 @ A22 ) ) )
=> ( ! [F5: code_natural > A,N5: code_natural,M6: code_natural] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F5 @ ( product_Pair @ code_natural @ code_natural @ N5 @ M6 ) ) )
=> ( ! [X8: product_unit] :
( ~ ( ord_less @ code_natural @ M6 @ N5 )
=> ( P @ F5 @ ( plus_plus @ code_natural @ N5 @ ( one_one @ code_natural ) ) @ M6 ) )
=> ( P @ F5 @ N5 @ M6 ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% Predicate.iterate_upto.pinduct
thf(fact_7719_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X3: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C ) )
= X3 ) ).
% scomp_Pair
thf(fact_7720_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X3: C,F3: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X3 ) @ F3 )
= ( F3 @ X3 ) ) ).
% Pair_scomp
thf(fact_7721_Random_Orange__def,axiom,
( range
= ( ^ [K4: code_natural] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )
@ ( iterate @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( log @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K4 )
@ ^ [L4: code_natural] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ next
@ ^ [V4: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ V4 @ ( times_times @ code_natural @ L4 @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ code_natural ) )
@ ^ [V4: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( modulo_modulo @ code_natural @ V4 @ K4 ) ) ) ) ) ).
% Random.range_def
thf(fact_7722_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X3: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( X3
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X3
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X3 @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) ) ) ).
% iterate.elims
thf(fact_7723_iterate_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( iterate @ B @ A )
= ( ^ [K4: code_natural,F7: B > A > ( product_prod @ B @ A ),X4: B] :
( if @ ( A > ( product_prod @ B @ A ) )
@ ( K4
= ( zero_zero @ code_natural ) )
@ ( product_Pair @ B @ A @ X4 )
@ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F7 @ X4 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K4 @ ( one_one @ code_natural ) ) @ F7 ) ) ) ) ) ).
% iterate.simps
thf(fact_7724_select__def,axiom,
! [A: $tType] :
( ( select @ A )
= ( ^ [Xs3: list @ A] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) @ ( range @ ( code_natural_of_nat @ ( size_size @ ( list @ A ) @ Xs3 ) ) )
@ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth @ A @ Xs3 @ ( code_nat_of_natural @ K4 ) ) ) ) ) ) ).
% select_def
thf(fact_7725_pick__same,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( pick @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) @ ( code_natural_of_nat @ L ) )
= ( nth @ A @ Xs @ L ) ) ) ).
% pick_same
thf(fact_7726_less__natural_Oabs__eq,axiom,
! [Xa: nat,X3: nat] :
( ( ord_less @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X3 ) )
= ( ord_less @ nat @ Xa @ X3 ) ) ).
% less_natural.abs_eq
thf(fact_7727_log_Opelims,axiom,
! [X3: code_natural,Xa: code_natural,Y: code_natural] :
( ( ( log @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X3 @ Xa ) )
=> ~ ( ( ( ( ( ord_less_eq @ code_natural @ X3 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X3 ) )
=> ( Y
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X3 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X3 ) )
=> ( Y
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X3 @ ( divide_divide @ code_natural @ Xa @ X3 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X3 @ Xa ) ) ) ) ) ).
% log.pelims
thf(fact_7728_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X3: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X3 @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X3 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) )
=> ~ ( ( ( ( X3
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X3
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X3 @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X3 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) ) ) ) ) ).
% iterate.pelims
thf(fact_7729_iterate_Ocases,axiom,
! [A: $tType,B: $tType,X3: product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B )] :
~ ! [K: code_natural,F5: B > A > ( product_prod @ B @ A ),X: B] :
( X3
!= ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ K @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ F5 @ X ) ) ) ).
% iterate.cases
thf(fact_7730_iter_H_Ocases,axiom,
! [A: $tType] :
( ( quickcheck_random @ A )
=> ! [X3: product_prod @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) )] :
~ ! [T5: itself @ A,Nrandom: code_natural,Sz: code_natural,Seed: product_prod @ code_natural @ code_natural] :
( X3
!= ( product_Pair @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ) @ T5 @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ Nrandom @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ Sz @ Seed ) ) ) ) ) ).
% iter'.cases
thf(fact_7731_in__range_Osimps,axiom,
! [H2: heap_ext @ product_unit,As2: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ As2 )
=> ( ord_less @ nat @ X4 @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% in_range.simps
thf(fact_7732_in__range_Oelims_I3_J,axiom,
! [X3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( in_range @ X3 )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( X3
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ! [X: nat] :
( ( member @ nat @ X @ As )
=> ( ord_less @ nat @ X @ ( lim @ product_unit @ H ) ) ) ) ) ).
% in_range.elims(3)
thf(fact_7733_in__range_Oelims_I2_J,axiom,
! [X3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ X3 )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( X3
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ~ ! [X8: nat] :
( ( member @ nat @ X8 @ As )
=> ( ord_less @ nat @ X8 @ ( lim @ product_unit @ H ) ) ) ) ) ).
% in_range.elims(2)
thf(fact_7734_in__range_Oelims_I1_J,axiom,
! [X3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( in_range @ X3 )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( X3
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( Y
= ( ~ ! [X4: nat] :
( ( member @ nat @ X4 @ As )
=> ( ord_less @ nat @ X4 @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ).
% in_range.elims(1)
thf(fact_7735_new__addrs__def,axiom,
( hoare_new_addrs
= ( ^ [H3: heap_ext @ product_unit,As4: set @ nat,H8: heap_ext @ product_unit] :
( sup_sup @ ( set @ nat ) @ As4
@ ( collect @ nat
@ ^ [A5: nat] :
( ( ord_less_eq @ nat @ ( lim @ product_unit @ H3 ) @ A5 )
& ( ord_less @ nat @ A5 @ ( lim @ product_unit @ H8 ) ) ) ) ) ) ) ).
% new_addrs_def
thf(fact_7736_sngr__assn__raw_Osimps,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,X3: A,H2: heap_ext @ product_unit,As2: set @ nat] :
( ( sngr_assn_raw @ A @ R2 @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
= ( ( ( ref_get @ A @ H2 @ R2 )
= X3 )
& ( As2
= ( insert3 @ nat @ ( addr_of_ref @ A @ R2 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ R2 ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% sngr_assn_raw.simps
thf(fact_7737_relH__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [As2: set @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R2: ref @ A] :
( ( relH @ As2 @ H2 @ H4 )
=> ( ( member @ nat @ ( addr_of_ref @ A @ R2 ) @ As2 )
=> ( ( ref_get @ A @ H2 @ R2 )
= ( ref_get @ A @ H4 @ R2 ) ) ) ) ) ).
% relH_ref
thf(fact_7738_sngr__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( sngr_assn_raw @ A @ X3 @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ( ( ref_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_ref @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ).
% sngr_assn_raw.elims(3)
thf(fact_7739_sngr__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( sngr_assn_raw @ A @ X3 @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ~ ( ( ( ref_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_ref @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ).
% sngr_assn_raw.elims(2)
thf(fact_7740_sngr__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( sngr_assn_raw @ A @ X3 @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( Y
= ( ~ ( ( ( ref_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_ref @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.elims(1)
thf(fact_7741_sngr__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( sngr_assn_raw @ A @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ 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 ) ) ) @ X3 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) )
=> ( ( ( ref_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_ref @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(3)
thf(fact_7742_sngr__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( sngr_assn_raw @ A @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ 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 ) ) ) @ X3 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) )
=> ~ ( ( ( ref_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_ref @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(2)
thf(fact_7743_sngr__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( sngr_assn_raw @ A @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ( Y
= ( ( ( ref_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_ref @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X3 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(1)
thf(fact_7744_relH__set__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,As2: set @ nat,H2: heap_ext @ product_unit,X3: A] :
( ~ ( member @ nat @ ( addr_of_ref @ A @ R2 ) @ As2 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ( relH @ As2 @ H2 @ ( ref_set @ A @ R2 @ X3 @ H2 ) ) ) ) ) ).
% relH_set_ref
thf(fact_7745_nth__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,Xs: list @ A,A4: array @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( hoare_hoare_triple @ A @ ( snga_assn @ A @ A4 @ Xs ) @ ( array_nth @ A @ A4 @ I )
@ ^ [R5: A] :
( times_times @ assn @ ( snga_assn @ A @ A4 @ Xs )
@ ( pure_assn
@ ( R5
= ( nth @ A @ Xs @ I ) ) ) ) ) ) ) ).
% nth_rule
thf(fact_7746_false__rule,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,Q: A > assn] : ( hoare_hoare_triple @ A @ ( bot_bot @ assn ) @ C2 @ Q ) ).
% false_rule
thf(fact_7747_norm__pre__pure__iff,axiom,
! [A: $tType,P: assn,B3: $o,F3: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ F3 @ Q )
= ( B3
=> ( hoare_hoare_triple @ A @ P @ F3 @ Q ) ) ) ).
% norm_pre_pure_iff
thf(fact_7748_norm__pre__pure__iff__sng,axiom,
! [A: $tType,B3: $o,F3: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ ( pure_assn @ B3 ) @ F3 @ Q )
= ( B3
=> ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F3 @ Q ) ) ) ).
% norm_pre_pure_iff_sng
thf(fact_7749_is__hoare__triple,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).
% is_hoare_triple
thf(fact_7750_frame__rule__left,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,R: assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ R @ P ) @ C2
@ ^ [X4: A] : ( times_times @ assn @ R @ ( Q @ X4 ) ) ) ) ).
% frame_rule_left
thf(fact_7751_fi__rule,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,Ps: assn,F: assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( entails @ Ps @ ( times_times @ assn @ P @ F ) )
=> ( hoare_hoare_triple @ A @ Ps @ C2
@ ^ [X4: A] : ( times_times @ assn @ ( Q @ X4 ) @ F ) ) ) ) ).
% fi_rule
thf(fact_7752_frame__rule,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,R: assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ R ) @ C2
@ ^ [X4: A] : ( times_times @ assn @ ( Q @ X4 ) @ R ) ) ) ).
% frame_rule
thf(fact_7753_norm__pre__pure__rule1,axiom,
! [A: $tType,B3: $o,P: assn,F3: heap_Time_Heap @ A,Q: A > assn] :
( ( B3
=> ( hoare_hoare_triple @ A @ P @ F3 @ Q ) )
=> ( hoare_hoare_triple @ A @ ( times_times @ assn @ P @ ( pure_assn @ B3 ) ) @ F3 @ Q ) ) ).
% norm_pre_pure_rule1
thf(fact_7754_cons__post__rule,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,Q3: A > assn] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ! [X: A] : ( entails @ ( Q @ X ) @ ( Q3 @ X ) )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q3 ) ) ) ).
% cons_post_rule
thf(fact_7755_cons__rule,axiom,
! [A: $tType,P: assn,P3: assn,Q: A > assn,Q3: A > assn,C2: heap_Time_Heap @ A] :
( ( entails @ P @ P3 )
=> ( ! [X: A] : ( entails @ ( Q @ X ) @ ( Q3 @ X ) )
=> ( ( hoare_hoare_triple @ A @ P3 @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q3 ) ) ) ) ).
% cons_rule
thf(fact_7756_hoare__triple__preI,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).
% hoare_triple_preI
thf(fact_7757_cons__pre__rule,axiom,
! [A: $tType,P: assn,P3: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
( ( entails @ P @ P3 )
=> ( ( hoare_hoare_triple @ A @ P3 @ C2 @ Q )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ) ).
% cons_pre_rule
thf(fact_7758_norm__pre__pure__rule2,axiom,
! [A: $tType,B3: $o,F3: heap_Time_Heap @ A,Q: A > assn] :
( ( B3
=> ( hoare_hoare_triple @ A @ ( one_one @ assn ) @ F3 @ Q ) )
=> ( hoare_hoare_triple @ A @ ( pure_assn @ B3 ) @ F3 @ Q ) ) ).
% norm_pre_pure_rule2
thf(fact_7759_case__prod__rule,axiom,
! [A: $tType,B: $tType,C: $tType,X3: product_prod @ A @ B,P: assn,F3: A > B > ( heap_Time_Heap @ C ),Q: C > assn] :
( ! [A6: A,B5: B] :
( ( X3
= ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( hoare_hoare_triple @ C @ P @ ( F3 @ A6 @ B5 ) @ Q ) )
=> ( hoare_hoare_triple @ C @ P @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F3 @ X3 ) @ Q ) ) ).
% case_prod_rule
thf(fact_7760_cons__rulet,axiom,
! [A: $tType,P: assn,P3: assn,Q: A > assn,Q3: A > assn,C2: heap_Time_Heap @ A] :
( ( entailst @ P @ P3 )
=> ( ! [X: A] : ( entailst @ ( Q @ X ) @ ( Q3 @ X ) )
=> ( ( hoare_hoare_triple @ A @ P3 @ C2
@ ^ [R5: A] : ( times_times @ assn @ ( Q @ R5 ) @ ( top_top @ assn ) ) )
=> ( hoare_hoare_triple @ A @ P @ C2
@ ^ [R5: A] : ( times_times @ assn @ ( Q3 @ R5 ) @ ( top_top @ assn ) ) ) ) ) ) ).
% cons_rulet
thf(fact_7761_cons__pre__rulet,axiom,
! [A: $tType,P: assn,P3: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
( ( entailst @ P @ P3 )
=> ( ( hoare_hoare_triple @ A @ P3 @ C2
@ ^ [R5: A] : ( times_times @ assn @ ( Q @ R5 ) @ ( top_top @ assn ) ) )
=> ( hoare_hoare_triple @ A @ P @ C2
@ ^ [R5: A] : ( times_times @ assn @ ( Q @ R5 ) @ ( top_top @ assn ) ) ) ) ) ).
% cons_pre_rulet
thf(fact_7762_cons__post__rulet,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,Q3: A > assn] :
( ( hoare_hoare_triple @ A @ P @ C2
@ ^ [R5: A] : ( times_times @ assn @ ( Q @ R5 ) @ ( top_top @ assn ) ) )
=> ( ! [X: A] : ( entailst @ ( Q @ X ) @ ( Q3 @ X ) )
=> ( hoare_hoare_triple @ A @ P @ C2
@ ^ [R5: A] : ( times_times @ assn @ ( Q3 @ R5 ) @ ( top_top @ assn ) ) ) ) ) ).
% cons_post_rulet
thf(fact_7763_if__rule__split,axiom,
! [A: $tType,B3: $o,P: assn,F3: heap_Time_Heap @ A,Q1: A > assn,G: heap_Time_Heap @ A,Q22: A > assn,Q: A > assn] :
( ( B3
=> ( hoare_hoare_triple @ A @ P @ F3 @ Q1 ) )
=> ( ( ~ B3
=> ( hoare_hoare_triple @ A @ P @ G @ Q22 ) )
=> ( ! [X: A] : ( entails @ ( sup_sup @ assn @ ( times_times @ assn @ ( Q1 @ X ) @ ( pure_assn @ B3 ) ) @ ( times_times @ assn @ ( Q22 @ X ) @ ( pure_assn @ ~ B3 ) ) ) @ ( Q @ X ) )
=> ( hoare_hoare_triple @ A @ P @ ( if @ ( heap_Time_Heap @ A ) @ B3 @ F3 @ G ) @ Q ) ) ) ) ).
% if_rule_split
thf(fact_7764_upd__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,Xs: list @ A,A4: array @ A,X3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( hoare_hoare_triple @ ( array @ A ) @ ( snga_assn @ A @ A4 @ Xs ) @ ( array_upd @ A @ I @ X3 @ A4 )
@ ^ [R5: array @ A] : ( times_times @ assn @ ( snga_assn @ A @ A4 @ ( list_update @ A @ Xs @ I @ X3 ) ) @ ( pure_assn @ ( R5 = A4 ) ) ) ) ) ) ).
% upd_rule
thf(fact_7765_make__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F3: nat > A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_make @ A @ N @ F3 )
@ ^ [R5: array @ A] : ( snga_assn @ A @ R5 @ ( map @ nat @ A @ F3 @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% make_rule
thf(fact_7766_new__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X3: A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_new @ A @ N @ X3 )
@ ^ [R5: array @ A] : ( snga_assn @ A @ R5 @ ( replicate @ A @ N @ X3 ) ) ) ) ).
% new_rule
thf(fact_7767_length__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,Xs: list @ A] :
( hoare_hoare_triple @ nat @ ( snga_assn @ A @ A4 @ Xs ) @ ( array_len @ A @ A4 )
@ ^ [R5: nat] :
( times_times @ assn @ ( snga_assn @ A @ A4 @ Xs )
@ ( pure_assn
@ ( R5
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% length_rule
thf(fact_7768_update__wp__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,Y: A,X3: A,Q: product_unit > assn] : ( hoare_hoare_triple @ product_unit @ ( times_times @ assn @ ( sngr_assn @ A @ R2 @ Y ) @ ( wand_assn @ ( sngr_assn @ A @ R2 @ X3 ) @ ( Q @ product_Unity ) ) ) @ ( ref_update @ A @ R2 @ X3 ) @ Q ) ) ).
% update_wp_rule
thf(fact_7769_lookup__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: ref @ A,X3: A] :
( hoare_hoare_triple @ A @ ( sngr_assn @ A @ P4 @ X3 ) @ ( ref_lookup @ A @ P4 )
@ ^ [R5: A] : ( times_times @ assn @ ( sngr_assn @ A @ P4 @ X3 ) @ ( pure_assn @ ( R5 = X3 ) ) ) ) ) ).
% lookup_rule
thf(fact_7770_update__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P4: ref @ A,Y: A,X3: A] :
( hoare_hoare_triple @ product_unit @ ( sngr_assn @ A @ P4 @ Y ) @ ( ref_update @ A @ P4 @ X3 )
@ ^ [R5: product_unit] : ( sngr_assn @ A @ P4 @ X3 ) ) ) ).
% update_rule
thf(fact_7771_hoare__triple__def,axiom,
! [A: $tType] :
( ( hoare_hoare_triple @ A )
= ( ^ [P2: assn,C6: heap_Time_Heap @ A,Q2: A > assn] :
! [H3: heap_ext @ product_unit,As4: set @ nat] :
( ( rep_assn @ P2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As4 ) )
=> ? [R5: A,H8: heap_ext @ product_unit] :
( ? [T3: nat] :
( ( heap_Time_execute @ A @ C6 @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ T3 ) ) ) )
& ( rep_assn @ ( Q2 @ R5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ ( hoare_new_addrs @ H3 @ As4 @ H8 ) ) )
& ( relH
@ ( collect @ nat
@ ^ [A5: nat] :
( ( ord_less @ nat @ A5 @ ( lim @ product_unit @ H3 ) )
& ~ ( member @ nat @ A5 @ As4 ) ) )
@ H3
@ H8 )
& ( ord_less_eq @ nat @ ( lim @ product_unit @ H3 ) @ ( lim @ product_unit @ H8 ) ) ) ) ) ) ).
% hoare_triple_def
thf(fact_7772_hoare__tripleI,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
( ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ? [R14: A,H9: heap_ext @ product_unit,T8: nat] :
( ( ( heap_Time_execute @ A @ C2 @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R14 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ T8 ) ) ) )
& ( rep_assn @ ( Q @ R14 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( hoare_new_addrs @ H @ As @ H9 ) ) )
& ( relH
@ ( collect @ nat
@ ^ [A5: nat] :
( ( ord_less @ nat @ A5 @ ( lim @ product_unit @ H ) )
& ~ ( member @ nat @ A5 @ As ) ) )
@ H
@ H9 )
& ( ord_less_eq @ nat @ ( lim @ product_unit @ H ) @ ( lim @ product_unit @ H9 ) ) ) )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).
% hoare_tripleI
thf(fact_7773_Heap__cases,axiom,
! [A: $tType,F3: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
( ! [X: A,H5: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F3 @ H2 )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ H5 ) ) )
=> ( ( heap_Time_execute @ A @ F3 @ H2 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% Heap_cases
thf(fact_7774_timeFrame_Ocases,axiom,
! [A: $tType,X3: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ! [N5: nat,R3: A,H: heap_ext @ product_unit,N4: nat] :
( X3
!= ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N5 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ N4 ) ) ) ) )
=> ~ ! [N5: nat] :
( X3
!= ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N5 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% timeFrame.cases
thf(fact_7775_hoare__tripleE,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H2: heap_ext @ product_unit,As2: set @ nat] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ~ ! [R3: A,H5: heap_ext @ product_unit] :
( ? [T9: nat] :
( ( heap_Time_execute @ A @ C2 @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ T9 ) ) ) )
=> ( ( rep_assn @ ( Q @ R3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( hoare_new_addrs @ H2 @ As2 @ H5 ) ) )
=> ( ( relH
@ ( collect @ nat
@ ^ [A5: nat] :
( ( ord_less @ nat @ A5 @ ( lim @ product_unit @ H2 ) )
& ~ ( member @ nat @ A5 @ As2 ) ) )
@ H2
@ H5 )
=> ~ ( ord_less_eq @ nat @ ( lim @ product_unit @ H2 ) @ ( lim @ product_unit @ H5 ) ) ) ) ) ) ) ).
% hoare_tripleE
thf(fact_7776_execute__lookup,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( ref_lookup @ A @ R2 ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( ref_get @ A @ H2 @ R2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_lookup
thf(fact_7777_execute__update,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,V2: A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ product_unit @ ( ref_update @ A @ R2 @ V2 ) @ H2 )
= ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R2 @ V2 @ H2 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_update
thf(fact_7778_execute__change,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [F3: A > A,R2: ref @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( ref_change @ A @ F3 @ R2 ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F3 @ ( ref_get @ A @ H2 @ R2 ) ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R2 @ ( F3 @ ( ref_get @ A @ H2 @ R2 ) ) @ H2 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% execute_change
thf(fact_7779_Ref__Time_Oupdate__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_update @ A )
= ( ^ [R5: ref @ A,V4: A] :
( heap_Time_heap @ product_unit
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R5 @ V4 @ H3 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% Ref_Time.update_def
thf(fact_7780_execute__tap,axiom,
! [A: $tType,F3: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( heap_Time_tap @ A @ F3 ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F3 @ H2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ).
% execute_tap
thf(fact_7781_execute__assert_H,axiom,
! [P: $o,H2: heap_ext @ product_unit] :
( ( P
=> ( ( heap_Time_execute @ product_unit @ ( refine_Imp_assert @ P ) @ H2 )
= ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( zero_zero @ nat ) ) ) ) ) )
& ( ~ P
=> ( ( heap_Time_execute @ product_unit @ ( refine_Imp_assert @ P ) @ H2 )
= ( none @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% execute_assert'
thf(fact_7782_assert_H__rule,axiom,
! [P: assn,Phi2: $o] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> Phi2 )
=> ( hoare_hoare_triple @ product_unit @ P @ ( refine_Imp_assert @ Phi2 )
@ ^ [Uu3: product_unit] : P ) ) ).
% assert'_rule
thf(fact_7783_execute__ureturn,axiom,
! [A: $tType,X3: A] :
( ( heap_Time_execute @ A @ ( heap_Time_ureturn @ A @ X3 ) )
= ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( zero_zero @ nat ) ) ) ) ) ).
% execute_ureturn
thf(fact_7784_execute__assert_I1_J,axiom,
! [A: $tType,P: A > $o,X3: A,H2: heap_ext @ product_unit] :
( ( P @ X3 )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X3 ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_assert(1)
thf(fact_7785_ureturn__def,axiom,
! [A: $tType] :
( ( heap_Time_ureturn @ A )
= ( ^ [X4: A] :
( heap_Time_heap @ A
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% ureturn_def
thf(fact_7786_execute__return,axiom,
! [A: $tType,X3: A] :
( ( heap_Time_execute @ A @ ( heap_Time_return @ A @ X3 ) )
= ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ).
% execute_return
thf(fact_7787_of__list__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_of_list @ A @ Xs )
@ ^ [R5: array @ A] : ( snga_assn @ A @ R5 @ Xs ) ) ) ).
% of_list_rule
thf(fact_7788_return__cons__rule,axiom,
! [A: $tType,P: assn,Q: A > assn,X3: A] :
( ( entails @ P @ ( Q @ X3 ) )
=> ( hoare_hoare_triple @ A @ P @ ( heap_Time_return @ A @ X3 ) @ Q ) ) ).
% return_cons_rule
thf(fact_7789_return__sp__rule,axiom,
! [A: $tType,P: assn,X3: A] :
( hoare_hoare_triple @ A @ P @ ( heap_Time_return @ A @ X3 )
@ ^ [R5: A] : ( times_times @ assn @ P @ ( pure_assn @ ( R5 = X3 ) ) ) ) ).
% return_sp_rule
thf(fact_7790_return__def,axiom,
! [A: $tType] :
( ( heap_Time_return @ A )
= ( ^ [X4: A] :
( heap_Time_heap @ A
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ).
% return_def
thf(fact_7791_array__of__list__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_of_list @ A )
= ( ^ [Xs3: list @ A] : ( array_make @ A @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( nth @ A @ Xs3 ) ) ) ) ) ).
% array_of_list_make
thf(fact_7792_timeFrame_Oelims,axiom,
! [A: $tType,X3: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( ( heap_Time_timeFrame @ A @ X3 @ Xa )
= Y )
=> ( ! [R3: A,H: heap_ext @ product_unit,N4: nat] :
( ( Xa
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ N4 ) ) ) )
=> ( Y
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( plus_plus @ nat @ X3 @ N4 ) ) ) ) ) )
=> ~ ( ( Xa
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( Y
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% timeFrame.elims
thf(fact_7793_timeFrame_Osimps_I1_J,axiom,
! [A: $tType,N: nat,R2: A,H2: heap_ext @ product_unit,N6: nat] :
( ( heap_Time_timeFrame @ A @ N @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N6 ) ) ) )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ N @ N6 ) ) ) ) ) ).
% timeFrame.simps(1)
thf(fact_7794_execute__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F3: nat > A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_make @ A @ N @ F3 ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H8: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( map @ nat @ A @ F3 @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) ) ) ) ) ).
% execute_make
thf(fact_7795_Array__Time_Omake__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_make @ A )
= ( ^ [N2: nat,F7: nat > A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H3: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H8: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( map @ nat @ A @ F7 @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) @ H3 ) ) ) ) ) ) ).
% Array_Time.make_def
thf(fact_7796_of__list__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_of_list @ A )
= ( ^ [Xs3: list @ A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H3: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H8: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) )
@ ( array_alloc @ A @ Xs3 @ H3 ) ) ) ) ) ) ).
% of_list_def
thf(fact_7797_execute__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H8: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) )
@ ( array_alloc @ A @ Xs @ H2 ) ) ) ) ) ).
% execute_of_list
thf(fact_7798_new__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_new @ A )
= ( ^ [N2: nat,X4: A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H3: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H8: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( replicate @ A @ N2 @ X4 ) @ H3 ) ) ) ) ) ) ).
% new_def
thf(fact_7799_execute__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X3: A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_new @ A @ N @ X3 ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H8: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( replicate @ A @ N @ X3 ) @ H2 ) ) ) ) ) ).
% execute_new
thf(fact_7800_freeze__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,Xs: list @ A] :
( hoare_hoare_triple @ ( list @ A ) @ ( snga_assn @ A @ A4 @ Xs ) @ ( array_freeze @ A @ A4 )
@ ^ [R5: list @ A] : ( times_times @ assn @ ( snga_assn @ A @ A4 @ Xs ) @ ( pure_assn @ ( R5 = Xs ) ) ) ) ) ).
% freeze_rule
thf(fact_7801_effect__makeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,H4: heap_ext @ product_unit,F3: nat > A,N: nat,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A4 @ H4 )
= ( array_alloc @ A @ ( map @ nat @ A @ F3 @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F3 ) @ H2 @ H4 @ A4 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% effect_makeI
thf(fact_7802_effectI,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,R2: A,H4: heap_ext @ product_unit,N: nat] :
( ( ( heap_Time_execute @ A @ C2 @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
=> ( heap_Time_effect @ A @ C2 @ H2 @ H4 @ R2 @ N ) ) ).
% effectI
thf(fact_7803_effect__def,axiom,
! [A: $tType] :
( ( heap_Time_effect @ A )
= ( ^ [C6: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H8: heap_ext @ product_unit,R5: A,N2: nat] :
( ( heap_Time_execute @ A @ C6 @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ N2 ) ) ) ) ) ) ).
% effect_def
thf(fact_7804_hoare__triple__effect,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H2: heap_ext @ product_unit,As2: set @ nat] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ? [H5: heap_ext @ product_unit,R3: A,T9: nat] :
( ( heap_Time_effect @ A @ C2 @ H2 @ H5 @ R3 @ T9 )
& ( rep_assn @ ( Q @ R3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( hoare_new_addrs @ H2 @ As2 @ H5 ) ) ) ) ) ) ).
% hoare_triple_effect
thf(fact_7805_effect__of__listI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,H4: heap_ext @ product_unit,Xs: list @ A,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A4 @ H4 )
= ( array_alloc @ A @ Xs @ H2 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 @ H4 @ A4 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% effect_of_listI
thf(fact_7806_effect__newI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,H4: heap_ext @ product_unit,N: nat,X3: A,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A4 @ H4 )
= ( array_alloc @ A @ ( replicate @ A @ N @ X3 ) @ H2 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X3 ) @ H2 @ H4 @ A4 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% effect_newI
thf(fact_7807_tap__def,axiom,
! [A: $tType] :
( ( heap_Time_tap @ A )
= ( ^ [F7: ( heap_ext @ product_unit ) > A] :
( heap_Time_Heap2 @ A
@ ^ [H3: heap_ext @ product_unit] : ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F7 @ H3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% tap_def
thf(fact_7808_timeFrame_Opelims,axiom,
! [A: $tType,X3: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( ( heap_Time_timeFrame @ A @ X3 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X3 @ Xa ) )
=> ( ! [R3: A,H: heap_ext @ product_unit,N4: nat] :
( ( Xa
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ N4 ) ) ) )
=> ( ( Y
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( plus_plus @ nat @ X3 @ N4 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X3 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ N4 ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( ( Y
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X3 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ) ) ) ).
% timeFrame.pelims
thf(fact_7809_wait__def,axiom,
( heap_Time_wait
= ( ^ [N2: nat] :
( heap_Time_Heap2 @ product_unit
@ ^ [H3: heap_ext @ product_unit] : ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N2 ) ) ) ) ) ) ).
% wait_def
thf(fact_7810_execute__bind_I1_J,axiom,
! [A: $tType,B: $tType,F3: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X3: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F3 @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F3 @ G ) @ H2 )
= ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G @ X3 ) @ H4 ) ) ) ) ).
% execute_bind(1)
thf(fact_7811_bind__rule_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: assn,F3: heap_Time_Heap @ B,R: B > assn,G: B > ( heap_Time_Heap @ C ),Q: C > assn] :
( ! [R3: A] :
( hoare_hoare_triple @ B @ P @ F3
@ ^ [S7: B] : ( times_times @ assn @ P @ ( R @ S7 ) ) )
=> ( ! [X: B] : ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( R @ X ) @ P ) @ ( G @ X ) @ Q )
=> ( hoare_hoare_triple @ C @ P @ ( heap_Time_bind @ B @ C @ F3 @ G ) @ Q ) ) ) ).
% bind_rule'
thf(fact_7812_assert_H__bind__rule,axiom,
! [A: $tType,P: assn,Phi2: $o,C2: heap_Time_Heap @ A,Q: A > assn] :
( ! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> Phi2 )
=> ( ( Phi2
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) )
=> ( hoare_hoare_triple @ A @ P
@ ( heap_Time_bind @ product_unit @ A @ ( refine_Imp_assert @ Phi2 )
@ ^ [Uu3: product_unit] : C2 )
@ Q ) ) ) ).
% assert'_bind_rule
thf(fact_7813_execute__bind__eq__SomeI,axiom,
! [A: $tType,B: $tType,F3: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X3: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B ),Y: B,H10: heap_ext @ product_unit,N6: nat] :
( ( ( heap_Time_execute @ A @ F3 @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
=> ( ( ( heap_Time_execute @ B @ ( G @ X3 ) @ H4 )
= ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H10 @ N6 ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F3 @ G ) @ H2 )
= ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H10 @ ( plus_plus @ nat @ N @ N6 ) ) ) ) ) ) ) ).
% execute_bind_eq_SomeI
thf(fact_7814_success__bind__executeI,axiom,
! [A: $tType,B: $tType,F3: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X3: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F3 @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
=> ( ( heap_Time_success @ B @ ( G @ X3 ) @ H4 )
=> ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F3 @ G ) @ H2 ) ) ) ).
% success_bind_executeI
thf(fact_7815_ref__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: A] :
( hoare_hoare_triple @ ( ref @ A ) @ ( one_one @ assn ) @ ( ref_ref @ A @ X3 )
@ ^ [R5: ref @ A] : ( sngr_assn @ A @ R5 @ X3 ) ) ) ).
% ref_rule
thf(fact_7816_hoare__triple__success,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H2: heap_ext @ product_unit,As2: set @ nat] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ( heap_Time_success @ A @ C2 @ H2 ) ) ) ).
% hoare_triple_success
thf(fact_7817_successE,axiom,
! [A: $tType,F3: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_success @ A @ F3 @ H2 )
=> ~ ! [R3: A,H5: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F3 @ H2 )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ H5 ) ) ) ) ).
% successE
thf(fact_7818_execute__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [V2: A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( ref @ A ) @ ( ref_ref @ A @ V2 ) @ H2 )
= ( some @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: ref @ A,H8: heap_ext @ product_unit] : ( product_Pair @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ ( one_one @ nat ) ) )
@ ( ref_alloc @ A @ V2 @ H2 ) ) ) ) ) ).
% execute_ref
thf(fact_7819_ref__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_ref @ A )
= ( ^ [V4: A] :
( heap_Time_heap @ ( ref @ A )
@ ^ [H3: heap_ext @ product_unit] :
( product_case_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: ref @ A,H8: heap_ext @ product_unit] : ( product_Pair @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ ( one_one @ nat ) ) )
@ ( ref_alloc @ A @ V4 @ H3 ) ) ) ) ) ) ).
% ref_def
thf(fact_7820_effect__refI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,H4: heap_ext @ product_unit,V2: A,H2: heap_ext @ product_unit,N: nat] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R2 @ H4 )
= ( ref_alloc @ A @ V2 @ H2 ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ ( ref @ A ) @ ( ref_ref @ A @ V2 ) @ H2 @ H4 @ R2 @ N ) ) ) ) ).
% effect_refI
thf(fact_7821_Ref__Time_Opresent__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_present @ A )
= ( ^ [H3: heap_ext @ product_unit,R5: ref @ A] : ( ord_less @ nat @ ( addr_of_ref @ A @ R5 ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% Ref_Time.present_def
thf(fact_7822_execute__upd_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,X3: A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( heap_Time_execute @ ( array @ A ) @ ( array_upd @ A @ I @ X3 @ A4 ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A4 @ I @ X3 @ H2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% execute_upd(1)
thf(fact_7823_next__present,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,H4: heap_ext @ product_unit,X3: A,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R2 @ H4 )
= ( ref_alloc @ A @ X3 @ H2 ) )
=> ( ref_present @ A @ H4 @ R2 ) ) ) ).
% next_present
thf(fact_7824_next__fresh,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,H4: heap_ext @ product_unit,X3: A,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R2 @ H4 )
= ( ref_alloc @ A @ X3 @ H2 ) )
=> ~ ( ref_present @ A @ H2 @ R2 ) ) ) ).
% next_fresh
thf(fact_7825_Array__Time_Olength__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H2: heap_ext @ product_unit,Ys: list @ A] :
( ( array_length @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys @ H2 ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% Array_Time.length_alloc
thf(fact_7826_effect__updI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,H4: heap_ext @ product_unit,V2: A,N: nat] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( H4
= ( array_update @ A @ A4 @ I @ V2 @ H2 ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_upd @ A @ I @ V2 @ A4 ) @ H2 @ H4 @ A4 @ N ) ) ) ) ) ).
% effect_updI
thf(fact_7827_effect__updE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,V2: A,A4: array @ A,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R2: array @ A,N: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_upd @ A @ I @ V2 @ A4 ) @ H2 @ H4 @ R2 @ N )
=> ~ ( ( R2 = A4 )
=> ( ( H4
= ( array_update @ A @ A4 @ I @ V2 @ H2 ) )
=> ( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ) ).
% effect_updE
thf(fact_7828_success__updI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,X3: A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( heap_Time_success @ ( array @ A ) @ ( array_upd @ A @ I @ X3 @ A4 ) @ H2 ) ) ) ).
% success_updI
thf(fact_7829_success__nthI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( heap_Time_success @ A @ ( array_nth @ A @ A4 @ I ) @ H2 ) ) ) ).
% success_nthI
thf(fact_7830_execute__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ nat @ ( array_len @ A @ A4 ) @ H2 )
= ( some @ ( product_prod @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_length @ A @ H2 @ A4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_len
thf(fact_7831_upd__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_upd @ A )
= ( ^ [I4: nat,X4: A,A5: array @ A] :
( heap_Time_guard @ ( array @ A )
@ ^ [H3: heap_ext @ product_unit] : ( ord_less @ nat @ I4 @ ( array_length @ A @ H3 @ A5 ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I4 @ X4 @ H3 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% upd_def
thf(fact_7832_execute__nth_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( heap_Time_execute @ A @ ( array_nth @ A @ A4 @ I ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% execute_nth(1)
thf(fact_7833_get__update__elem__neqIndex,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,J: nat,A4: array @ A,V2: A,H2: heap_ext @ product_unit] :
( ( I != J )
=> ( ( nth @ A @ ( array_get @ A @ ( array_update @ A @ A4 @ J @ V2 @ H2 ) @ A4 ) @ I )
= ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) ) ) ) ).
% get_update_elem_neqIndex
thf(fact_7834_get__update__eq,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,I: nat,V2: A,H2: heap_ext @ product_unit] :
( ( array_get @ A @ ( array_update @ A @ A4 @ I @ V2 @ H2 ) @ A4 )
= ( list_update @ A @ ( array_get @ A @ H2 @ A4 ) @ I @ V2 ) ) ) ).
% get_update_eq
thf(fact_7835_length__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_length @ A )
= ( ^ [H3: heap_ext @ product_unit,A5: array @ A] : ( size_size @ ( list @ A ) @ ( array_get @ A @ H3 @ A5 ) ) ) ) ) ).
% length_def
thf(fact_7836_Array__Time_Onth__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_nth @ A )
= ( ^ [A5: array @ A,I4: nat] :
( heap_Time_guard @ A
@ ^ [H3: heap_ext @ product_unit] : ( ord_less @ nat @ I4 @ ( array_length @ A @ H3 @ A5 ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H3 @ A5 ) @ I4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% Array_Time.nth_def
thf(fact_7837_execute__freeze,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( list @ A ) @ ( array_freeze @ A @ A4 ) @ H2 )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H2 @ A4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H2 @ A4 ) ) ) ) ) ) ) ).
% execute_freeze
thf(fact_7838_freeze__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_freeze @ A )
= ( ^ [A5: array @ A] :
( heap_Time_heap @ ( list @ A )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H3 @ A5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H3 @ A5 ) ) ) ) ) ) ) ) ).
% freeze_def
thf(fact_7839_effect__nthI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,H4: heap_ext @ product_unit,R2: A,N: nat] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( H4 = H2 )
=> ( ( R2
= ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ A @ ( array_nth @ A @ A4 @ I ) @ H2 @ H4 @ R2 @ N ) ) ) ) ) ) ).
% effect_nthI
thf(fact_7840_effect__nthE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A4: array @ A,I: nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R2: A,N: nat] :
( ( heap_Time_effect @ A @ ( array_nth @ A @ A4 @ I ) @ H2 @ H4 @ R2 @ N )
=> ~ ( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( R2
= ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) )
=> ( ( H4 = H2 )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ) ).
% effect_nthE
thf(fact_7841_execute__map__entry_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,F3: A > A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( heap_Time_execute @ ( array @ A ) @ ( array_map_entry @ A @ I @ F3 @ A4 ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A4 @ I @ ( F3 @ ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) ) @ H2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% execute_map_entry(1)
thf(fact_7842_execute__swap_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,X3: A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( heap_Time_execute @ A @ ( array_swap @ A @ I @ X3 @ A4 ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A4 @ I @ X3 @ H2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% execute_swap(1)
thf(fact_7843_success__swapI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,X3: A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( heap_Time_success @ A @ ( array_swap @ A @ I @ X3 @ A4 ) @ H2 ) ) ) ).
% success_swapI
thf(fact_7844_success__map__entryI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,F3: A > A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( heap_Time_success @ ( array @ A ) @ ( array_map_entry @ A @ I @ F3 @ A4 ) @ H2 ) ) ) ).
% success_map_entryI
thf(fact_7845_effect__map__entryI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,H4: heap_ext @ product_unit,F3: A > A,R2: array @ A,N: nat] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( H4
= ( array_update @ A @ A4 @ I @ ( F3 @ ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) ) @ H2 ) )
=> ( ( R2 = A4 )
=> ( ( N
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_map_entry @ A @ I @ F3 @ A4 ) @ H2 @ H4 @ R2 @ N ) ) ) ) ) ) ).
% effect_map_entryI
thf(fact_7846_effect__map__entryE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,F3: A > A,A4: array @ A,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R2: array @ A,N: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_map_entry @ A @ I @ F3 @ A4 ) @ H2 @ H4 @ R2 @ N )
=> ~ ( ( R2 = A4 )
=> ( ( H4
= ( array_update @ A @ A4 @ I @ ( F3 @ ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) ) @ H2 ) )
=> ( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( N
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% effect_map_entryE
thf(fact_7847_effect__swapI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H2: heap_ext @ product_unit,A4: array @ A,H4: heap_ext @ product_unit,X3: A,R2: A,N: nat] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( ( H4
= ( array_update @ A @ A4 @ I @ X3 @ H2 ) )
=> ( ( R2
= ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) )
=> ( ( N
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( heap_Time_effect @ A @ ( array_swap @ A @ I @ X3 @ A4 ) @ H2 @ H4 @ R2 @ N ) ) ) ) ) ) ).
% effect_swapI
thf(fact_7848_effect__swapE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X3: A,A4: array @ A,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R2: A,N: nat] :
( ( heap_Time_effect @ A @ ( array_swap @ A @ I @ X3 @ A4 ) @ H2 @ H4 @ R2 @ N )
=> ~ ( ( R2
= ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) )
=> ( ( H4
= ( array_update @ A @ A4 @ I @ X3 @ H2 ) )
=> ( ( ord_less @ nat @ I @ ( array_length @ A @ H2 @ A4 ) )
=> ( N
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% effect_swapE
thf(fact_7849_swap__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_swap @ A )
= ( ^ [I4: nat,X4: A,A5: array @ A] :
( heap_Time_guard @ A
@ ^ [H3: heap_ext @ product_unit] : ( ord_less @ nat @ I4 @ ( array_length @ A @ H3 @ A5 ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H3 @ A5 ) @ I4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I4 @ X4 @ H3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% swap_def
thf(fact_7850_map__entry__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_map_entry @ A )
= ( ^ [I4: nat,F7: A > A,A5: array @ A] :
( heap_Time_guard @ ( array @ A )
@ ^ [H3: heap_ext @ product_unit] : ( ord_less @ nat @ I4 @ ( array_length @ A @ H3 @ A5 ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I4 @ ( F7 @ ( nth @ A @ ( array_get @ A @ H3 @ A5 ) @ I4 ) ) @ H3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% map_entry_def
thf(fact_7851_effect__of__listE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R2: array @ A,N6: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 @ H4 @ R2 @ N6 )
=> ~ ( ( R2
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) )
=> ( ( H4
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) )
=> ( ( ( array_get @ A @ H4 @ R2 )
= Xs )
=> ( ( array_present @ A @ H4 @ R2 )
=> ( ~ ( array_present @ A @ H2 @ R2 )
=> ( N6
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ) ) ) ).
% effect_of_listE
thf(fact_7852_snga__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( snga_assn_raw @ A @ X3 @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ( ( array_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_array @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ).
% snga_assn_raw.elims(3)
thf(fact_7853_Array__Time_Opresent__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_present @ A )
= ( ^ [H3: heap_ext @ product_unit,A5: array @ A] : ( ord_less @ nat @ ( addr_of_array @ A @ A5 ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% Array_Time.present_def
thf(fact_7854_relH__array,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [As2: set @ nat,H2: heap_ext @ product_unit,H4: heap_ext @ product_unit,R2: array @ A] :
( ( relH @ As2 @ H2 @ H4 )
=> ( ( member @ nat @ ( addr_of_array @ A @ R2 ) @ As2 )
=> ( ( array_get @ A @ H2 @ R2 )
= ( array_get @ A @ H4 @ R2 ) ) ) ) ) ).
% relH_array
thf(fact_7855_snga__assn__raw_Osimps,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: array @ A,X3: list @ A,H2: heap_ext @ product_unit,As2: set @ nat] :
( ( snga_assn_raw @ A @ R2 @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
= ( ( ( array_get @ A @ H2 @ R2 )
= X3 )
& ( As2
= ( insert3 @ nat @ ( addr_of_array @ A @ R2 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ R2 ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% snga_assn_raw.simps
thf(fact_7856_snga__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( snga_assn_raw @ A @ X3 @ Xa @ Xb )
= Y )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( Y
= ( ~ ( ( ( array_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_array @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% snga_assn_raw.elims(1)
thf(fact_7857_snga__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( snga_assn_raw @ A @ X3 @ Xa @ Xb )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ~ ( ( ( array_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_array @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ).
% snga_assn_raw.elims(2)
thf(fact_7858_snga__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( snga_assn_raw @ A @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ 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 ) ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) )
=> ( ( ( array_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_array @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(3)
thf(fact_7859_snga__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( snga_assn_raw @ A @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ 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 ) ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) )
=> ~ ( ( ( array_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_array @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(2)
thf(fact_7860_snga__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X3: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( snga_assn_raw @ A @ X3 @ 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 ) ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ( Y
= ( ( ( array_get @ A @ H @ X3 )
= Xa )
& ( As
= ( insert3 @ nat @ ( addr_of_array @ A @ X3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X3 ) @ ( lim @ product_unit @ H ) ) ) )
=> ~ ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(1)
thf(fact_7861_nth__update__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A4: array @ A,B3: array @ B,J: nat,V2: B,H2: heap_ext @ product_unit,I: nat] :
( ( array_noteq @ A @ B @ A4 @ B3 )
=> ( ( nth @ A @ ( array_get @ A @ ( array_update @ B @ B3 @ J @ V2 @ H2 ) @ A4 ) @ I )
= ( nth @ A @ ( array_get @ A @ H2 @ A4 ) @ I ) ) ) ) ).
% nth_update_neq
thf(fact_7862_relH__set__array,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: array @ A,As2: set @ nat,H2: heap_ext @ product_unit,X3: list @ A] :
( ~ ( member @ nat @ ( addr_of_array @ A @ R2 ) @ As2 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As2 ) )
=> ( relH @ As2 @ H2 @ ( array_set @ A @ R2 @ X3 @ H2 ) ) ) ) ) ).
% relH_set_array
thf(fact_7863_Array__Time_Oupdate__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_update @ A )
= ( ^ [A5: array @ A,I4: nat,X4: A,H3: heap_ext @ product_unit] : ( array_set @ A @ A5 @ ( list_update @ A @ ( array_get @ A @ H3 @ A5 ) @ I4 @ X4 ) @ H3 ) ) ) ) ).
% Array_Time.update_def
thf(fact_7864_Array__Time_Oalloc__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_alloc @ A )
= ( ^ [Xs3: list @ A,H3: heap_ext @ product_unit] :
( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array2 @ A @ ( lim @ product_unit @ H3 ) )
@ ( array_set @ A @ ( array2 @ A @ ( lim @ product_unit @ H3 ) ) @ Xs3
@ ( lim_update @ product_unit
@ ^ [Uu3: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H3 ) @ ( one_one @ nat ) )
@ H3 ) ) ) ) ) ) ).
% Array_Time.alloc_def
thf(fact_7865_Ref__Time_Oalloc__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_alloc @ A )
= ( ^ [X4: A,H3: heap_ext @ product_unit] :
( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( ref2 @ A @ ( lim @ product_unit @ H3 ) )
@ ( ref_set @ A @ ( ref2 @ A @ ( lim @ product_unit @ H3 ) ) @ X4
@ ( lim_update @ product_unit
@ ^ [Uu3: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H3 ) @ ( one_one @ nat ) )
@ H3 ) ) ) ) ) ) ).
% Ref_Time.alloc_def
thf(fact_7866_raise__iff,axiom,
! [A: $tType,P: assn,S3: list @ char,Q: A > assn] :
( ( hoare_hoare_triple @ A @ P @ ( heap_Time_raise @ A @ S3 ) @ Q )
= ( P
= ( bot_bot @ assn ) ) ) ).
% raise_iff
thf(fact_7867_raise__rule,axiom,
! [A: $tType,S3: list @ char,Q: A > assn] : ( hoare_hoare_triple @ A @ ( bot_bot @ assn ) @ ( heap_Time_raise @ A @ S3 ) @ Q ) ).
% raise_rule
thf(fact_7868_Random__Pred_Ounion__def,axiom,
! [A: $tType] :
( ( random_union @ A )
= ( ^ [R15: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),R24: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),S7: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P14: pred @ A,S9: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P26: pred @ A] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( sup_sup @ ( pred @ A ) @ P14 @ P26 ) )
@ ( R24 @ S9 ) )
@ ( R15 @ S7 ) ) ) ) ).
% Random_Pred.union_def
thf(fact_7869_Nitpick_Otranclp__unfold,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [R5: A > A > $o,A5: A,B4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ ( transitive_trancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% Nitpick.tranclp_unfold
thf(fact_7870_less__nat__rel,axiom,
( ( ord_less @ nat )
= ( transitive_tranclp @ nat
@ ^ [M: nat,N2: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% less_nat_rel
thf(fact_7871_tranclp__power,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [P2: A > A > $o,X4: A,Y5: A] :
? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( compow @ ( A > A > $o ) @ N2 @ P2 @ X4 @ Y5 ) ) ) ) ).
% tranclp_power
thf(fact_7872_tranclp__induct2,axiom,
! [A: $tType,B: $tType,R2: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R2 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ! [A6: A,B5: B] :
( ( R2 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: B,Aa2: A,Ba: B] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R2 @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( ( R2 @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P @ A6 @ B5 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% tranclp_induct2
thf(fact_7873_tranclp__trancl__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_tranclp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R2 ) )
= ( ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% tranclp_trancl_eq
thf(fact_7874_trancl__def,axiom,
! [A: $tType] :
( ( transitive_trancl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_tranclp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ).
% trancl_def
thf(fact_7875_Random__Pred_Oempty__def,axiom,
! [A: $tType] :
( ( random_empty @ A )
= ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( bot_bot @ ( pred @ A ) ) ) ) ).
% Random_Pred.empty_def
thf(fact_7876_Random__Pred_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( random_bind @ A @ B )
= ( ^ [R4: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),F7: A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ),S7: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P2: pred @ A,S9: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [S12: product_prod @ code_natural @ code_natural] :
( product_Pair @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural )
@ ( bind2 @ A @ B @ P2
@ ^ [A5: A] : ( product_fst @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) @ ( F7 @ A5 @ S12 ) ) ) )
@ ( split_seed @ S9 ) )
@ ( R4 @ S7 ) ) ) ) ).
% Random_Pred.bind_def
thf(fact_7877_reflp__refl__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( reflp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R2 ) )
= ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R2 ) ) ).
% reflp_refl_eq
thf(fact_7878_list_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R4: A > B > $o] :
( relcompp @ ( list @ A ) @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ B )
@ ( conversep @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ A )
@ ( bNF_Grp @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ A )
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ B )
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R4 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% list.rel_compp_Grp
thf(fact_7879_relcompp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S: A > C > $o,T4: A > C > $o,R: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( sup_sup @ ( A > C > $o ) @ S @ T4 ) @ R )
= ( sup_sup @ ( A > B > $o ) @ ( relcompp @ A @ C @ B @ S @ R ) @ ( relcompp @ A @ C @ B @ T4 @ R ) ) ) ).
% relcompp_distrib2
thf(fact_7880_relcompp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R: A > C > $o,S: C > B > $o,T4: C > B > $o] :
( ( relcompp @ A @ C @ B @ R @ ( sup_sup @ ( C > B > $o ) @ S @ T4 ) )
= ( sup_sup @ ( A > B > $o ) @ ( relcompp @ A @ C @ B @ R @ S ) @ ( relcompp @ A @ C @ B @ R @ T4 ) ) ) ).
% relcompp_distrib
thf(fact_7881_relcompp__bot1,axiom,
! [C: $tType,B: $tType,A: $tType,R: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( bot_bot @ ( A > C > $o ) ) @ R )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot1
thf(fact_7882_relcompp__bot2,axiom,
! [C: $tType,B: $tType,A: $tType,R: A > C > $o] :
( ( relcompp @ A @ C @ B @ R @ ( bot_bot @ ( C > B > $o ) ) )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot2
thf(fact_7883_relcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcomp @ A @ B @ C )
= ( ^ [R5: set @ ( product_prod @ A @ B ),S7: set @ ( product_prod @ B @ C )] :
( collect @ ( product_prod @ A @ C )
@ ( product_case_prod @ A @ C @ $o
@ ( relcompp @ A @ B @ C
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 )
@ ^ [X4: B,Y5: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X4 @ Y5 ) @ S7 ) ) ) ) ) ) ).
% relcomp_def
thf(fact_7884_list_Orel__Grp,axiom,
! [B: $tType,A: $tType,A3: set @ A,F3: A > B] :
( ( list_all2 @ A @ B @ ( bNF_Grp @ A @ B @ A3 @ F3 ) )
= ( bNF_Grp @ ( list @ A ) @ ( list @ B )
@ ( collect @ ( list @ A )
@ ^ [X4: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ X4 ) @ A3 ) )
@ ( map @ A @ B @ F3 ) ) ) ).
% list.rel_Grp
thf(fact_7885_transp__relcompp__less__eq,axiom,
! [A: $tType,R2: A > A > $o] :
( ( transp @ A @ R2 )
=> ( ord_less_eq @ ( A > A > $o ) @ ( relcompp @ A @ A @ A @ R2 @ R2 ) @ R2 ) ) ).
% transp_relcompp_less_eq
thf(fact_7886_relcompp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R7: A > B > $o,R2: A > B > $o,S5: B > C > $o,S3: B > C > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R7 @ R2 )
=> ( ( ord_less_eq @ ( B > C > $o ) @ S5 @ S3 )
=> ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R7 @ S5 ) @ ( relcompp @ A @ B @ C @ R2 @ S3 ) ) ) ) ).
% relcompp_mono
thf(fact_7887_reflp__inf,axiom,
! [A: $tType,R2: A > A > $o,S3: A > A > $o] :
( ( reflp @ A @ R2 )
=> ( ( reflp @ A @ S3 )
=> ( reflp @ A @ ( inf_inf @ ( A > A > $o ) @ R2 @ S3 ) ) ) ) ).
% reflp_inf
thf(fact_7888_converse__relcompp,axiom,
! [A: $tType,C: $tType,B: $tType,R2: B > C > $o,S3: C > A > $o] :
( ( conversep @ B @ A @ ( relcompp @ B @ C @ A @ R2 @ S3 ) )
= ( relcompp @ A @ C @ B @ ( conversep @ C @ A @ S3 ) @ ( conversep @ B @ C @ R2 ) ) ) ).
% converse_relcompp
thf(fact_7889_relcompp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,S3: B > C > $o,A1: A,A22: C] :
( ( relcompp @ A @ B @ C @ R2 @ S3 @ A1 @ A22 )
=> ~ ! [B5: B] :
( ( R2 @ A1 @ B5 )
=> ~ ( S3 @ B5 @ A22 ) ) ) ).
% relcompp.cases
thf(fact_7890_relcompp_Osimps,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R5: A > B > $o,S7: B > C > $o,A13: A,A23: C] :
? [A5: A,B4: B,C6: C] :
( ( A13 = A5 )
& ( A23 = C6 )
& ( R5 @ A5 @ B4 )
& ( S7 @ B4 @ C6 ) ) ) ) ).
% relcompp.simps
thf(fact_7891_OO__eq,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( relcompp @ A @ B @ B @ R
@ ^ [Y4: B,Z4: B] : Y4 = Z4 )
= R ) ).
% OO_eq
thf(fact_7892_eq__OO,axiom,
! [B: $tType,A: $tType,R: A > B > $o] :
( ( relcompp @ A @ A @ B
@ ^ [Y4: A,Z4: A] : Y4 = Z4
@ R )
= R ) ).
% eq_OO
thf(fact_7893_reflpD,axiom,
! [A: $tType,R2: A > A > $o,X3: A] :
( ( reflp @ A @ R2 )
=> ( R2 @ X3 @ X3 ) ) ).
% reflpD
thf(fact_7894_reflpE,axiom,
! [A: $tType,R2: A > A > $o,X3: A] :
( ( reflp @ A @ R2 )
=> ( R2 @ X3 @ X3 ) ) ).
% reflpE
thf(fact_7895_reflpI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X: A] : ( R2 @ X @ X )
=> ( reflp @ A @ R2 ) ) ).
% reflpI
thf(fact_7896_reflp__def,axiom,
! [A: $tType] :
( ( reflp @ A )
= ( ^ [R5: A > A > $o] :
! [X4: A] : ( R5 @ X4 @ X4 ) ) ) ).
% reflp_def
thf(fact_7897_relcomppE,axiom,
! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,S3: B > C > $o,A4: A,C2: C] :
( ( relcompp @ A @ B @ C @ R2 @ S3 @ A4 @ C2 )
=> ~ ! [B5: B] :
( ( R2 @ A4 @ B5 )
=> ~ ( S3 @ B5 @ C2 ) ) ) ).
% relcomppE
thf(fact_7898_relcomppI,axiom,
! [A: $tType,B: $tType,C: $tType,R2: A > B > $o,A4: A,B3: B,S3: B > C > $o,C2: C] :
( ( R2 @ A4 @ B3 )
=> ( ( S3 @ B3 @ C2 )
=> ( relcompp @ A @ B @ C @ R2 @ S3 @ A4 @ C2 ) ) ) ).
% relcomppI
thf(fact_7899_reflp__mono,axiom,
! [A: $tType,R: A > A > $o,Q: A > A > $o] :
( ( reflp @ A @ R )
=> ( ! [X: A,Y2: A] :
( ( R @ X @ Y2 )
=> ( Q @ X @ Y2 ) )
=> ( reflp @ A @ Q ) ) ) ).
% reflp_mono
thf(fact_7900_reflp__equality,axiom,
! [A: $tType] :
( reflp @ A
@ ^ [Y4: A,Z4: A] : Y4 = Z4 ) ).
% reflp_equality
thf(fact_7901_relcompp__apply,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R4: A > B > $o,S2: B > C > $o,A5: A,C6: C] :
? [B4: B] :
( ( R4 @ A5 @ B4 )
& ( S2 @ B4 @ C6 ) ) ) ) ).
% relcompp_apply
thf(fact_7902_relcompp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R2: A > D > $o,S3: D > C > $o,T2: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R2 @ S3 ) @ T2 )
= ( relcompp @ A @ D @ B @ R2 @ ( relcompp @ D @ C @ B @ S3 @ T2 ) ) ) ).
% relcompp_assoc
thf(fact_7903_list_Orel__reflp,axiom,
! [A: $tType,R: A > A > $o] :
( ( reflp @ A @ R )
=> ( reflp @ ( list @ A ) @ ( list_all2 @ A @ A @ R ) ) ) ).
% list.rel_reflp
thf(fact_7904_list_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R: A > B > $o,S: B > C > $o] :
( ( list_all2 @ A @ C @ ( relcompp @ A @ B @ C @ R @ S ) )
= ( relcompp @ ( list @ A ) @ ( list @ B ) @ ( list @ C ) @ ( list_all2 @ A @ B @ R ) @ ( list_all2 @ B @ C @ S ) ) ) ).
% list.rel_compp
thf(fact_7905_rel__filter__distr,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A > C > $o,B2: C > B > $o] :
( ( relcompp @ ( filter @ A ) @ ( filter @ C ) @ ( filter @ B ) @ ( rel_filter @ A @ C @ A3 ) @ ( rel_filter @ C @ B @ B2 ) )
= ( rel_filter @ A @ B @ ( relcompp @ A @ C @ B @ A3 @ B2 ) ) ) ).
% rel_filter_distr
thf(fact_7906_rel__filter__Grp,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( rel_filter @ A @ B @ ( bNF_Grp @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 ) )
= ( bNF_Grp @ ( filter @ A ) @ ( filter @ B ) @ ( top_top @ ( set @ ( filter @ A ) ) ) @ ( filtermap @ A @ B @ F3 ) ) ) ).
% rel_filter_Grp
thf(fact_7907_relcompp__SUP__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R2: D > A > C > $o,I3: set @ D,S3: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( complete_Sup_Sup @ ( A > C > $o ) @ ( image2 @ D @ ( A > C > $o ) @ R2 @ I3 ) ) @ S3 )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I4: D] : ( relcompp @ A @ C @ B @ ( R2 @ I4 ) @ S3 )
@ I3 ) ) ) ).
% relcompp_SUP_distrib2
thf(fact_7908_relcompp__SUP__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S3: A > C > $o,R2: D > C > B > $o,I3: set @ D] :
( ( relcompp @ A @ C @ B @ S3 @ ( complete_Sup_Sup @ ( C > B > $o ) @ ( image2 @ D @ ( C > B > $o ) @ R2 @ I3 ) ) )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I4: D] : ( relcompp @ A @ C @ B @ S3 @ ( R2 @ I4 ) )
@ I3 ) ) ) ).
% relcompp_SUP_distrib
thf(fact_7909_relcompp__relcomp__eq,axiom,
! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( relcompp @ A @ B @ C
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R2 )
@ ^ [X4: B,Y5: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X4 @ Y5 ) @ S3 ) )
= ( ^ [X4: A,Y5: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y5 ) @ ( relcomp @ A @ B @ C @ R2 @ S3 ) ) ) ) ).
% relcompp_relcomp_eq
thf(fact_7910_reflp__sup,axiom,
! [A: $tType,R2: A > A > $o,S3: A > A > $o] :
( ( reflp @ A @ R2 )
=> ( ( reflp @ A @ S3 )
=> ( reflp @ A @ ( sup_sup @ ( A > A > $o ) @ R2 @ S3 ) ) ) ) ).
% reflp_sup
thf(fact_7911_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_7912_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_7913_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X3 @ Y )
| ( ~ ( Less @ Y @ X3 )
& ( lexordp_eq @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_7914_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_7915_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_7916_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X3 @ Y )
=> ( ~ ( Less @ Y @ X3 )
=> ( ( lexordp_eq @ A @ Less @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_7917_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X3 @ Y )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_7918_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A13: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A13
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A13
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ( Less2 @ X4 @ Y5 ) )
| ? [X4: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A13
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( Less2 @ X4 @ Y5 )
& ~ ( Less2 @ Y5 @ X4 )
& ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_7919_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 ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys5: list @ A] :
( A22
= ( cons @ A @ Y2 @ Ys5 ) )
=> ~ ( Less @ X @ Y2 ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys5: list @ A] :
( ( A22
= ( cons @ A @ Y2 @ Ys5 ) )
=> ( ~ ( Less @ X @ Y2 )
=> ( ~ ( Less @ Y2 @ X )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys5 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_7920_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_7921_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_7922_ord_Olexordp__eq__pref,axiom,
! [A: $tType,Less: A > A > $o,U: list @ A,V2: list @ A] : ( lexordp_eq @ A @ Less @ U @ ( append @ A @ U @ V2 ) ) ).
% ord.lexordp_eq_pref
thf(fact_7923_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_7924_ord_Olexordp__eq_Ocong,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( lexordp_eq @ A ) ) ).
% ord.lexordp_eq.cong
thf(fact_7925_pure__assn__raw_Opelims_I3_J,axiom,
! [B: $tType,A: $tType,X3: $o,Xa: product_prod @ A @ ( set @ B )] :
( ~ ( pure_assn_raw @ A @ B @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X3 @ Xa ) )
=> ~ ! [H: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As ) )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X3 @ ( product_Pair @ A @ ( set @ B ) @ H @ As ) ) )
=> ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X3 ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
thf(fact_7926_pure__assn__raw_Opelims_I2_J,axiom,
! [B: $tType,A: $tType,X3: $o,Xa: product_prod @ A @ ( set @ B )] :
( ( pure_assn_raw @ A @ B @ X3 @ Xa )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X3 @ Xa ) )
=> ~ ! [H: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As ) )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X3 @ ( product_Pair @ A @ ( set @ B ) @ H @ As ) ) )
=> ~ ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X3 ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
thf(fact_7927_pure__assn__raw_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X3: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
( ( ( pure_assn_raw @ A @ B @ X3 @ 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 ) ) @ X3 @ Xa ) )
=> ~ ! [H: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H @ As ) )
=> ( ( Y
= ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X3 ) )
=> ~ ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X3 @ ( product_Pair @ A @ ( set @ B ) @ H @ As ) ) ) ) ) ) ) ).
% pure_assn_raw.pelims(1)
thf(fact_7928_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_7929_is__arg__max__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic501386751176901750rg_max @ B @ A )
= ( ^ [F7: B > A,P2: B > $o,X4: B] :
( ( P2 @ X4 )
& ~ ? [Y5: B] :
( ( P2 @ Y5 )
& ( ord_less @ A @ ( F7 @ X4 ) @ ( F7 @ Y5 ) ) ) ) ) ) ) ).
% is_arg_max_def
thf(fact_7930_preordering__bdd_Oempty,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( condit622319405099724424ng_bdd @ A @ Less_eq @ Less )
=> ( condit16957441358409770ng_bdd @ A @ Less_eq @ ( bot_bot @ ( set @ A ) ) ) ) ).
% preordering_bdd.empty
thf(fact_7931_lazI,axiom,
! [A: $tType,B: $tType,A4: list @ A,B3: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ B ) @ B3 ) )
=> ( P @ ( nth @ A @ A4 @ I5 ) @ ( nth @ B @ B3 @ I5 ) ) )
=> ( list_all_zip @ A @ B @ P @ A4 @ B3 ) ) ) ).
% lazI
thf(fact_7932_laz__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,A4: list @ A,B3: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [X4: A,Y5: B] :
( ( P @ X4 @ Y5 )
& ( Q @ X4 @ Y5 ) )
@ A4
@ B3 )
= ( ( list_all_zip @ A @ B @ P @ A4 @ B3 )
& ( list_all_zip @ A @ B @ Q @ A4 @ B3 ) ) ) ).
% laz_conj
thf(fact_7933_laz__weak__Pa,axiom,
! [B: $tType,A: $tType,P: A > $o,A3: list @ A,B2: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A5: A,B4: B] : ( P @ A5 )
@ A3
@ B2 )
= ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ A3 ) )
=> ( P @ X4 ) ) ) ) ).
% laz_weak_Pa
thf(fact_7934_laz__weak__Pb,axiom,
! [A: $tType,B: $tType,P: B > $o,A3: list @ A,B2: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A5: A] : P
@ A3
@ B2 )
= ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
& ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ B2 ) )
=> ( P @ X4 ) ) ) ) ).
% laz_weak_Pb
thf(fact_7935_laz__len,axiom,
! [A: $tType,B: $tType,P: A > B > $o,A4: list @ A,B3: list @ B] :
( ( list_all_zip @ A @ B @ P @ A4 @ B3 )
=> ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) ) ) ).
% laz_len
thf(fact_7936_laz__eq,axiom,
! [A: $tType,A4: list @ A,B3: list @ A] :
( ( list_all_zip @ A @ A
@ ^ [Y4: A,Z4: A] : Y4 = Z4
@ A4
@ B3 )
= ( A4 = B3 ) ) ).
% laz_eq
thf(fact_7937_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_7938_laz__swap__ex,axiom,
! [B: $tType,A: $tType,C: $tType,P: A > B > C > $o,A3: list @ A,B2: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A5: A,B4: B] :
? [X9: C] : ( P @ A5 @ B4 @ X9 )
@ A3
@ B2 )
=> ~ ! [C9: list @ C] :
( ( list_all_zip @ A @ C
@ ^ [A5: A,C6: C] :
? [B4: B] : ( P @ A5 @ B4 @ C6 )
@ A3
@ C9 )
=> ~ ( list_all_zip @ B @ C
@ ^ [B4: B,C6: C] :
? [A5: A] : ( P @ A5 @ B4 @ C6 )
@ B2
@ C9 ) ) ) ).
% laz_swap_ex
thf(fact_7939_list__all__zip_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,A4: A,As2: list @ A,B3: B,Bs: list @ B] :
( ( list_all_zip @ A @ B @ P @ ( cons @ A @ A4 @ As2 ) @ ( cons @ B @ B3 @ Bs ) )
= ( ( P @ A4 @ B3 )
& ( list_all_zip @ A @ B @ P @ As2 @ Bs ) ) ) ).
% list_all_zip.simps(2)
thf(fact_7940_list__all__zip_Oelims_I3_J,axiom,
! [A: $tType,B: $tType,X3: A > B > $o,Xa: list @ A,Xb: list @ B] :
( ~ ( list_all_zip @ A @ B @ X3 @ Xa @ Xb )
=> ( ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ( ( X3 @ A6 @ B5 )
& ( list_all_zip @ A @ B @ X3 @ As @ Bs2 ) ) ) )
=> ( ( ? [V3: A,Va: list @ A] :
( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( Xb
!= ( nil @ B ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: B,Va: list @ B] :
( Xb
!= ( cons @ B @ V3 @ Va ) ) ) ) ) ) ).
% list_all_zip.elims(3)
thf(fact_7941_list__all__zip_Oelims_I2_J,axiom,
! [A: $tType,B: $tType,X3: A > B > $o,Xa: list @ A,Xb: list @ B] :
( ( list_all_zip @ A @ B @ X3 @ Xa @ Xb )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Xb
!= ( nil @ B ) ) )
=> ~ ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ~ ( ( X3 @ A6 @ B5 )
& ( list_all_zip @ A @ B @ X3 @ As @ Bs2 ) ) ) ) ) ) ).
% list_all_zip.elims(2)
thf(fact_7942_list__all__zip_Oelims_I1_J,axiom,
! [B: $tType,A: $tType,X3: A > B > $o,Xa: list @ A,Xb: list @ B,Y: $o] :
( ( ( list_all_zip @ A @ B @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ~ Y ) )
=> ( ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ( Y
= ( ~ ( ( X3 @ A6 @ B5 )
& ( list_all_zip @ A @ B @ X3 @ As @ Bs2 ) ) ) ) ) )
=> ( ( ? [V3: A,Va: list @ A] :
( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> Y ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V3: B,Va: list @ B] :
( Xb
= ( cons @ B @ V3 @ Va ) )
=> Y ) ) ) ) ) ) ).
% list_all_zip.elims(1)
thf(fact_7943_list__all__zip_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,V2: B,Va2: list @ B] :
~ ( list_all_zip @ A @ B @ P @ ( nil @ A ) @ ( cons @ B @ V2 @ Va2 ) ) ).
% list_all_zip.simps(4)
thf(fact_7944_list__all__zip_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,V2: A,Va2: list @ A] :
~ ( list_all_zip @ A @ B @ P @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ B ) ) ).
% list_all_zip.simps(3)
thf(fact_7945_list__all__zip__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F3: C > A,As2: list @ C,Bs: list @ B] :
( ( list_all_zip @ A @ B @ P @ ( map @ C @ A @ F3 @ As2 ) @ Bs )
= ( list_all_zip @ C @ B
@ ^ [A5: C] : ( P @ ( F3 @ A5 ) )
@ As2
@ Bs ) ) ).
% list_all_zip_map1
thf(fact_7946_list__all__zip__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As2: list @ A,F3: C > B,Bs: list @ C] :
( ( list_all_zip @ A @ B @ P @ As2 @ ( map @ C @ B @ F3 @ Bs ) )
= ( list_all_zip @ A @ C
@ ^ [A5: A,B4: C] : ( P @ A5 @ ( F3 @ B4 ) )
@ As2
@ Bs ) ) ).
% list_all_zip_map2
thf(fact_7947_list__all__zip__alt,axiom,
! [B: $tType,A: $tType] :
( ( list_all_zip @ A @ B )
= ( ^ [P2: A > B > $o,As4: list @ A,Bs3: list @ B] :
( ( ( size_size @ ( list @ A ) @ As4 )
= ( size_size @ ( list @ B ) @ Bs3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ As4 ) )
=> ( P2 @ ( nth @ A @ As4 @ I4 ) @ ( nth @ B @ Bs3 @ I4 ) ) ) ) ) ) ).
% list_all_zip_alt
thf(fact_7948_list__all__zip_Opelims_I3_J,axiom,
! [A: $tType,B: $tType,X3: A > B > $o,Xa: list @ A,Xb: list @ B] :
( ~ ( list_all_zip @ A @ B @ X3 @ 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ ( cons @ B @ B5 @ Bs2 ) ) ) )
=> ( ( X3 @ A6 @ B5 )
& ( list_all_zip @ A @ B @ X3 @ As @ Bs2 ) ) ) ) )
=> ( ! [V3: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: B,Va: list @ B] :
( ( Xb
= ( cons @ B @ V3 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(3)
thf(fact_7949_list__all__zip_Opelims_I2_J,axiom,
! [A: $tType,B: $tType,X3: A > B > $o,Xa: list @ A,Xb: list @ B] :
( ( list_all_zip @ A @ B @ X3 @ 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 ) ) @ X3 @ ( 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) )
=> ~ ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ ( cons @ B @ B5 @ Bs2 ) ) ) )
=> ~ ( ( X3 @ A6 @ B5 )
& ( list_all_zip @ A @ B @ X3 @ As @ Bs2 ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(2)
thf(fact_7950_list__all__zip_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X3: A > B > $o,Xa: list @ A,Xb: list @ B,Y: $o] :
( ( ( list_all_zip @ A @ B @ X3 @ 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 ) ) @ X3 @ ( 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
=> ( ! [A6: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ( ( Y
= ( ( X3 @ A6 @ B5 )
& ( list_all_zip @ A @ B @ X3 @ 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 ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As ) @ ( cons @ B @ B5 @ Bs2 ) ) ) ) ) ) )
=> ( ! [V3: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V3 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V3 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V3: B,Va: list @ B] :
( ( Xb
= ( cons @ B @ V3 @ Va ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(1)
thf(fact_7951_less__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ ( option @ A ) )
= ( ^ [X4: option @ A] :
( case_option @ $o @ A @ $false
@ ^ [Y5: A] :
( case_option @ $o @ A @ $true
@ ^ [Z6: A] : ( ord_less @ A @ Z6 @ Y5 )
@ X4 ) ) ) ) ) ).
% less_option_def
thf(fact_7952_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B2: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( C > ( product_prod @ A @ C ) ) @ ( D > ( product_prod @ B @ D ) ) @ A3 @ ( bNF_rel_fun @ C @ D @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ B2 @ ( basic_rel_prod @ A @ B @ C @ D @ A3 @ B2 ) ) @ ( product_Pair @ A @ C ) @ ( product_Pair @ B @ D ) ) ).
% Pair_transfer
thf(fact_7953_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R1: A > B > $o,R22: C > D > $o,A4: A,B3: C,C2: B,D3: D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R1 @ R22 @ ( product_Pair @ A @ C @ A4 @ B3 ) @ ( product_Pair @ B @ D @ C2 @ D3 ) )
= ( ( R1 @ A4 @ C2 )
& ( R22 @ B3 @ D3 ) ) ) ).
% rel_prod_inject
thf(fact_7954_prod__filter__parametric,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R: A > B > $o,S: C > D > $o] : ( bNF_rel_fun @ ( filter @ A ) @ ( filter @ B ) @ ( ( filter @ C ) > ( filter @ ( product_prod @ A @ C ) ) ) @ ( ( filter @ D ) > ( filter @ ( product_prod @ B @ D ) ) ) @ ( rel_filter @ A @ B @ R ) @ ( bNF_rel_fun @ ( filter @ C ) @ ( filter @ D ) @ ( filter @ ( product_prod @ A @ C ) ) @ ( filter @ ( product_prod @ B @ D ) ) @ ( rel_filter @ C @ D @ S ) @ ( rel_filter @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( basic_rel_prod @ A @ B @ C @ D @ R @ S ) ) ) @ ( prod_filter @ A @ C ) @ ( prod_filter @ B @ D ) ) ).
% prod_filter_parametric
thf(fact_7955_product__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B2: 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 @ A3 ) @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ ( list @ ( product_prod @ A @ C ) ) @ ( list @ ( product_prod @ B @ D ) ) @ ( list_all2 @ C @ D @ B2 ) @ ( list_all2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( basic_rel_prod @ A @ B @ C @ D @ A3 @ B2 ) ) ) @ ( product @ A @ C ) @ ( product @ B @ D ) ) ).
% product_transfer
thf(fact_7956_zip__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A3: A > B > $o,B2: 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 @ A3 ) @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ ( list @ ( product_prod @ A @ C ) ) @ ( list @ ( product_prod @ B @ D ) ) @ ( list_all2 @ C @ D @ B2 ) @ ( list_all2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( basic_rel_prod @ A @ B @ C @ D @ A3 @ B2 ) ) ) @ ( zip @ A @ C ) @ ( zip @ B @ D ) ) ).
% zip_transfer
thf(fact_7957_rtrancl__parametric,axiom,
! [A: $tType,B: $tType,A3: A > B > $o] :
( ( bi_unique @ A @ B @ A3 )
=> ( ( bi_total @ A @ B @ A3 )
=> ( 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 @ A3 @ A3 ) ) @ ( bNF_rel_set @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( basic_rel_prod @ A @ B @ A @ B @ A3 @ A3 ) ) @ ( transitive_rtrancl @ A ) @ ( transitive_rtrancl @ B ) ) ) ) ).
% rtrancl_parametric
thf(fact_7958_partition__transfer,axiom,
! [A: $tType,B: $tType,A3: 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 @ A3
@ ^ [Y4: $o,Z4: $o] : Y4 = Z4 )
@ ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( list_all2 @ A @ B @ A3 ) @ ( basic_rel_prod @ ( list @ A ) @ ( list @ B ) @ ( list @ A ) @ ( list @ B ) @ ( list_all2 @ A @ B @ A3 ) @ ( list_all2 @ A @ B @ A3 ) ) )
@ ( partition @ A )
@ ( partition @ B ) ) ).
% partition_transfer
thf(fact_7959_rel__prod_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R1: A > B > $o,R22: C > D > $o,A1: product_prod @ A @ C,A22: product_prod @ B @ D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R1 @ R22 @ A1 @ A22 )
=> ~ ! [A6: A,B5: B,C5: C] :
( ( A1
= ( product_Pair @ A @ C @ A6 @ C5 ) )
=> ! [D4: D] :
( ( A22
= ( product_Pair @ B @ D @ B5 @ D4 ) )
=> ( ( R1 @ A6 @ B5 )
=> ~ ( R22 @ C5 @ D4 ) ) ) ) ) ).
% rel_prod.cases
thf(fact_7960_rel__prod_Osimps,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( basic_rel_prod @ A @ B @ C @ D )
= ( ^ [R15: A > B > $o,R24: C > D > $o,A13: product_prod @ A @ C,A23: product_prod @ B @ D] :
? [A5: A,B4: B,C6: C,D6: D] :
( ( A13
= ( product_Pair @ A @ C @ A5 @ C6 ) )
& ( A23
= ( product_Pair @ B @ D @ B4 @ D6 ) )
& ( R15 @ A5 @ B4 )
& ( R24 @ C6 @ D6 ) ) ) ) ).
% rel_prod.simps
thf(fact_7961_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > B > $o,A4: A,B3: B,R22: C > D > $o,C2: C,D3: D] :
( ( R1 @ A4 @ B3 )
=> ( ( R22 @ C2 @ D3 )
=> ( basic_rel_prod @ A @ B @ C @ D @ R1 @ R22 @ ( product_Pair @ A @ C @ A4 @ C2 ) @ ( product_Pair @ B @ D @ B3 @ D3 ) ) ) ) ).
% rel_prod.intros
thf(fact_7962_refines__def,axiom,
! [A: $tType] :
( ( refine_Imp_refines @ A )
= ( ^ [P7: heap_Time_Heap @ A,Q8: heap_Time_Heap @ A] :
! [H3: heap_ext @ product_unit] :
( case_option @ $o @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ $true
@ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ $o
@ ^ [R5: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ $o
@ ^ [H8: heap_ext @ product_unit,T3: nat] :
( ( heap_Time_execute @ A @ P7 @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H8 @ T3 ) ) ) ) ) )
@ ( heap_Time_execute @ A @ Q8 @ H3 ) ) ) ) ).
% refines_def
thf(fact_7963_group_Oaxioms_I2_J,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F3 @ Z2 @ Inverse )
=> ( group_axioms @ A @ F3 @ Z2 @ Inverse ) ) ).
% group.axioms(2)
thf(fact_7964_group__axioms__def,axiom,
! [A: $tType] :
( ( group_axioms @ A )
= ( ^ [F7: A > A > A,Z6: A,Inverse2: A > A] :
( ! [A5: A] :
( ( F7 @ Z6 @ A5 )
= A5 )
& ! [A5: A] :
( ( F7 @ ( Inverse2 @ A5 ) @ A5 )
= Z6 ) ) ) ) ).
% group_axioms_def
thf(fact_7965_group__axioms_Ointro,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A] :
( ! [A6: A] :
( ( F3 @ Z2 @ A6 )
= A6 )
=> ( ! [A6: A] :
( ( F3 @ ( Inverse @ A6 ) @ A6 )
= Z2 )
=> ( group_axioms @ A @ F3 @ Z2 @ Inverse ) ) ) ).
% group_axioms.intro
thf(fact_7966_group_Ointro,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A] :
( ( semigroup @ A @ F3 )
=> ( ( group_axioms @ A @ F3 @ Z2 @ Inverse )
=> ( group @ A @ F3 @ Z2 @ Inverse ) ) ) ).
% group.intro
thf(fact_7967_group__def,axiom,
! [A: $tType] :
( ( group @ A )
= ( ^ [F7: A > A > A,Z6: A,Inverse2: A > A] :
( ( semigroup @ A @ F7 )
& ( group_axioms @ A @ F7 @ Z6 @ Inverse2 ) ) ) ) ).
% group_def
thf(fact_7968_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.semigroup_axioms
thf(fact_7969_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ( semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.semigroup_axioms
thf(fact_7970_inf_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semigroup @ A @ ( inf_inf @ A ) ) ) ).
% inf.semigroup_axioms
thf(fact_7971_max_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_max @ A ) ) ) ).
% max.semigroup_axioms
thf(fact_7972_append_Osemigroup__axioms,axiom,
! [A: $tType] : ( semigroup @ ( list @ A ) @ ( append @ A ) ) ).
% append.semigroup_axioms
thf(fact_7973_semigroup_Oassoc,axiom,
! [A: $tType,F3: A > A > A,A4: A,B3: A,C2: A] :
( ( semigroup @ A @ F3 )
=> ( ( F3 @ ( F3 @ A4 @ B3 ) @ C2 )
= ( F3 @ A4 @ ( F3 @ B3 @ C2 ) ) ) ) ).
% semigroup.assoc
thf(fact_7974_semigroup_Ointro,axiom,
! [A: $tType,F3: A > A > A] :
( ! [A6: A,B5: A,C5: A] :
( ( F3 @ ( F3 @ A6 @ B5 ) @ C5 )
= ( F3 @ A6 @ ( F3 @ B5 @ C5 ) ) )
=> ( semigroup @ A @ F3 ) ) ).
% semigroup.intro
thf(fact_7975_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F7: A > A > A] :
! [A5: A,B4: A,C6: A] :
( ( F7 @ ( F7 @ A5 @ B4 ) @ C6 )
= ( F7 @ A5 @ ( F7 @ B4 @ C6 ) ) ) ) ) ).
% semigroup_def
thf(fact_7976_min_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_min @ A ) ) ) ).
% min.semigroup_axioms
thf(fact_7977_group_Oaxioms_I1_J,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F3 @ Z2 @ Inverse )
=> ( semigroup @ A @ F3 ) ) ).
% group.axioms(1)
thf(fact_7978_sup_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semigroup @ A @ ( sup_sup @ A ) ) ) ).
% sup.semigroup_axioms
thf(fact_7979_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_7980_bot__pred__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( pred @ A ) )
= ( pred2 @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_pred_def
thf(fact_7981_ordering__top__axioms__def,axiom,
! [A: $tType] :
( ( ordering_top_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Top2: A] :
! [A5: A] : ( Less_eq2 @ A5 @ Top2 ) ) ) ).
% ordering_top_axioms_def
thf(fact_7982_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_7983_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_7984_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_7985_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A
@ ^ [X4: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X4 )
@ ^ [X4: A,Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ) ).
% dual_order.ordering_axioms
thf(fact_7986_order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.ordering_axioms
thf(fact_7987_ordering_Oeq__iff,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A4 = B3 )
= ( ( Less_eq @ A4 @ B3 )
& ( Less_eq @ B3 @ A4 ) ) ) ) ).
% ordering.eq_iff
thf(fact_7988_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
=> ( ( Less_eq @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ).
% ordering.antisym
thf(fact_7989_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A4 @ B3 )
= ( ( Less @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% ordering.order_iff_strict
thf(fact_7990_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
= ( ( Less_eq @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% ordering.strict_iff_order
thf(fact_7991_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A4 @ B3 )
=> ( A4 != B3 ) ) ) ).
% ordering.strict_implies_not_eq
thf(fact_7992_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A4: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A4 != B3 )
=> ( ( Less_eq @ A4 @ B3 )
=> ( Less @ A4 @ B3 ) ) ) ) ).
% ordering.not_eq_order_implies_strict
thf(fact_7993_ordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A6: A,B5: A] :
( ( Less_eq @ A6 @ B5 )
= ( ( Less @ A6 @ B5 )
| ( A6 = B5 ) ) )
=> ( ! [A6: A,B5: A] :
( ( Less @ A6 @ B5 )
=> ~ ( Less @ B5 @ A6 ) )
=> ( ! [A6: A] :
~ ( Less @ A6 @ A6 )
=> ( ! [A6: A,B5: A,C5: A] :
( ( Less @ A6 @ B5 )
=> ( ( Less @ B5 @ C5 )
=> ( Less @ A6 @ C5 ) ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% ordering_strictI
thf(fact_7994_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_7995_ordering__dualI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( ordering @ A
@ ^ [A5: A,B4: A] : ( Less_eq @ B4 @ A5 )
@ ^ [A5: A,B4: A] : ( Less @ B4 @ A5 ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ).
% ordering_dualI
thf(fact_7996_ID_Opred__set,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o ) )
= ( ^ [P2: A > $o,X4: A] :
! [Y5: A] :
( ( member @ A @ Y5 @ ( insert3 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( P2 @ Y5 ) ) ) ) ).
% ID.pred_set
thf(fact_7997_Predicate_Oiterate__upto_Opelims,axiom,
! [A: $tType,X3: code_natural > A,Xa: code_natural,Xb: code_natural,Y: pred @ A] :
( ( ( iterate_upto @ A @ X3 @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ X3 @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) )
=> ~ ( ( Y
= ( seq2 @ A
@ ^ [U3: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ Xb @ Xa ) @ ( empty @ A ) @ ( insert2 @ A @ ( X3 @ Xa ) @ ( iterate_upto @ A @ X3 @ ( plus_plus @ code_natural @ Xa @ ( one_one @ code_natural ) ) @ Xb ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ X3 @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) ) ) ) ) ).
% Predicate.iterate_upto.pelims
thf(fact_7998_ID_Opred__cong,axiom,
! [A: $tType,X3: A,Ya: A,P: A > $o,Pa: A > $o] :
( ( X3 = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( insert3 @ A @ Ya @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( bNF_id_bnf @ ( A > $o ) @ P @ X3 )
= ( bNF_id_bnf @ ( A > $o ) @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_7999_ID_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X3: A,Pa: A > $o] :
( ( bNF_id_bnf @ ( A > $o ) @ P @ X3 )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( insert3 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( bNF_id_bnf @ ( A > $o ) @ Pa @ X3 ) ) ) ).
% ID.pred_mono_strong
thf(fact_8000_Predicate_Oiterate__upto_Opsimps,axiom,
! [A: $tType,F3: code_natural > A,N: code_natural,M2: code_natural] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F3 @ ( product_Pair @ code_natural @ code_natural @ N @ M2 ) ) )
=> ( ( iterate_upto @ A @ F3 @ N @ M2 )
= ( seq2 @ A
@ ^ [U3: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M2 @ N ) @ ( empty @ A ) @ ( insert2 @ A @ ( F3 @ N ) @ ( iterate_upto @ A @ F3 @ ( plus_plus @ code_natural @ N @ ( one_one @ code_natural ) ) @ M2 ) ) ) ) ) ) ).
% Predicate.iterate_upto.psimps
thf(fact_8001_Random__Pred_Oiterate__upto__def,axiom,
! [A: $tType] :
( ( random_iterate_upto @ A )
= ( ^ [F7: code_natural > A,N2: code_natural,M: code_natural] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( iterate_upto @ A @ F7 @ N2 @ M ) ) ) ) ).
% Random_Pred.iterate_upto_def
thf(fact_8002_of__seq__code_I1_J,axiom,
! [A: $tType] :
( ( set_of_seq @ A @ ( empty @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% of_seq_code(1)
thf(fact_8003_of__seq__code_I2_J,axiom,
! [B: $tType,X3: B,P: pred @ B] :
( ( set_of_seq @ B @ ( insert2 @ B @ X3 @ P ) )
= ( insert3 @ B @ X3 @ ( set_of_pred @ B @ P ) ) ) ).
% of_seq_code(2)
thf(fact_8004_of__pred__code,axiom,
! [A: $tType,F3: product_unit > ( seq @ A )] :
( ( set_of_pred @ A @ ( seq2 @ A @ F3 ) )
= ( case_seq @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X4: A,P2: pred @ A] : ( insert3 @ A @ X4 @ ( set_of_pred @ A @ P2 ) )
@ ^ [P2: pred @ A,Xq: seq @ A] : ( sup_sup @ ( set @ A ) @ ( set_of_pred @ A @ P2 ) @ ( set_of_seq @ A @ Xq ) )
@ ( F3 @ product_Unity ) ) ) ).
% of_pred_code
% Type constructors (831)
thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( heap_Time_Heap @ A19 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( typerep @ ( heap_Time_Heap @ A19 ) ) ) ).
thf(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of_1,axiom,
code_term_of @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_2,axiom,
typerep @ code_natural ).
thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_3,axiom,
code_term_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_4,axiom,
typerep @ code_integer ).
thf(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_5,axiom,
code_term_of @ code_term ).
thf(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_6,axiom,
typerep @ code_term ).
thf(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_7,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( heap_ext @ A19 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_8,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( typerep @ ( heap_ext @ A19 ) ) ) ).
thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_9,axiom,
code_term_of @ product_unit ).
thf(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
enum @ product_unit ).
thf(tcon_Product__Type_Ounit___Typerep_Otyperep_10,axiom,
typerep @ product_unit ).
thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_11,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep @ A19 )
& ( typerep @ A20 ) )
=> ( code_term_of @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Enum_Oenum_12,axiom,
! [A19: $tType,A20: $tType] :
( ( ( enum @ A19 )
& ( enum @ A20 ) )
=> ( enum @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_13,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep @ A19 )
& ( typerep @ A20 ) )
=> ( typerep @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_14,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Typerep_Otyperep_15,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( typerep @ ( multiset @ A19 ) ) ) ).
thf(tcon_Assertions_Oassn___Typerep_Otyperep_16,axiom,
typerep @ assn ).
thf(tcon_Predicate_Opred___Code__Evaluation_Oterm__of_17,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( pred @ A19 ) ) ) ).
thf(tcon_Predicate_Opred___Typerep_Otyperep_18,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( typerep @ ( pred @ A19 ) ) ) ).
thf(tcon_Predicate_Oseq___Code__Evaluation_Oterm__of_19,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( seq @ A19 ) ) ) ).
thf(tcon_Predicate_Oseq___Typerep_Otyperep_20,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( typerep @ ( seq @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_21,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Enum_Oenum_22,axiom,
! [A19: $tType] :
( ( enum @ A19 )
=> ( enum @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Typerep_Otyperep_23,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( typerep @ ( option @ A19 ) ) ) ).
thf(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_24,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( filter @ A19 ) ) ) ).
thf(tcon_Filter_Ofilter___Typerep_Otyperep_25,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( typerep @ ( filter @ A19 ) ) ) ).
thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_26,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep @ A19 )
& ( typerep @ A20 ) )
=> ( code_term_of @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Enum_Oenum_27,axiom,
! [A19: $tType,A20: $tType] :
( ( ( enum @ A19 )
& ( enum @ A20 ) )
=> ( enum @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_28,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep @ A19 )
& ( typerep @ A20 ) )
=> ( typerep @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_29,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( array @ A19 ) ) ) ).
thf(tcon_Heap_Oarray___Typerep_Otyperep_30,axiom,
! [A19: $tType] : ( typerep @ ( array @ A19 ) ) ).
thf(tcon_List_Olist___Code__Evaluation_Oterm__of_31,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( list @ A19 ) ) ) ).
thf(tcon_List_Olist___Typerep_Otyperep_32,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( typerep @ ( list @ A19 ) ) ) ).
thf(tcon_Heap_Oref___Code__Evaluation_Oterm__of_33,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( ref @ A19 ) ) ) ).
thf(tcon_Heap_Oref___Typerep_Otyperep_34,axiom,
! [A19: $tType] : ( typerep @ ( ref @ A19 ) ) ).
thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_35,axiom,
code_term_of @ $o ).
thf(tcon_HOL_Obool___Enum_Oenum_36,axiom,
enum @ $o ).
thf(tcon_HOL_Obool___Typerep_Otyperep_37,axiom,
typerep @ $o ).
thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_38,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( set @ A19 ) ) ) ).
thf(tcon_Set_Oset___Enum_Oenum_39,axiom,
! [A19: $tType] :
( ( enum @ A19 )
=> ( enum @ ( set @ A19 ) ) ) ).
thf(tcon_Set_Oset___Typerep_Otyperep_40,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( typerep @ ( set @ A19 ) ) ) ).
thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_41,axiom,
code_term_of @ rat ).
thf(tcon_Rat_Orat___Typerep_Otyperep_42,axiom,
typerep @ rat ).
thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_43,axiom,
code_term_of @ num ).
thf(tcon_Num_Onum___Typerep_Otyperep_44,axiom,
typerep @ num ).
thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_45,axiom,
code_term_of @ nat ).
thf(tcon_Nat_Onat___Typerep_Otyperep_46,axiom,
typerep @ nat ).
thf(tcon_Int_Oint___Code__Evaluation_Oterm__of_47,axiom,
code_term_of @ int ).
thf(tcon_Int_Oint___Typerep_Otyperep_48,axiom,
typerep @ int ).
thf(tcon_itself___Code__Evaluation_Oterm__of_49,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( code_term_of @ ( itself @ A19 ) ) ) ).
thf(tcon_itself___Typerep_Otyperep_50,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( typerep @ ( itself @ A19 ) ) ) ).
thf(tcon_fun___Code__Evaluation_Oterm__of_51,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep @ A19 )
& ( typerep @ A20 ) )
=> ( code_term_of @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Enum_Oenum_52,axiom,
! [A19: $tType,A20: $tType] :
( ( ( enum @ A19 )
& ( enum @ A20 ) )
=> ( enum @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Typerep_Otyperep_53,axiom,
! [A19: $tType,A20: $tType] :
( ( ( typerep @ A19 )
& ( typerep @ A20 ) )
=> ( typerep @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A19: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( condit1219197933456340205attice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A19: $tType,A20: $tType] :
( ( comple592849572758109894attice @ A20 )
=> ( comple592849572758109894attice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
! [A19: $tType,A20: $tType] :
( ( ( cl_HOL_Oequal @ A19 )
& ( quickc3360725361186068524ustive @ A19 )
& ( quickc3360725361186068524ustive @ A20 ) )
=> ( quickc3360725361186068524ustive @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A19: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounde4967611905675639751up_bot @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A19: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounde4346867609351753570nf_top @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A19: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple6319245703460814977attice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
! [A19: $tType,A20: $tType] :
( ( ( cl_HOL_Oequal @ A19 )
& ( quickc658316121487927005ustive @ A19 )
& ( quickc658316121487927005ustive @ A20 ) )
=> ( quickc658316121487927005ustive @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A19: $tType,A20: $tType] :
( ( boolea8198339166811842893lgebra @ A20 )
=> ( boolea8198339166811842893lgebra @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A19: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounded_lattice_top @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A19: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounded_lattice_bot @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A19: $tType,A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple9053668089753744459l_ccpo @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Quickcheck__Random_Orandom,axiom,
! [A19: $tType,A20: $tType] :
( ( ( code_term_of @ A19 )
& ( cl_HOL_Oequal @ A19 )
& ( quickcheck_random @ A20 ) )
=> ( quickcheck_random @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A19: $tType,A20: $tType] :
( ( semilattice_sup @ A20 )
=> ( semilattice_sup @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A19: $tType,A20: $tType] :
( ( semilattice_inf @ A20 )
=> ( semilattice_inf @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A19: $tType,A20: $tType] :
( ( distrib_lattice @ A20 )
=> ( distrib_lattice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A19: $tType,A20: $tType] :
( ( bounded_lattice @ A20 )
=> ( bounded_lattice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A19: $tType,A20: $tType] :
( ( order_top @ A20 )
=> ( order_top @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A19: $tType,A20: $tType] :
( ( order_bot @ A20 )
=> ( order_bot @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A19: $tType,A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A19: $tType,A20: $tType] :
( ( ( finite_finite @ A19 )
& ( finite_finite @ A20 ) )
=> ( finite_finite @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A19: $tType,A20: $tType] :
( ( lattice @ A20 )
=> ( lattice @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A19: $tType,A20: $tType] :
( ( order @ A20 )
=> ( order @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A19: $tType,A20: $tType] :
( ( top @ A20 )
=> ( top @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A19: $tType,A20: $tType] :
( ( ord @ A20 )
=> ( ord @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A19: $tType,A20: $tType] :
( ( bot @ A20 )
=> ( bot @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A19: $tType,A20: $tType] :
( ( uminus @ A20 )
=> ( uminus @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A19: $tType,A20: $tType] :
( ( minus @ A20 )
=> ( minus @ ( A19 > A20 ) ) ) ).
thf(tcon_fun___HOL_Oequal,axiom,
! [A19: $tType,A20: $tType] :
( ( ( enum @ A19 )
& ( cl_HOL_Oequal @ A20 ) )
=> ( cl_HOL_Oequal @ ( A19 > A20 ) ) ) ).
thf(tcon_itself___Quickcheck__Random_Orandom_54,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( quickcheck_random @ ( itself @ A19 ) ) ) ).
thf(tcon_itself___HOL_Oequal_55,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( itself @ A19 ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_56,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative,axiom,
normal6328177297339901930cative @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Quickcheck__Exhaustive_Ofull__exhaustive_57,axiom,
quickc3360725361186068524ustive @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
euclid5891614535332579305n_ring @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Quickcheck__Exhaustive_Oexhaustive_58,axiom,
quickc658316121487927005ustive @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
normal8620421768224518004emidom @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Quickcheck__Random_Orandom_59,axiom,
quickcheck_random @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_60,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_61,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_62,axiom,
distrib_lattice @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel @ 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___Parity_Oring__parity,axiom,
ring_parity @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_63,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
idom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
idom_divide @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_64,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_65,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_66,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_67,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___Groups_Otimes,axiom,
times @ int ).
thf(tcon_Int_Oint___Groups_Ominus_68,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___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Int_Oint___Heap_Oheap,axiom,
heap @ int ).
thf(tcon_Int_Oint___HOL_Oequal_69,axiom,
cl_HOL_Oequal @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_70,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_71,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_72,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_73,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_74,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_75,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_76,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_77,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_78,axiom,
normal6328177297339901930cative @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_79,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_80,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_81,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_82,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_83,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_84,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_85,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_86,axiom,
quickc3360725361186068524ustive @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_87,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_88,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_89,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_90,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_91,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_92,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_93,axiom,
semido2269285787275462019factor @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_94,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_95,axiom,
quickc658316121487927005ustive @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_96,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_97,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_98,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_99,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_100,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_101,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_102,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_103,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_104,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom_105,axiom,
normal8620421768224518004emidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_106,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_107,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_108,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_109,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Random_Orandom_110,axiom,
quickcheck_random @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_111,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_112,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_113,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_114,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_115,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_116,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_117,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_118,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_119,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_120,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_121,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_122,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_123,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_124,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_125,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_126,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_127,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_128,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_129,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_130,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_131,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_132,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_133,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_134,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_135,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_136,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_137,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_138,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_139,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_140,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_141,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_142,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_143,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_144,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_145,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_146,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_147,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_148,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_149,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes_150,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_151,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_152,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_153,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_154,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_155,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_156,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_157,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_158,axiom,
heap @ nat ).
thf(tcon_Nat_Onat___HOL_Oequal_159,axiom,
cl_HOL_Oequal @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_160,axiom,
quickc3360725361186068524ustive @ num ).
thf(tcon_Num_Onum___Quickcheck__Random_Orandom_161,axiom,
quickcheck_random @ num ).
thf(tcon_Num_Onum___Orderings_Opreorder_162,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_163,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_164,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_165,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Otimes_166,axiom,
times @ num ).
thf(tcon_Num_Onum___Groups_Oplus_167,axiom,
plus @ num ).
thf(tcon_Num_Onum___HOL_Oequal_168,axiom,
cl_HOL_Oequal @ num ).
thf(tcon_Num_Onum___Nat_Osize_169,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_170,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_171,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_172,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_173,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_174,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_175,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_176,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_177,axiom,
quickc3360725361186068524ustive @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_178,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_179,axiom,
ordere8940638589300402666id_add @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
archim462609752435547400_field @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_180,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_181,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_182,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_183,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_184,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_185,axiom,
quickc658316121487927005ustive @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_186,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_187,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_188,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_189,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_190,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_191,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_192,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_193,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_194,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_195,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_196,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_197,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_198,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_199,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_200,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_201,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_202,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Random_Orandom_203,axiom,
quickcheck_random @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_204,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_205,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_206,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_207,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_208,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_209,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_210,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_211,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_212,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_213,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_214,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_215,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_216,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_217,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_218,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_219,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_220,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_221,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_222,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_223,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_224,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_225,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_226,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_227,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_228,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_229,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_230,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_231,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_232,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_233,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_234,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_235,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_236,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_237,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_238,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_239,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_240,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_241,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_242,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_243,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_244,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_245,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_246,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_247,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_248,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_249,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_250,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Groups_Otimes_251,axiom,
times @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_252,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_253,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_254,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_255,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_256,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_257,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_258,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_259,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_260,axiom,
dvd @ rat ).
thf(tcon_Rat_Orat___HOL_Oequal_261,axiom,
cl_HOL_Oequal @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_262,axiom,
! [A19: $tType] : ( condit1219197933456340205attice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_263,axiom,
! [A19: $tType] : ( comple592849572758109894attice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_264,axiom,
! [A19: $tType] :
( ( quickc3360725361186068524ustive @ A19 )
=> ( quickc3360725361186068524ustive @ ( set @ A19 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_265,axiom,
! [A19: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_266,axiom,
! [A19: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_267,axiom,
! [A19: $tType] : ( comple6319245703460814977attice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_268,axiom,
! [A19: $tType] :
( ( quickc658316121487927005ustive @ A19 )
=> ( quickc658316121487927005ustive @ ( set @ A19 ) ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_269,axiom,
! [A19: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_270,axiom,
! [A19: $tType] : ( bounded_lattice_top @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_271,axiom,
! [A19: $tType] : ( bounded_lattice_bot @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_272,axiom,
! [A19: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Random_Orandom_273,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( set @ A19 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_274,axiom,
! [A19: $tType] : ( semilattice_sup @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_275,axiom,
! [A19: $tType] : ( semilattice_inf @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_276,axiom,
! [A19: $tType] : ( distrib_lattice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_277,axiom,
! [A19: $tType] : ( bounded_lattice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_278,axiom,
! [A19: $tType] : ( order_top @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_279,axiom,
! [A19: $tType] : ( order_bot @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_280,axiom,
! [A19: $tType] : ( preorder @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_281,axiom,
! [A19: $tType] :
( ( finite_finite @ A19 )
=> ( finite_finite @ ( set @ A19 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_282,axiom,
! [A19: $tType] : ( lattice @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_283,axiom,
! [A19: $tType] : ( order @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_284,axiom,
! [A19: $tType] : ( top @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_285,axiom,
! [A19: $tType] : ( ord @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_286,axiom,
! [A19: $tType] : ( bot @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_287,axiom,
! [A19: $tType] : ( uminus @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_288,axiom,
! [A19: $tType] : ( minus @ ( set @ A19 ) ) ).
thf(tcon_Set_Oset___HOL_Oequal_289,axiom,
! [A19: $tType] :
( ( cl_HOL_Oequal @ A19 )
=> ( cl_HOL_Oequal @ ( set @ A19 ) ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_290,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_291,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_292,axiom,
quickc3360725361186068524ustive @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_293,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_294,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_295,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_296,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_297,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_298,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_299,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Random_Orandom_300,axiom,
quickcheck_random @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_301,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_302,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_303,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_304,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_305,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_306,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_307,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_308,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_309,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_310,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_311,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_312,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_313,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_314,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_315,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_316,axiom,
minus @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_317,axiom,
heap @ $o ).
thf(tcon_HOL_Obool___HOL_Oequal_318,axiom,
cl_HOL_Oequal @ $o ).
thf(tcon_Heap_Oref___Quickcheck__Exhaustive_Ofull__exhaustive_319,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( quickc3360725361186068524ustive @ ( ref @ A19 ) ) ) ).
thf(tcon_Heap_Oref___Quickcheck__Random_Orandom_320,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( quickcheck_random @ ( ref @ A19 ) ) ) ).
thf(tcon_Heap_Oref___Heap_Oheap_321,axiom,
! [A19: $tType] : ( heap @ ( ref @ A19 ) ) ).
thf(tcon_Heap_Oref___HOL_Oequal_322,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( ref @ A19 ) ) ).
thf(tcon_Heap_Oref___Nat_Osize_323,axiom,
! [A19: $tType] : ( size @ ( ref @ A19 ) ) ).
thf(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_324,axiom,
! [A19: $tType] :
( ( quickc3360725361186068524ustive @ A19 )
=> ( quickc3360725361186068524ustive @ ( list @ A19 ) ) ) ).
thf(tcon_List_Olist___Quickcheck__Random_Orandom_325,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( list @ A19 ) ) ) ).
thf(tcon_List_Olist___Heap_Oheap_326,axiom,
! [A19: $tType] :
( ( heap @ A19 )
=> ( heap @ ( list @ A19 ) ) ) ).
thf(tcon_List_Olist___HOL_Oequal_327,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( list @ A19 ) ) ).
thf(tcon_List_Olist___Nat_Osize_328,axiom,
! [A19: $tType] : ( size @ ( list @ A19 ) ) ).
thf(tcon_Heap_Oarray___Quickcheck__Exhaustive_Ofull__exhaustive_329,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( quickc3360725361186068524ustive @ ( array @ A19 ) ) ) ).
thf(tcon_Heap_Oarray___Quickcheck__Random_Orandom_330,axiom,
! [A19: $tType] :
( ( typerep @ A19 )
=> ( quickcheck_random @ ( array @ A19 ) ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_331,axiom,
! [A19: $tType] : ( heap @ ( array @ A19 ) ) ).
thf(tcon_Heap_Oarray___HOL_Oequal_332,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( array @ A19 ) ) ).
thf(tcon_Heap_Oarray___Nat_Osize_333,axiom,
! [A19: $tType] : ( size @ ( array @ A19 ) ) ).
thf(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_334,axiom,
! [A19: $tType,A20: $tType] :
( ( ( quickc3360725361186068524ustive @ A19 )
& ( quickc3360725361186068524ustive @ A20 ) )
=> ( quickc3360725361186068524ustive @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_335,axiom,
! [A19: $tType,A20: $tType] :
( ( ( quickcheck_random @ A19 )
& ( quickcheck_random @ A20 ) )
=> ( quickcheck_random @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_336,axiom,
! [A19: $tType,A20: $tType] :
( ( ( finite_finite @ A19 )
& ( finite_finite @ A20 ) )
=> ( finite_finite @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_337,axiom,
! [A19: $tType,A20: $tType] :
( ( ( heap @ A19 )
& ( heap @ A20 ) )
=> ( heap @ ( sum_sum @ A19 @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___HOL_Oequal_338,axiom,
! [A19: $tType,A20: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A19 @ A20 ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_339,axiom,
! [A19: $tType,A20: $tType] : ( size @ ( sum_sum @ A19 @ A20 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_340,axiom,
! [A19: $tType] : ( condit1219197933456340205attice @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_341,axiom,
! [A19: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_342,axiom,
! [A19: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_343,axiom,
! [A19: $tType] : ( comple6319245703460814977attice @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_344,axiom,
! [A19: $tType] : ( bounded_lattice_top @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_345,axiom,
! [A19: $tType] : ( bounded_lattice_bot @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_346,axiom,
! [A19: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_347,axiom,
! [A19: $tType] : ( semilattice_sup @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_348,axiom,
! [A19: $tType] : ( semilattice_inf @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_349,axiom,
! [A19: $tType] : ( distrib_lattice @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_350,axiom,
! [A19: $tType] : ( bounded_lattice @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_351,axiom,
! [A19: $tType] : ( order_top @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_352,axiom,
! [A19: $tType] : ( order_bot @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_353,axiom,
! [A19: $tType] : ( preorder @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_354,axiom,
! [A19: $tType] : ( lattice @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_355,axiom,
! [A19: $tType] : ( order @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_356,axiom,
! [A19: $tType] : ( top @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_357,axiom,
! [A19: $tType] : ( ord @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_358,axiom,
! [A19: $tType] : ( bot @ ( filter @ A19 ) ) ).
thf(tcon_Filter_Ofilter___HOL_Oequal_359,axiom,
! [A19: $tType] :
( ( cl_HOL_Oequal @ A19 )
=> ( cl_HOL_Oequal @ ( filter @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_360,axiom,
! [A19: $tType] :
( ( comple5582772986160207858norder @ A19 )
=> ( condit6923001295902523014norder @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_361,axiom,
! [A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( condit1219197933456340205attice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_362,axiom,
! [A19: $tType] :
( ( comple592849572758109894attice @ A19 )
=> ( comple592849572758109894attice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_363,axiom,
! [A19: $tType] :
( ( quickc3360725361186068524ustive @ A19 )
=> ( quickc3360725361186068524ustive @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_364,axiom,
! [A19: $tType] :
( ( lattice @ A19 )
=> ( bounde4967611905675639751up_bot @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_365,axiom,
! [A19: $tType] :
( ( bounded_lattice_top @ A19 )
=> ( bounde4346867609351753570nf_top @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
! [A19: $tType] :
( ( comple5582772986160207858norder @ A19 )
=> ( comple5582772986160207858norder @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_366,axiom,
! [A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( comple6319245703460814977attice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_367,axiom,
! [A19: $tType] :
( ( bounded_lattice_top @ A19 )
=> ( bounded_lattice_top @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_368,axiom,
! [A19: $tType] :
( ( lattice @ A19 )
=> ( bounded_lattice_bot @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_369,axiom,
! [A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( comple9053668089753744459l_ccpo @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Random_Orandom_370,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_371,axiom,
! [A19: $tType] :
( ( semilattice_sup @ A19 )
=> ( semilattice_sup @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_372,axiom,
! [A19: $tType] :
( ( semilattice_inf @ A19 )
=> ( semilattice_inf @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Odistrib__lattice_373,axiom,
! [A19: $tType] :
( ( distrib_lattice @ A19 )
=> ( distrib_lattice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice_374,axiom,
! [A19: $tType] :
( ( bounded_lattice_top @ A19 )
=> ( bounded_lattice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_375,axiom,
! [A19: $tType] :
( ( wellorder @ A19 )
=> ( wellorder @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_376,axiom,
! [A19: $tType] :
( ( order_top @ A19 )
=> ( order_top @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_377,axiom,
! [A19: $tType] :
( ( order @ A19 )
=> ( order_bot @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_378,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( preorder @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_379,axiom,
! [A19: $tType] :
( ( linorder @ A19 )
=> ( linorder @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_380,axiom,
! [A19: $tType] :
( ( finite_finite @ A19 )
=> ( finite_finite @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Olattice_381,axiom,
! [A19: $tType] :
( ( lattice @ A19 )
=> ( lattice @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_382,axiom,
! [A19: $tType] :
( ( order @ A19 )
=> ( order @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Otop_383,axiom,
! [A19: $tType] :
( ( order_top @ A19 )
=> ( top @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_384,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( ord @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Obot_385,axiom,
! [A19: $tType] :
( ( order @ A19 )
=> ( bot @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_386,axiom,
! [A19: $tType] :
( ( heap @ A19 )
=> ( heap @ ( option @ A19 ) ) ) ).
thf(tcon_Option_Ooption___HOL_Oequal_387,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( option @ A19 ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_388,axiom,
! [A19: $tType] : ( size @ ( option @ A19 ) ) ).
thf(tcon_Predicate_Oseq___HOL_Oequal_389,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( seq @ A19 ) ) ).
thf(tcon_Predicate_Opred___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_390,axiom,
! [A19: $tType] : ( condit1219197933456340205attice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__distrib__lattice_391,axiom,
! [A19: $tType] : ( comple592849572758109894attice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__semilattice__sup__bot_392,axiom,
! [A19: $tType] : ( bounde4967611905675639751up_bot @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__semilattice__inf__top_393,axiom,
! [A19: $tType] : ( bounde4346867609351753570nf_top @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_394,axiom,
! [A19: $tType] : ( comple6319245703460814977attice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_395,axiom,
! [A19: $tType] : ( boolea8198339166811842893lgebra @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_396,axiom,
! [A19: $tType] : ( bounded_lattice_top @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice__bot_397,axiom,
! [A19: $tType] : ( bounded_lattice_bot @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_398,axiom,
! [A19: $tType] : ( comple9053668089753744459l_ccpo @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__sup_399,axiom,
! [A19: $tType] : ( semilattice_sup @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__inf_400,axiom,
! [A19: $tType] : ( semilattice_inf @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Odistrib__lattice_401,axiom,
! [A19: $tType] : ( distrib_lattice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice_402,axiom,
! [A19: $tType] : ( bounded_lattice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder__top_403,axiom,
! [A19: $tType] : ( order_top @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder__bot_404,axiom,
! [A19: $tType] : ( order_bot @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Opreorder_405,axiom,
! [A19: $tType] : ( preorder @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Olattice_406,axiom,
! [A19: $tType] : ( lattice @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder_407,axiom,
! [A19: $tType] : ( order @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Otop_408,axiom,
! [A19: $tType] : ( top @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oord_409,axiom,
! [A19: $tType] : ( ord @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Obot_410,axiom,
! [A19: $tType] : ( bot @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Groups_Ouminus_411,axiom,
! [A19: $tType] : ( uminus @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___Groups_Ominus_412,axiom,
! [A19: $tType] : ( minus @ ( pred @ A19 ) ) ).
thf(tcon_Predicate_Opred___HOL_Oequal_413,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( pred @ A19 ) ) ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_414,axiom,
bounde4967611905675639751up_bot @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_415,axiom,
bounde4346867609351753570nf_top @ assn ).
thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_416,axiom,
boolea8198339166811842893lgebra @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_417,axiom,
bounded_lattice_top @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_418,axiom,
bounded_lattice_bot @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_419,axiom,
semilattice_sup @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_420,axiom,
semilattice_inf @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_421,axiom,
distrib_lattice @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice_422,axiom,
bounded_lattice @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_423,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_424,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_425,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__top_426,axiom,
order_top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_427,axiom,
order_bot @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Opreorder_428,axiom,
preorder @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_429,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Olattice_430,axiom,
lattice @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder_431,axiom,
order @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Otop_432,axiom,
top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oord_433,axiom,
ord @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Obot_434,axiom,
bot @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ouminus_435,axiom,
uminus @ assn ).
thf(tcon_Assertions_Oassn___Groups_Otimes_436,axiom,
times @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ominus_437,axiom,
minus @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_438,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_439,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_440,axiom,
dvd @ assn ).
thf(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_441,axiom,
! [A19: $tType] :
( ( quickc3360725361186068524ustive @ A19 )
=> ( quickc3360725361186068524ustive @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_442,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_443,axiom,
! [A19: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_444,axiom,
! [A19: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_445,axiom,
! [A19: $tType] : ( cancel_semigroup_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_446,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_447,axiom,
! [A19: $tType] : ( comm_monoid_diff @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_448,axiom,
! [A19: $tType] : ( ab_semigroup_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_449,axiom,
! [A19: $tType] : ( comm_monoid_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_450,axiom,
! [A19: $tType] : ( semigroup_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_451,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( preorder @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_452,axiom,
! [A19: $tType] : ( monoid_add @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_453,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( order @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_454,axiom,
! [A19: $tType] :
( ( preorder @ A19 )
=> ( ord @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ominus_455,axiom,
! [A19: $tType] : ( minus @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_456,axiom,
! [A19: $tType] : ( zero @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oplus_457,axiom,
! [A19: $tType] : ( plus @ ( multiset @ A19 ) ) ).
thf(tcon_Multiset_Omultiset___HOL_Oequal_458,axiom,
! [A19: $tType] :
( ( cl_HOL_Oequal @ A19 )
=> ( cl_HOL_Oequal @ ( multiset @ A19 ) ) ) ).
thf(tcon_Multiset_Omultiset___Nat_Osize_459,axiom,
! [A19: $tType] : ( size @ ( multiset @ A19 ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_460,axiom,
! [A19: $tType,A20: $tType] :
( ( ( quickc3360725361186068524ustive @ A19 )
& ( quickc3360725361186068524ustive @ A20 ) )
=> ( quickc3360725361186068524ustive @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_461,axiom,
! [A19: $tType,A20: $tType] :
( ( ( quickc658316121487927005ustive @ A19 )
& ( quickc658316121487927005ustive @ A20 ) )
=> ( quickc658316121487927005ustive @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_462,axiom,
! [A19: $tType,A20: $tType] :
( ( ( quickcheck_random @ A19 )
& ( quickcheck_random @ A20 ) )
=> ( quickcheck_random @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_463,axiom,
! [A19: $tType,A20: $tType] :
( ( ( finite_finite @ A19 )
& ( finite_finite @ A20 ) )
=> ( finite_finite @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_464,axiom,
! [A19: $tType,A20: $tType] :
( ( ( heap @ A19 )
& ( heap @ A20 ) )
=> ( heap @ ( product_prod @ A19 @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___HOL_Oequal_465,axiom,
! [A19: $tType,A20: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A19 @ A20 ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_466,axiom,
! [A19: $tType,A20: $tType] : ( size @ ( product_prod @ A19 @ A20 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_467,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_468,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_469,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_470,axiom,
quickc3360725361186068524ustive @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_471,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_472,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_473,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_474,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_475,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_476,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_477,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_478,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_479,axiom,
quickcheck_random @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_480,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_481,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_482,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_483,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_484,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_485,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_486,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_487,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_488,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_489,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_490,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_491,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_492,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_493,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_494,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_495,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_496,axiom,
minus @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_497,axiom,
heap @ product_unit ).
thf(tcon_Product__Type_Ounit___HOL_Oequal_498,axiom,
cl_HOL_Oequal @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_499,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( heap_ext @ A19 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_500,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( heap_ext @ A19 ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_501,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_502,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_503,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_504,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_505,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_506,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_507,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_508,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_509,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_510,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_511,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_512,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_513,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_514,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_515,axiom,
quickc3360725361186068524ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_516,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_517,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_518,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_519,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_520,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_521,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_522,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_523,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_524,axiom,
euclid5891614535332579305n_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_525,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_526,axiom,
quickc658316121487927005ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_527,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_528,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_529,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_530,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_531,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_532,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_533,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_534,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_535,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_536,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_537,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_538,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_539,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_540,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_541,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_542,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_543,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_544,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_545,axiom,
quickcheck_random @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_546,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_547,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_548,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_549,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_550,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_551,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_552,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_553,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_554,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_555,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_556,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_557,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_558,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_559,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_560,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_561,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_562,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_563,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_564,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_565,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_566,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_567,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_568,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_569,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_570,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_571,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_572,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_573,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_574,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_575,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_576,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_577,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_578,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_579,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_580,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_581,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_582,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_583,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_584,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_585,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_586,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_587,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_588,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_589,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_590,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_591,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_592,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Otimes_593,axiom,
times @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_594,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_595,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_596,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_597,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_598,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_599,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_600,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_601,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_602,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_603,axiom,
cl_HOL_Oequal @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_604,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_605,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_606,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_607,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_608,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_609,axiom,
semiri6575147826004484403cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_610,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_611,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_612,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_613,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_614,axiom,
linord2810124833399127020strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_615,axiom,
quickc3360725361186068524ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_616,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_617,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_618,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_619,axiom,
linord4140545234300271783up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_620,axiom,
semiri2026040879449505780visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_621,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_622,axiom,
linord8928482502909563296strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_623,axiom,
quickc658316121487927005ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_624,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_625,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_626,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_627,axiom,
cancel2418104881723323429up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_628,axiom,
cancel1802427076303600483id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_629,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_630,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_631,axiom,
ordere2520102378445227354miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_632,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_633,axiom,
linordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_634,axiom,
ordered_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_635,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_636,axiom,
quickcheck_random @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_637,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_638,axiom,
semiring_1_cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_639,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_640,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_641,axiom,
comm_monoid_diff @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_642,axiom,
ab_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_643,axiom,
ordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_644,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_645,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_646,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_647,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_648,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_649,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_650,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_651,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_652,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_653,axiom,
semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_654,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_655,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_656,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_657,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_658,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_659,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_660,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_661,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_662,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_663,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_664,axiom,
mult_zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_665,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_666,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_667,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Otimes_668,axiom,
times @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ominus_669,axiom,
minus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_670,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_671,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_672,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oplus_673,axiom,
plus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_674,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_675,axiom,
dvd @ code_natural ).
thf(tcon_Code__Numeral_Onatural___HOL_Oequal_676,axiom,
cl_HOL_Oequal @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_677,axiom,
size @ code_natural ).
thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Random_Orandom_678,axiom,
! [A19: $tType] :
( ( quickcheck_random @ A19 )
=> ( quickcheck_random @ ( heap_Time_Heap @ A19 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_679,axiom,
! [A19: $tType] : ( cl_HOL_Oequal @ ( heap_Time_Heap @ A19 ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_680,axiom,
! [A19: $tType] : ( size @ ( heap_Time_Heap @ A19 ) ) ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X9: A] : ( P @ X9 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
entails @ p @ q ).
%------------------------------------------------------------------------------