TPTP Problem File: ITP213^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP213^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Hoare_Triple 00154_004690
% 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 : 0025_Hoare_Triple_00154_004690 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 9787 (3222 unt; 802 typ; 0 def)
% Number of atoms : 27915 (9385 equ; 6 cnn)
% Maximal formula atoms : 48 ( 3 avg)
% Number of connectives : 196738 (2477 ~; 329 |;1953 &;179432 @)
% ( 0 <=>;12547 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 7291 (7291 >; 0 *; 0 +; 0 <<)
% Number of symbols : 795 ( 791 usr; 29 con; 0-9 aty)
% Number of variables : 35010 (4146 ^;29248 !; 765 ?;35010 :)
% ( 851 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 15:20:39.011
%------------------------------------------------------------------------------
% Could-be-implicit typings (23)
thf(ty_t_Heap__Time__Monad_OHeap,type,
heap_Time_Heap: $tType > $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_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_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_Heap_Oarray,type,
array: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Heap_Oref,type,
ref: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Rat_Orat,type,
rat: $tType ).
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (779)
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_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Otimes,type,
times:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oinf,type,
inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osup,type,
sup:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_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_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OInf,type,
complete_Inf:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_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_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_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_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[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_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_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__boolean__algebra,type,
comple489889107523837845lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[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_Assertions_Oassn_OAbs__assn,type,
abs_assn: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > assn ).
thf(sy_c_Assertions_Oassn_ORep__assn,type,
rep_assn: assn > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oentails,type,
entails: assn > assn > $o ).
thf(sy_c_Assertions_Oex__assn,type,
ex_assn:
!>[A: $tType] : ( ( A > assn ) > assn ) ).
thf(sy_c_Assertions_Oin__range,type,
in_range: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oin__range__rel,type,
in_range_rel: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oone__assn__raw,type,
one_assn_raw: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oone__assn__raw__rel,type,
one_assn_raw_rel: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oprecise,type,
precise:
!>[A: $tType,B: $tType] : ( ( A > B > assn ) > $o ) ).
thf(sy_c_Assertions_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_Osngr__assn,type,
sngr_assn:
!>[A: $tType] : ( ( ref @ A ) > A > assn ) ).
thf(sy_c_Assertions_Osngr__assn__raw,type,
sngr_assn_raw:
!>[A: $tType] : ( ( ref @ A ) > A > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) ).
thf(sy_c_Assertions_Osngr__assn__raw__rel,type,
sngr_assn_raw_rel:
!>[A: $tType] : ( ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ) ).
thf(sy_c_Assertions_Otimes__assn__raw,type,
times_assn_raw: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Otimes__assn__raw__rel,type,
times_assn_raw_rel: ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ).
thf(sy_c_Assertions_Owand__assn,type,
wand_assn: assn > assn > assn ).
thf(sy_c_Assertions_Owand__raw,type,
wand_raw: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Owand__raw__rel,type,
wand_raw_rel: ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ).
thf(sy_c_BNF__Cardinal__Arithmetic_OCsum,type,
bNF_Cardinal_Csum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
bNF_Cardinal_cexp:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocfinite,type,
bNF_Cardinal_cfinite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
bNF_Cardinal_cone: set @ ( product_prod @ product_unit @ product_unit ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
bNF_Cardinal_cprod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_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__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_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_BNF__Def_Orel__set,type,
bNF_rel_set:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
bNF_Gr5436034075474128252omCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > B > A ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr4221423524335903396lImage:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
bNF_Gr7122648621184425601vImage:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
bNF_Greatest_toCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > A > B ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
bNF_Gr1419584066657907630d_pred:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
bNF_We4925052301507509544nc_map:
!>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set @ B ) > ( C > A ) > ( B > D ) > ( D > C ) > B > A ) ).
thf(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
bNF_Wellorder_bsqr:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
bNF_Wellorder_curr:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ A ) > ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
bNF_We2720479622203943262_image:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > A2 ) > ( set @ ( product_prod @ A2 @ A2 ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
bNF_We413866401316099525erIncl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordIso,type,
bNF_Wellorder_ordIso:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
bNF_Wellorder_ordLeq:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordLess,type,
bNF_We4044943003108391690rdLess:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__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__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
thf(sy_c_Basic__BNFs_Opred__fun,type,
basic_pred_fun:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_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_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_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
bit_un4731106466462545111um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
bit_un2901131394128224187um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_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_Odup,type,
code_dup: code_integer > code_integer ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: code_natural > nat ).
thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: nat > code_natural ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
comple1908693960933563346ssible:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
comple7512665784863727008ratesp:
!>[A: $tType] : ( ( A > A ) > A > $o ) ).
thf(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
condit8047198070973881523_above:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
condit8119078960628432327_below:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_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_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_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofilter_OAbs__filter,type,
abs_filter:
!>[A: $tType] : ( ( ( A > $o ) > $o ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofiltercomap,type,
filtercomap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofinite__subsets__at__top,type,
finite5375528669736107172at_top:
!>[A: $tType] : ( ( set @ A ) > ( filter @ ( set @ A ) ) ) ).
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_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
thf(sy_c_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_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__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_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
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_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__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__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_Odefault__class_Odefault,type,
default_default:
!>[A: $tType] : A ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Heap_Oaddr__of__ref,type,
addr_of_ref:
!>[A: $tType] : ( ( ref @ A ) > nat ) ).
thf(sy_c_Heap_Oheap_Olim,type,
lim:
!>[Z: $tType] : ( ( heap_ext @ Z ) > nat ) ).
thf(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
heap_Time_Heap2:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OHeap__lub,type,
heap_Time_Heap_lub:
!>[A: $tType] : ( ( set @ ( heap_Time_Heap @ A ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oassert,type,
heap_Time_assert:
!>[A: $tType] : ( ( A > $o ) > A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Obind,type,
heap_Time_bind:
!>[A: $tType,B: $tType] : ( ( heap_Time_Heap @ A ) > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ B ) ) ).
thf(sy_c_Heap__Time__Monad_Oeffect,type,
heap_Time_effect:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o ) ).
thf(sy_c_Heap__Time__Monad_Oexecute,type,
heap_Time_execute:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
thf(sy_c_Heap__Time__Monad_Oguard,type,
heap_Time_guard:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > $o ) > ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oheap,type,
heap_Time_heap:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Olift,type,
heap_Time_lift:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > ( heap_Time_Heap @ B ) ) ).
thf(sy_c_Heap__Time__Monad_Oraise,type,
heap_Time_raise:
!>[A: $tType] : ( ( list @ char ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oreturn,type,
heap_Time_return:
!>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_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_Ogfp,type,
complete_lattice_gfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Ointrel,type,
intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opcr__int,type,
pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( A > int > A ) ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMax,type,
lattices_Max:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMin,type,
lattices_Min:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
lattices_ord_arg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf_OInf__fin,type,
lattic8678736583308907530nf_fin:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic4895041142388067077er_set:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup_OSup__fin,type,
lattic4630905495605216202up_fin:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lifting_Orel__pred__comp,type,
rel_pred_comp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( B > $o ) > A > $o ) ).
thf(sy_c_List_OBleast,type,
bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oabort__Bleast,type,
abort_Bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Oarg__min__list__rel,type,
arg_min_list_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type,
linord144544945434240204of_set:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olinorder__class_Ostable__sort__key,type,
linord3483353639454293061rt_key:
!>[B: $tType,A: $tType] : ( ( ( B > A ) > ( list @ B ) > ( list @ B ) ) > $o ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Olist__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__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__project,type,
map_project:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev__rel,type,
map_tailrec_rev_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_Omin__list,type,
min_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Omin__list__rel,type,
min_list_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Oord_Olexordp,type,
lexordp2:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osorted__wrt__rel,type,
sorted_wrt_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_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_Oinv__on,type,
inv_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > B > A ) ).
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_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__mult__class_Oprod__mset,type,
comm_m9189036328036947845d_mset:
!>[A: $tType] : ( ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ofilter__mset,type,
filter_mset:
!>[A: $tType] : ( ( A > $o ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Ofold__mset,type,
fold_mset:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( multiset @ A ) > B ) ).
thf(sy_c_Multiset_Oimage__mset,type,
image_mset:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( multiset @ A ) > ( multiset @ B ) ) ).
thf(sy_c_Multiset_Ointer__mset,type,
inter_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).
thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset,type,
linord6283353356039996273ltiset:
!>[A: $tType] : ( ( multiset @ A ) > ( list @ A ) ) ).
thf(sy_c_Multiset_Oms__strict,type,
ms_strict: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Multiset_Oms__weak,type,
ms_weak: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Multiset_Omset,type,
mset:
!>[A: $tType] : ( ( list @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Omset__set,type,
mset_set:
!>[B: $tType] : ( ( set @ B ) > ( multiset @ B ) ) ).
thf(sy_c_Multiset_Omult,type,
mult:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omult1,type,
mult1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omulteqp__code,type,
multeqp_code:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Omultiset_OAbs__multiset,type,
abs_multiset:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Omultiset_Ocount,type,
count:
!>[A: $tType] : ( ( multiset @ A ) > A > nat ) ).
thf(sy_c_Multiset_Omultp,type,
multp:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Omultp__code,type,
multp_code:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Opcr__multiset,type,
pcr_multiset:
!>[C: $tType,B: $tType] : ( ( C > B > $o ) > ( C > nat ) > ( multiset @ B ) > $o ) ).
thf(sy_c_Multiset_Opw__leq,type,
pw_leq: ( multiset @ ( product_prod @ nat @ nat ) ) > ( multiset @ ( product_prod @ nat @ nat ) ) > $o ).
thf(sy_c_Multiset_Orel__mset,type,
rel_mset:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( multiset @ A ) > ( multiset @ B ) > $o ) ).
thf(sy_c_Multiset_Orepeat__mset,type,
repeat_mset:
!>[A: $tType] : ( nat > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oreplicate__mset,type,
replicate_mset:
!>[A: $tType] : ( nat > A > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
thf(sy_c_Multiset_Osize__multiset,type,
size_multiset:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > nat ) ).
thf(sy_c_Multiset_Osubset__eq__mset__impl,type,
subset_eq_mset_impl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( option @ $o ) ) ).
thf(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
subset751672762298770561pl_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Multiset_Osubset__mset,type,
subset_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Osubseteq__mset,type,
subseteq_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Ounion__mset,type,
union_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Owcount,type,
wcount:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > A > nat ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : ( nat > ( A > A ) > A > A ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
rec_set_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_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_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Option_Ocombine__options,type,
combine_options:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
thf(sy_c_Option_Ooption_Orec__option,type,
rec_option:
!>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder_OGreatest,type,
greatest:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder_Omono,type,
mono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( 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_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > 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_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_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( T > product_unit > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
product_Abs_unit: $o > product_unit ).
thf(sy_c_Product__Type_Ounit_ORep__unit,type,
product_Rep_unit: product_unit > $o ).
thf(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
linorder_quicksort:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
linord6200660962353139674rt_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_Random_Oinc__shift,type,
inc_shift: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Oiterate,type,
iterate:
!>[B: $tType,A: $tType] : ( code_natural > ( B > A > ( product_prod @ B @ A ) ) > B > A > ( product_prod @ B @ A ) ) ).
thf(sy_c_Random_Oiterate__rel,type,
iterate_rel:
!>[B: $tType,A: $tType] : ( ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > $o ) ).
thf(sy_c_Random_Olog,type,
log: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Olog__rel,type,
log_rel: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ code_natural ) > $o ).
thf(sy_c_Random_Ominus__shift,type,
minus_shift: code_natural > code_natural > code_natural > code_natural ).
thf(sy_c_Random_Onext,type,
next: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Opick,type,
pick:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > code_natural > A ) ).
thf(sy_c_Random_Orange,type,
range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( ( list @ A ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random_Osplit__seed,type,
split_seed: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Ocr__rat,type,
cr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
field_char_0_Rats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oratrel,type,
ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
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_Oset,type,
ref_set:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Ref__Time_Oupdate,type,
ref_update:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_Time_Heap @ product_unit ) ) ).
thf(sy_c_Relation_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_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_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_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
thf(sy_c_Relation_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_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_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_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_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,
insert:
!>[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_OatLeast,type,
set_atLeast:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
set_atLeastAtMost:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeastLessThan,type,
set_atLeastLessThan:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatMost,type,
set_atMost:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThan,type,
set_greaterThan:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost,type,
set_gr3752724095348155675AtMost:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
set_gr287244882034783167ssThan:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OlessThan,type,
set_lessThan:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_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_Transfer_Oright__total,type,
right_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oright__unique,type,
right_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Otransfer__bforall,type,
transfer_bforall:
!>[A: $tType] : ( ( A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_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_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_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: assn ).
thf(sy_v_Q,type,
q: a > assn ).
thf(sy_v_R,type,
r: assn ).
thf(sy_v_as1____,type,
as1: set @ nat ).
thf(sy_v_as2____,type,
as2: set @ nat ).
thf(sy_v_as____,type,
as: set @ nat ).
thf(sy_v_c,type,
c: heap_Time_Heap @ a ).
thf(sy_v_h_H____,type,
h: heap_ext @ product_unit ).
thf(sy_v_h____,type,
h2: heap_ext @ product_unit ).
thf(sy_v_r____,type,
r2: a ).
thf(sy_v_t____,type,
t: nat ).
% Relevant facts (8187)
thf(fact_0_DJ,axiom,
( ( inf_inf @ ( set @ nat ) @ as1 @ as2 )
= ( bot_bot @ ( set @ nat ) ) ) ).
% DJ
thf(fact_1__092_060open_062_Ih_H_M_Aas2_J_A_092_060Turnstile_062_AR_092_060close_062,axiom,
rep_assn @ r @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ h @ as2 ) ).
% \<open>(h', as2) \<Turnstile> R\<close>
thf(fact_2_M2,axiom,
rep_assn @ r @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ h2 @ as2 ) ).
% M2
thf(fact_3_DJN,axiom,
( ( inf_inf @ ( set @ nat ) @ ( hoare_new_addrs @ h2 @ as1 @ h ) @ as2 )
= ( bot_bot @ ( set @ nat ) ) ) ).
% DJN
thf(fact_4__092_060open_062new__addrs_Ah_Aas_Ah_H_A_061_Anew__addrs_Ah_Aas1_Ah_H_A_092_060union_062_Aas2_092_060close_062,axiom,
( ( hoare_new_addrs @ h2 @ as @ h )
= ( sup_sup @ ( set @ nat ) @ ( hoare_new_addrs @ h2 @ as1 @ h ) @ as2 ) ) ).
% \<open>new_addrs h as h' = new_addrs h as1 h' \<union> as2\<close>
thf(fact_5_new__addr__refl,axiom,
! [H: heap_ext @ product_unit,As: set @ nat] :
( ( hoare_new_addrs @ H @ As @ H )
= As ) ).
% new_addr_refl
thf(fact_6_M1,axiom,
rep_assn @ p @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ h2 @ as1 ) ).
% M1
thf(fact_7_MDL,axiom,
rep_assn @ ( q @ r2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ h @ ( hoare_new_addrs @ h2 @ as1 @ h ) ) ).
% MDL
thf(fact_8__092_060open_062_Ih_M_Aas_J_A_092_060Turnstile_062_AP_A_K_AR_092_060close_062,axiom,
rep_assn @ ( times_times @ assn @ p @ r ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ h2 @ as ) ).
% \<open>(h, as) \<Turnstile> P * R\<close>
thf(fact_9__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062as1_Aas2_O_A_092_060lbrakk_062as_A_061_Aas1_A_092_060union_062_Aas2_059_Aas1_A_092_060inter_062_Aas2_A_061_A_123_125_059_A_Ih_M_Aas1_J_A_092_060Turnstile_062_AP_059_A_Ih_M_Aas2_J_A_092_060Turnstile_062_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [As1: set @ nat,As2: set @ nat] :
( ( as
= ( sup_sup @ ( set @ nat ) @ As1 @ As2 ) )
=> ( ( ( inf_inf @ ( set @ nat ) @ As1 @ As2 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( rep_assn @ p @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ h2 @ As1 ) )
=> ~ ( rep_assn @ r @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ h2 @ As2 ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>as1 as2. \<lbrakk>as = as1 \<union> as2; as1 \<inter> as2 = {}; (h, as1) \<Turnstile> P; (h, as2) \<Turnstile> R\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_10_Rep__assn__inject,axiom,
! [X: assn,Y: assn] :
( ( ( rep_assn @ X )
= ( rep_assn @ Y ) )
= ( X = Y ) ) ).
% Rep_assn_inject
thf(fact_11_mod__h__bot__iff_I5_J,axiom,
! [P: assn,Q: assn,H: heap_ext @ product_unit] :
( ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(5)
thf(fact_12__092_060open_062relH_Aas2_Ah_Ah_H_092_060close_062,axiom,
relH @ as2 @ h2 @ h ).
% \<open>relH as2 h h'\<close>
thf(fact_13_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( ( A3 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_15_mod__starD,axiom,
! [A5: assn,B4: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A5 @ B4 ) @ H )
=> ? [H1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),H2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ A5 @ H1 )
& ( rep_assn @ B4 @ H2 ) ) ) ).
% mod_starD
thf(fact_16_mod__starE,axiom,
! [A3: assn,B2: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A3 @ B2 ) @ H )
=> ~ ( ? [X_1: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : ( rep_assn @ A3 @ X_1 )
=> ! [H_2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ B2 @ H_2 ) ) ) ).
% mod_starE
thf(fact_17_assms,axiom,
hoare_hoare_triple @ a @ p @ c @ q ).
% assms
thf(fact_18__092_060open_062lim_Ah_A_092_060le_062_Alim_Ah_H_092_060close_062,axiom,
ord_less_eq @ nat @ ( lim @ product_unit @ h2 ) @ ( lim @ product_unit @ h ) ).
% \<open>lim h \<le> lim h'\<close>
thf(fact_19__092_060open_062as_A_061_Aas1_A_092_060union_062_Aas2_092_060close_062,axiom,
( as
= ( sup_sup @ ( set @ nat ) @ as1 @ as2 ) ) ).
% \<open>as = as1 \<union> as2\<close>
thf(fact_20_mod__h__bot__indep,axiom,
! [P: assn,H: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_indep
thf(fact_21_relH__dist__union,axiom,
! [As: set @ nat,As3: set @ nat,H: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( relH @ ( sup_sup @ ( set @ nat ) @ As @ As3 ) @ H @ H3 )
= ( ( relH @ As @ H @ H3 )
& ( relH @ As3 @ H @ H3 ) ) ) ).
% relH_dist_union
thf(fact_22_snga__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R: array @ A,X3: list @ A,H4: heap_ext @ product_unit,As4: set @ nat] :
( X
!= ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) ) ) ).
% snga_assn_raw.cases
thf(fact_23_sngr__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R: ref @ A,X3: A,H4: heap_ext @ product_unit,As4: set @ nat] :
( X
!= ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) ) ) ).
% sngr_assn_raw.cases
thf(fact_24_times__assn__raw_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [P2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q2: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H4: heap_ext @ product_unit,As4: set @ nat] :
( X
!= ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ P2 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Q2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) ) ).
% times_assn_raw.cases
thf(fact_25_relH__sym,axiom,
! [As: set @ nat,H: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( relH @ As @ H @ H3 )
=> ( relH @ As @ H3 @ H ) ) ).
% relH_sym
thf(fact_26_relH__trans,axiom,
! [As: set @ nat,H12: heap_ext @ product_unit,H22: heap_ext @ product_unit,H32: heap_ext @ product_unit] :
( ( relH @ As @ H12 @ H22 )
=> ( ( relH @ As @ H22 @ H32 )
=> ( relH @ As @ H12 @ H32 ) ) ) ).
% relH_trans
thf(fact_27_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
~ ! [B5: $o,H4: A,As4: set @ B] :
( X
!= ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B5 @ ( product_Pair @ A @ ( set @ B ) @ H4 @ As4 ) ) ) ).
% pure_assn_raw.cases
thf(fact_28_mod__relH,axiom,
! [As: set @ nat,H: heap_ext @ product_unit,H3: heap_ext @ product_unit,P: assn] :
( ( relH @ As @ H @ H3 )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ) ).
% mod_relH
thf(fact_29_hoare__triple__preI,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H4 )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).
% hoare_triple_preI
thf(fact_30_mod__star__conv,axiom,
! [A5: assn,B4: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A5 @ B4 ) @ H )
= ( ? [Hr: heap_ext @ product_unit,As12: set @ nat,As22: set @ nat] :
( ( H
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ ( sup_sup @ ( set @ nat ) @ As12 @ As22 ) ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( rep_assn @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As12 ) )
& ( rep_assn @ B4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As22 ) ) ) ) ) ).
% mod_star_conv
thf(fact_31_star__assnI,axiom,
! [P: assn,H: heap_ext @ product_unit,As: set @ nat,Q: assn,As3: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As3 ) )
=> ( ( ( inf_inf @ ( set @ nat ) @ As @ As3 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( sup_sup @ ( set @ nat ) @ As @ As3 ) ) ) ) ) ) ).
% star_assnI
thf(fact_32_prod__induct7,axiom,
! [G: $tType,F: $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 @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_33_prod__induct6,axiom,
! [F: $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 @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_34_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_35_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A6: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_36_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A6: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_37_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F2: F,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_38_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_39_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,C3: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_40_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,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_41_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B5: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_42_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ~ ( ( A3 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_43_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_44_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P3
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ! [X3: A] :
( ( F3 @ X3 )
= ( G3 @ X3 ) )
=> ( F3 = G3 ) ) ).
% ext
thf(fact_49_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_50_one__assn__raw_Ocases,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( X
!= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ).
% one_assn_raw.cases
thf(fact_51_assn__times__assoc,axiom,
! [P: assn,Q: assn,R2: assn] :
( ( times_times @ assn @ ( times_times @ assn @ P @ Q ) @ R2 )
= ( times_times @ assn @ P @ ( times_times @ assn @ Q @ R2 ) ) ) ).
% assn_times_assoc
thf(fact_52_assn__times__comm,axiom,
( ( times_times @ assn )
= ( ^ [P4: assn,Q3: assn] : ( times_times @ assn @ Q3 @ P4 ) ) ) ).
% assn_times_comm
thf(fact_53_Int__Un__eq_I4_J,axiom,
! [A: $tType,T2: set @ A,S: set @ A] :
( ( sup_sup @ ( set @ A ) @ T2 @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_54_Int__Un__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( sup_sup @ ( set @ A ) @ S @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_55_Int__Un__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_56_Int__Un__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_57_Un__Int__eq_I4_J,axiom,
! [A: $tType,T2: set @ A,S: set @ A] :
( ( inf_inf @ ( set @ A ) @ T2 @ ( sup_sup @ ( set @ A ) @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_58_Un__Int__eq_I3_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( inf_inf @ ( set @ A ) @ S @ ( sup_sup @ ( set @ A ) @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_59_Un__Int__eq_I2_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_60_Un__Int__eq_I1_J,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_61_Un__empty,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_62_inf__sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= X ) ) ).
% inf_sup_absorb
thf(fact_63_sup__inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= X ) ) ).
% sup_inf_absorb
thf(fact_64_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_65_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_66_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_67_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_68_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_69_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B,X4: A] : ( inf_inf @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% inf_apply
thf(fact_70_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_right_idem
thf(fact_71_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 )
= ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_72_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_left_idem
thf(fact_73_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B2 ) )
= ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_74_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% inf_idem
thf(fact_75_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ A3 )
= A3 ) ) ).
% inf.idem
thf(fact_76_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B,X4: A] : ( sup_sup @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% sup_apply
thf(fact_77_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ B2 )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_78_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_79_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_80_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_81_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_82_Int__iff,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( ( member @ A @ C2 @ A5 )
& ( member @ A @ C2 @ B4 ) ) ) ).
% Int_iff
thf(fact_83_IntI,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A5 )
=> ( ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% IntI
thf(fact_84_Un__iff,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ( member @ A @ C2 @ A5 )
| ( member @ A @ C2 @ B4 ) ) ) ).
% Un_iff
thf(fact_85_UnCI,axiom,
! [A: $tType,C2: A,B4: set @ A,A5: set @ A] :
( ( ~ ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ A5 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnCI
thf(fact_86_mod__or__dist,axiom,
! [P: assn,Q: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( sup_sup @ assn @ P @ Q ) @ H )
= ( ( rep_assn @ P @ H )
| ( rep_assn @ Q @ H ) ) ) ).
% mod_or_dist
thf(fact_87_star__false__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( bot_bot @ assn ) @ P )
= ( bot_bot @ assn ) ) ).
% star_false_left
thf(fact_88_star__false__right,axiom,
! [P: assn] :
( ( times_times @ assn @ P @ ( bot_bot @ assn ) )
= ( bot_bot @ assn ) ) ).
% star_false_right
thf(fact_89_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_90_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_91_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_92_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_93_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_94_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_95_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
= A3 ) ) ).
% sup_bot.right_neutral
thf(fact_96_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A3 @ B2 ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_97_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
= A3 ) ) ).
% sup_bot.left_neutral
thf(fact_98_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( sup_sup @ A @ A3 @ B2 )
= ( bot_bot @ A ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_99_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_100_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_101_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_102_mod__h__bot__iff_I6_J,axiom,
! [P: assn,Q: assn,H: heap_ext @ product_unit] :
( ( rep_assn @ ( inf_inf @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(6)
thf(fact_103_mod__h__bot__iff_I7_J,axiom,
! [P: assn,Q: assn,H: heap_ext @ product_unit] :
( ( rep_assn @ ( sup_sup @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
| ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(7)
thf(fact_104_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_105_mod__and__dist,axiom,
! [P: assn,Q: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( inf_inf @ assn @ P @ Q ) @ H )
= ( ( rep_assn @ P @ H )
& ( rep_assn @ Q @ H ) ) ) ).
% mod_and_dist
thf(fact_106_mod__false,axiom,
! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ ( bot_bot @ assn ) @ H ) ).
% mod_false
thf(fact_107_star__or__dist1,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( times_times @ assn @ ( sup_sup @ assn @ A5 @ B4 ) @ C4 )
= ( sup_sup @ assn @ ( times_times @ assn @ A5 @ C4 ) @ ( times_times @ assn @ B4 @ C4 ) ) ) ).
% star_or_dist1
thf(fact_108_star__or__dist2,axiom,
! [C4: assn,A5: assn,B4: assn] :
( ( times_times @ assn @ C4 @ ( sup_sup @ assn @ A5 @ B4 ) )
= ( sup_sup @ assn @ ( times_times @ assn @ C4 @ A5 ) @ ( times_times @ assn @ C4 @ B4 ) ) ) ).
% star_or_dist2
thf(fact_109_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_110_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_111_equals0D,axiom,
! [A: $tType,A5: set @ A,A3: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A5 ) ) ).
% equals0D
thf(fact_112_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_113_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B,X4: A] : ( inf_inf @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% inf_fun_def
thf(fact_114_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_left_commute
thf(fact_115_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A3 @ C2 ) )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.left_commute
thf(fact_116_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X4: A,Y4: A] : ( inf_inf @ A @ Y4 @ X4 ) ) ) ) ).
% inf_commute
thf(fact_117_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A7: A,B6: A] : ( inf_inf @ A @ B6 @ A7 ) ) ) ) ).
% inf.commute
thf(fact_118_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z2 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ).
% inf_assoc
thf(fact_119_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.assoc
thf(fact_120_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X4: A,Y4: A] : ( inf_inf @ A @ Y4 @ X4 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_121_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z2 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_122_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_123_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_sup_aci(4)
thf(fact_124_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B,X4: A] : ( sup_sup @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% sup_fun_def
thf(fact_125_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% sup_left_commute
thf(fact_126_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A3 @ C2 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_127_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X4: A,Y4: A] : ( sup_sup @ A @ Y4 @ X4 ) ) ) ) ).
% sup_commute
thf(fact_128_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A7: A,B6: A] : ( sup_sup @ A @ B6 @ A7 ) ) ) ) ).
% sup.commute
thf(fact_129_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% sup_assoc
thf(fact_130_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ C2 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_131_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X4: A,Y4: A] : ( sup_sup @ A @ Y4 @ X4 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_132_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_133_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_134_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_135_Int__left__commute,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Int_left_commute
thf(fact_136_Int__left__absorb,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ).
% Int_left_absorb
thf(fact_137_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A8 ) ) ) ).
% Int_commute
thf(fact_138_Int__absorb,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ A5 )
= A5 ) ).
% Int_absorb
thf(fact_139_Int__assoc,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Int_assoc
thf(fact_140_IntD2,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
=> ( member @ A @ C2 @ B4 ) ) ).
% IntD2
thf(fact_141_IntD1,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
=> ( member @ A @ C2 @ A5 ) ) ).
% IntD1
thf(fact_142_IntE,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
=> ~ ( ( member @ A @ C2 @ A5 )
=> ~ ( member @ A @ C2 @ B4 ) ) ) ).
% IntE
thf(fact_143_Un__left__commute,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_144_Un__left__absorb,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_left_absorb
thf(fact_145_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A8 ) ) ) ).
% Un_commute
thf(fact_146_Un__absorb,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ A5 )
= A5 ) ).
% Un_absorb
thf(fact_147_Un__assoc,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Un_assoc
thf(fact_148_ball__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( P @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_149_bex__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
& ( P @ X4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) )
| ? [X4: A] :
( ( member @ A @ X4 @ B4 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_150_UnI2,axiom,
! [A: $tType,C2: A,B4: set @ A,A5: set @ A] :
( ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnI2
thf(fact_151_UnI1,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnI1
thf(fact_152_UnE,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( ~ ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% UnE
thf(fact_153_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI2
thf(fact_154_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI1
thf(fact_155_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A7: A] :
( ( inf_inf @ A @ A7 @ B6 )
= B6 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_156_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B6: A] :
( ( inf_inf @ A @ A7 @ B6 )
= A7 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_157_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 ) ) ).
% inf.cobounded2
thf(fact_158_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ A3 ) ) ).
% inf.cobounded1
thf(fact_159_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B6: A] :
( A7
= ( inf_inf @ A @ A7 @ B6 ) ) ) ) ) ).
% inf.order_iff
thf(fact_160_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Z2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ) ).
% inf_greatest
thf(fact_161_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ).
% inf.boundedI
thf(fact_162_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% inf.boundedE
thf(fact_163_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( inf_inf @ A @ X @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_164_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( inf_inf @ A @ X @ Y )
= X ) ) ) ).
% inf_absorb1
thf(fact_165_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
thf(fact_166_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb1
thf(fact_167_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y4: A] :
( ( inf_inf @ A @ X4 @ Y4 )
= X4 ) ) ) ) ).
% le_iff_inf
thf(fact_168_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F3: A > A > A,X: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Z3 )
=> ( ord_less_eq @ A @ X3 @ ( F3 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_169_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( inf_inf @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% inf.orderI
thf(fact_170_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( inf_inf @ A @ A3 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_171_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI2
thf(fact_172_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI1
thf(fact_173_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ ( inf_inf @ A @ C2 @ D3 ) ) ) ) ) ).
% inf_mono
thf(fact_174_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_175_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A3 )
=> ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% le_infE
thf(fact_176_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_177_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_le1
thf(fact_178_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_sup_ord(1)
thf(fact_179_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_180_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_181_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_182_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B6: A] :
( ( sup_sup @ A @ A7 @ B6 )
= B6 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_183_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A7: A] :
( ( sup_sup @ A @ A7 @ B6 )
= A7 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_184_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_185_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_186_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A7: A] :
( A7
= ( sup_sup @ A @ A7 @ B6 ) ) ) ) ) ).
% sup.order_iff
thf(fact_187_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_188_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_189_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_190_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_191_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_192_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_193_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F3: A > A > A,X: A,Y: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F3 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ Z3 @ X3 )
=> ( ord_less_eq @ A @ ( F3 @ Y3 @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_194_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% sup.orderI
thf(fact_195_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_196_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y4: A] :
( ( sup_sup @ A @ X4 @ Y4 )
= Y4 ) ) ) ) ).
% le_iff_sup
thf(fact_197_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A,Z2: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X ) ) ) ) ).
% sup_least
thf(fact_198_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ ( sup_sup @ A @ C2 @ D3 ) ) ) ) ) ).
% sup_mono
thf(fact_199_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D3 ) @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_200_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_201_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_202_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_203_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_204_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_205_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A3 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_206_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_207_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_208_disjoint__iff__not__equal,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ B4 )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_209_Int__empty__right,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_210_Int__empty__left,axiom,
! [A: $tType,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_211_disjoint__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ~ ( member @ A @ X4 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_212_Int__emptyI,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ~ ( member @ A @ X3 @ B4 ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_213_sup__inf__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z2 ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z2 @ X ) ) ) ) ).
% sup_inf_distrib2
thf(fact_214_sup__inf__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% sup_inf_distrib1
thf(fact_215_inf__sup__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z2 @ X ) ) ) ) ).
% inf_sup_distrib2
thf(fact_216_inf__sup__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_distrib1
thf(fact_217_distrib__imp2,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ! [X3: A,Y3: A,Z3: A] :
( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z3 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ ( sup_sup @ A @ X3 @ Z3 ) ) )
=> ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ) ).
% distrib_imp2
thf(fact_218_distrib__imp1,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] :
( ! [X3: A,Y3: A,Z3: A] :
( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y3 @ Z3 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ ( inf_inf @ A @ X3 @ Z3 ) ) )
=> ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ) ).
% distrib_imp1
thf(fact_219_Un__empty__right,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Un_empty_right
thf(fact_220_Un__empty__left,axiom,
! [A: $tType,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_221_Un__Int__distrib2,axiom,
! [A: $tType,B4: set @ A,C4: set @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) @ A5 )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B4 @ A5 ) @ ( sup_sup @ ( set @ A ) @ C4 @ A5 ) ) ) ).
% Un_Int_distrib2
thf(fact_222_Int__Un__distrib2,axiom,
! [A: $tType,B4: set @ A,C4: set @ A,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) @ A5 )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ B4 @ A5 ) @ ( inf_inf @ ( set @ A ) @ C4 @ A5 ) ) ) ).
% Int_Un_distrib2
thf(fact_223_Un__Int__distrib,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ ( sup_sup @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Un_Int_distrib
thf(fact_224_Int__Un__distrib,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Int_Un_distrib
thf(fact_225_Un__Int__crazy,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) @ ( inf_inf @ ( set @ A ) @ C4 @ A5 ) )
= ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) @ ( sup_sup @ ( set @ A ) @ C4 @ A5 ) ) ) ).
% Un_Int_crazy
thf(fact_226_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% distrib_sup_le
thf(fact_227_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% distrib_inf_le
thf(fact_228_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_right
thf(fact_229_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_left
thf(fact_230_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( ( heap @ B )
=> ! [Q4: array @ B,Y: list @ B,H: heap_ext @ product_unit] :
~ ( rep_assn @ ( snga_assn @ B @ Q4 @ Y ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(4)
thf(fact_231_mod__h__bot__iff_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: ref @ A,X: A,H: heap_ext @ product_unit] :
~ ( rep_assn @ ( sngr_assn @ A @ P3 @ X ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(3)
thf(fact_232_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( F1 @ A3 @ B2 ) ) ).
% old.prod.rec
thf(fact_233_EX,axiom,
( ( heap_Time_execute @ a @ c @ 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 @ h @ t ) ) ) ) ).
% EX
thf(fact_234_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_235_times__assn__raw_Oelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ? [As13: set @ nat,As23: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As13 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As23 ) ) ) ) ) ).
% times_assn_raw.elims(3)
thf(fact_236_times__assn__raw_Oelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ~ ? [As1: set @ nat,As2: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As1 @ As2 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As2 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ) ).
% times_assn_raw.elims(2)
thf(fact_237_times__assn__raw_Oelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( Y
= ( ~ ? [As12: set @ nat,As22: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As12 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As22 ) ) ) ) ) ) ) ).
% times_assn_raw.elims(1)
thf(fact_238_times__assn__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H: heap_ext @ product_unit,As: set @ nat] :
( ( times_assn_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
= ( ? [As12: set @ nat,As22: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As12 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As12 ) )
& ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As22 ) ) ) ) ) ).
% times_assn_raw.simps
thf(fact_239_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_240_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_241_subset__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_242_empty__subsetI,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% empty_subsetI
thf(fact_243_Int__subset__iff,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C4 @ A5 )
& ( ord_less_eq @ ( set @ A ) @ C4 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_244_Un__subset__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_245_sngr__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: ref @ A,X: A,Y: A] :
( ( times_times @ assn @ ( sngr_assn @ A @ P3 @ X ) @ ( sngr_assn @ A @ P3 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% sngr_same_false
thf(fact_246_snga__same__false,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: array @ A,X: list @ A,Y: list @ A] :
( ( times_times @ assn @ ( snga_assn @ A @ P3 @ X ) @ ( snga_assn @ A @ P3 @ Y ) )
= ( bot_bot @ assn ) ) ) ).
% snga_same_false
thf(fact_247_less__eq__assn__def,axiom,
( ( ord_less_eq @ assn )
= ( ^ [A7: assn,B6: assn] :
( A7
= ( inf_inf @ assn @ A7 @ B6 ) ) ) ) ).
% less_eq_assn_def
thf(fact_248_Int__mono,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B4: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( inf_inf @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% Int_mono
thf(fact_249_Int__lower1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ A5 ) ).
% Int_lower1
thf(fact_250_Int__lower2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_251_Int__absorb1,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_252_Int__absorb2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= A5 ) ) ).
% Int_absorb2
thf(fact_253_Int__greatest,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_254_Int__Collect__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B4 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_255_Un__mono,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B4: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_256_Un__least,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C4 ) ) ) ).
% Un_least
thf(fact_257_Un__upper1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_upper1
thf(fact_258_Un__upper2,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_upper2
thf(fact_259_Un__absorb1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_260_Un__absorb2,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= A5 ) ) ).
% Un_absorb2
thf(fact_261_subset__UnE,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ~ ! [A9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A9 @ A5 )
=> ! [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ B4 )
=> ( C4
!= ( sup_sup @ ( set @ A ) @ A9 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_262_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( ( sup_sup @ ( set @ A ) @ A8 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_263_relH__subset,axiom,
! [Bs: set @ nat,H: heap_ext @ product_unit,H3: heap_ext @ product_unit,As: set @ nat] :
( ( relH @ Bs @ H @ H3 )
=> ( ( ord_less_eq @ ( set @ nat ) @ As @ Bs )
=> ( relH @ As @ H @ H3 ) ) ) ).
% relH_subset
thf(fact_264_Un__Int__assoc__eq,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C4 @ A5 ) ) ).
% Un_Int_assoc_eq
thf(fact_265_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( ord_less_eq @ A @ A3 @ B2 ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( B2 != A3 ) ) ) ) ).
% nle_le
thf(fact_266_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_267_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_268_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_269_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_270_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% order_antisym
thf(fact_271_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_272_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_273_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: 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 @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_274_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [A7: A,B6: A] :
( ( ord_less_eq @ A @ B6 @ A7 )
& ( ord_less_eq @ A @ A7 @ B6 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_275_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_276_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_277_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% antisym
thf(fact_278_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funD
thf(fact_279_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funE
thf(fact_280_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% le_funI
thf(fact_281_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_282_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [A7: A,B6: A] :
( ( ord_less_eq @ A @ A7 @ B6 )
& ( ord_less_eq @ A @ B6 @ A7 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_283_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_284_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F3: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_285_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_eq_refl
thf(fact_286_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_linear
thf(fact_287_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C2: B] :
( ( A3
= ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_288_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F3: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( F3 @ B2 )
= C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_289_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_le_cases
thf(fact_290_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_291_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_292_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5
= ( inf_inf @ A @ K @ A3 ) )
=> ( ( inf_inf @ A @ A5 @ B2 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_293_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B4: A,K: A,B2: A,A3: A] :
( ( B4
= ( inf_inf @ A @ K @ B2 ) )
=> ( ( inf_inf @ A @ A3 @ B4 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_294_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5
= ( sup_sup @ A @ K @ A3 ) )
=> ( ( sup_sup @ A @ A5 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_295_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B4: A,K: A,B2: A,A3: A] :
( ( B4
= ( sup_sup @ A @ K @ B2 ) )
=> ( ( sup_sup @ A @ A3 @ B4 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_296_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_297_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_298_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_299_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% boolean_algebra.disj_zero_right
thf(fact_300_boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_301_boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_302_boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z2 @ X ) ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_303_boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z2 ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z2 @ X ) ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_304_sup__Some,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( some @ A @ ( sup_sup @ A @ X @ Y ) ) ) ) ).
% sup_Some
thf(fact_305_inf__Some,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( some @ A @ ( inf_inf @ A @ X @ Y ) ) ) ) ).
% inf_Some
thf(fact_306_less__eq__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_eq_option_Some
thf(fact_307__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062r_Ah_H_At_O_A_092_060lbrakk_062execute_Ac_Ah_A_061_ASome_A_Ir_M_Ah_H_M_At_J_059_A_Ih_H_M_Anew__addrs_Ah_Aas1_Ah_H_J_A_092_060Turnstile_062_AQ_Ar_059_ArelH_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_Ah_Ah_H_059_Alim_Ah_A_092_060le_062_Alim_Ah_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [R: a,H5: heap_ext @ product_unit] :
( ? [T3: nat] :
( ( heap_Time_execute @ a @ c @ h2 )
= ( some @ ( product_prod @ a @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ a @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ T3 ) ) ) )
=> ( ( rep_assn @ ( q @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( hoare_new_addrs @ h2 @ as1 @ H5 ) ) )
=> ( ( relH
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ h2 ) )
& ~ ( member @ nat @ A7 @ as1 ) ) )
@ h2
@ H5 )
=> ~ ( ord_less_eq @ nat @ ( lim @ product_unit @ h2 ) @ ( lim @ product_unit @ H5 ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>r h' t. \<lbrakk>execute c h = Some (r, h', t); (h', new_addrs h as1 h') \<Turnstile> Q r; relH {a. a < lim h \<and> a \<notin> as1} h h'; lim h \<le> lim h'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_308_option_Oinject,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( ( some @ A @ X2 )
= ( some @ A @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_309_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_310_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_311_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F2: A > B,A6: A,B5: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F2 @ ( product_Pair @ A @ A @ A6 @ B5 ) ) ) ).
% pairself.cases
thf(fact_312_disjoint__mono,axiom,
! [A: $tType,A3: set @ A,A4: set @ A,B2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ B3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% disjoint_mono
thf(fact_313_disjointI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ~ ( member @ A @ X3 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjointI
thf(fact_314_le__some__optE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M: A,X: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M ) @ X )
=> ~ ! [M2: A] :
( ( X
= ( some @ A @ M2 ) )
=> ~ ( ord_less_eq @ A @ M @ M2 ) ) ) ) ).
% le_some_optE
thf(fact_315_subsetI,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% subsetI
thf(fact_316_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( A5 = B4 ) ) ) ).
% subset_antisym
thf(fact_317_RH1,axiom,
( relH
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ h2 ) )
& ~ ( member @ nat @ A7 @ as1 ) ) )
@ h2
@ h ) ).
% RH1
thf(fact_318_less__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ).
% less_option_Some
thf(fact_319__092_060open_062relH_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas_125_Ah_Ah_H_092_060close_062,axiom,
( relH
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ h2 ) )
& ~ ( member @ nat @ A7 @ as ) ) )
@ h2
@ h ) ).
% \<open>relH {a. a < lim h \<and> a \<notin> as} h h'\<close>
thf(fact_320__092_060open_062as2_A_092_060subseteq_062_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_092_060close_062,axiom,
( ord_less_eq @ ( set @ nat ) @ as2
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ h2 ) )
& ~ ( member @ nat @ A7 @ as1 ) ) ) ) ).
% \<open>as2 \<subseteq> {a. a < lim h \<and> a \<notin> as1}\<close>
thf(fact_321__092_060open_062_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas_125_A_092_060subseteq_062_A_123a_O_Aa_A_060_Alim_Ah_A_092_060and_062_Aa_A_092_060notin_062_Aas1_125_092_060close_062,axiom,
( ord_less_eq @ ( set @ nat )
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ h2 ) )
& ~ ( member @ nat @ A7 @ as ) ) )
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ h2 ) )
& ~ ( member @ nat @ A7 @ as1 ) ) ) ) ).
% \<open>{a. a < lim h \<and> a \<notin> as} \<subseteq> {a. a < lim h \<and> a \<notin> as1}\<close>
thf(fact_322_in__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B4 ) ) ) ).
% in_mono
thf(fact_323_subsetD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_324_equalityE,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% equalityE
thf(fact_325_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ A @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_326_equalityD1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% equalityD1
thf(fact_327_equalityD2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ).
% equalityD2
thf(fact_328_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
! [T4: A] :
( ( member @ A @ T4 @ A8 )
=> ( member @ A @ T4 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_329_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_330_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_331_subset__trans,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_trans
thf(fact_332_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4 )
= ( ^ [A8: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% set_eq_subset
thf(fact_333_sup__Un__eq,axiom,
! [A: $tType,R2: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R2 )
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_334_Collect__subset,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_335_inf__Int__eq,axiom,
! [A: $tType,R2: set @ A,S: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R2 )
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_336_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_337_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A8 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_338_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_339_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_340_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_341_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_342_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% pred_subset_eq2
thf(fact_343_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_344_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_345_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_346_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_347_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order.asym
thf(fact_348_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_349_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_350_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_351_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_352_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_353_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_354_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_355_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P5: A > $o] :
? [X5: A] : ( P5 @ X5 ) )
= ( ^ [P4: A > $o] :
? [N: A] :
( ( P4 @ N )
& ! [M3: A] :
( ( ord_less @ A @ M3 @ N )
=> ~ ( P4 @ M3 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_356_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: 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 @ A3 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_357_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_358_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_359_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_360_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_361_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_362_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE
thf(fact_363_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_asym
thf(fact_364_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neq_iff
thf(fact_365_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order_less_asym'
thf(fact_366_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% order_less_trans
thf(fact_367_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C2: B] :
( ( A3
= ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_368_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F3: A > B,C2: B] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ( F3 @ B2 )
= C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ B @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_369_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_370_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_371_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F3: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F3 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_372_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_not_sym
thf(fact_373_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_374_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_less_linear
thf(fact_375_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_376_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
thf(fact_377_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_imp_not_less
thf(fact_378_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_379_pred__subset__eq,axiom,
! [A: $tType,R2: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R2 )
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_380_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% Set.empty_def
thf(fact_381_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A8 )
& ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% Int_def
thf(fact_382_Int__Collect,axiom,
! [A: $tType,X: A,A5: set @ A,P: A > $o] :
( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X @ A5 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_383_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A8 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% inf_set_def
thf(fact_384_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_385_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A8 )
| ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_386_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A8 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_387_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_388_exists__leI,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ N2 )
=> ~ ( P @ N3 ) )
=> ( P @ N2 ) )
=> ? [N4: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
& ( P @ N4 ) ) ) ).
% exists_leI
thf(fact_389_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_390_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_le_less_linear
thf(fact_391_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F3: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_392_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_393_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F3: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F3 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A3 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_394_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F3: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_395_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% order_less_le_trans
thf(fact_396_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% order_le_less_trans
thf(fact_397_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 != B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_neq_le_trans
thf(fact_398_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_le_neq_trans
thf(fact_399_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_less_imp_le
thf(fact_400_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_not_less
thf(fact_401_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_not_le
thf(fact_402_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ) ).
% order_less_le
thf(fact_403_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ) ).
% order_le_less
thf(fact_404_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_405_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_406_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B6: A,A7: A] :
( ( ord_less_eq @ A @ B6 @ A7 )
& ~ ( ord_less_eq @ A @ A7 @ B6 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_407_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_408_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_409_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B6: A,A7: A] :
( ( ord_less_eq @ A @ B6 @ A7 )
& ( A7 != B6 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_410_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A7: A] :
( ( ord_less @ A @ B6 @ A7 )
| ( A7 = B6 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_411_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_412_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_413_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A,B6: A] :
( ( ord_less_eq @ A @ A7 @ B6 )
& ~ ( ord_less_eq @ A @ B6 @ A7 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_414_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_415_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_416_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A,B6: A] :
( ( ord_less_eq @ A @ A7 @ B6 )
& ( A7 != B6 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_417_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B6: A] :
( ( ord_less @ A @ A7 @ B6 )
| ( A7 = B6 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_418_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_419_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ~ ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_420_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z2: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_421_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_422_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_423_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_424_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( ord_less @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% nless_le
thf(fact_425_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_426_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_427_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_428_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_429_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_430_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_431_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A,B6: A] :
( ( A7
= ( inf_inf @ A @ A7 @ B6 ) )
& ( A7 != B6 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_432_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_433_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb4
thf(fact_434_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb3
thf(fact_435_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A3: A] :
( ( ord_less @ A @ B2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% less_infI2
thf(fact_436_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less @ A @ A3 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% less_infI1
thf(fact_437_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_438_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ A3 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_439_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B6: A,A7: A] :
( ( A7
= ( sup_sup @ A @ A7 @ B6 ) )
& ( A7 != B6 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_440_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_441_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb4
thf(fact_442_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb3
thf(fact_443_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less @ A @ X @ B2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_444_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_445_new__addrs__def,axiom,
( hoare_new_addrs
= ( ^ [H6: heap_ext @ product_unit,As5: set @ nat,H7: heap_ext @ product_unit] :
( sup_sup @ ( set @ nat ) @ As5
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less_eq @ nat @ ( lim @ product_unit @ H6 ) @ A7 )
& ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H7 ) ) ) ) ) ) ) ).
% new_addrs_def
thf(fact_446_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ( C2 = D3 )
=> ( ord_less_eq @ A @ A3 @ D3 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_447_bex2I,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,S: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( P @ A3 @ B2 ) )
=> ? [A6: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ S )
& ( P @ A6 @ B5 ) ) ) ) ).
% bex2I
thf(fact_448_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ).
% subrelI
thf(fact_449_memb__imp__not__empty,axiom,
! [A: $tType,X: A,S: set @ A] :
( ( member @ A @ X @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_450_set__notEmptyE,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
~ ( member @ A @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_451_inter__eq__subsetI,axiom,
! [A: $tType,S: set @ A,S3: set @ A,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ S3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ S3 )
= ( inf_inf @ ( set @ A ) @ B4 @ S3 ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ S )
= ( inf_inf @ ( set @ A ) @ B4 @ S ) ) ) ) ).
% inter_eq_subsetI
thf(fact_452_hoare__triple__def,axiom,
! [A: $tType] :
( ( hoare_hoare_triple @ A )
= ( ^ [P4: assn,C5: heap_Time_Heap @ A,Q3: A > assn] :
! [H6: heap_ext @ product_unit,As5: set @ nat] :
( ( rep_assn @ P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As5 ) )
=> ? [R4: A,H7: heap_ext @ product_unit] :
( ? [T4: nat] :
( ( heap_Time_execute @ A @ C5 @ H6 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ T4 ) ) ) )
& ( rep_assn @ ( Q3 @ R4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( hoare_new_addrs @ H6 @ As5 @ H7 ) ) )
& ( relH
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H6 ) )
& ~ ( member @ nat @ A7 @ As5 ) ) )
@ H6
@ H7 )
& ( ord_less_eq @ nat @ ( lim @ product_unit @ H6 ) @ ( lim @ product_unit @ H7 ) ) ) ) ) ) ).
% hoare_triple_def
thf(fact_453_hoare__tripleI,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
( ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ? [R5: A,H8: heap_ext @ product_unit,T5: nat] :
( ( ( heap_Time_execute @ A @ C2 @ H4 )
= ( 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 @ T5 ) ) ) )
& ( rep_assn @ ( Q @ R5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ ( hoare_new_addrs @ H4 @ As4 @ H8 ) ) )
& ( relH
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H4 ) )
& ~ ( member @ nat @ A7 @ As4 ) ) )
@ H4
@ H8 )
& ( ord_less_eq @ nat @ ( lim @ product_unit @ H4 ) @ ( lim @ product_unit @ H8 ) ) ) )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).
% hoare_tripleI
thf(fact_454_hoare__tripleE,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H: heap_ext @ product_unit,As: set @ nat] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ~ ! [R: A,H5: heap_ext @ product_unit] :
( ? [T3: 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 ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ T3 ) ) ) )
=> ( ( rep_assn @ ( Q @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( hoare_new_addrs @ H @ As @ H5 ) ) )
=> ( ( relH
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H ) )
& ~ ( member @ nat @ A7 @ As ) ) )
@ H
@ H5 )
=> ~ ( ord_less_eq @ nat @ ( lim @ product_unit @ H ) @ ( lim @ product_unit @ H5 ) ) ) ) ) ) ) ).
% hoare_tripleE
thf(fact_455_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_456_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F3: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F3 @ Y3 ) @ B2 ) )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ ( F3 @ Y6 ) @ ( F3 @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_457_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less @ nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N2 )
=> ( ! [I: nat] :
( ( ord_less @ nat @ K2 @ I )
=> ( P @ I ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_458_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ord_less @ nat @ ( F3 @ I3 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ ( F3 @ I2 ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_459_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less @ nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_460_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less @ nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_461_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N: nat] :
( ( ord_less @ nat @ M3 @ N )
| ( M3 = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_462_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_463_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
& ( M3 != N ) ) ) ) ).
% nat_less_le
thf(fact_464_subset__emptyI,axiom,
! [A: $tType,A5: set @ A] :
( ! [X3: A] :
~ ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_465_preciseD,axiom,
! [B: $tType,A: $tType,R2: A > B > assn,A3: A,P3: B,F5: assn,A4: A,F6: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( precise @ A @ B @ R2 )
=> ( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R2 @ A3 @ P3 ) @ F5 ) @ ( times_times @ assn @ ( R2 @ A4 @ P3 ) @ F6 ) ) @ H )
=> ( A3 = A4 ) ) ) ).
% preciseD
thf(fact_466_psubsetI,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( A5 != B4 )
=> ( ord_less @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% psubsetI
thf(fact_467_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_468_sup2CI,axiom,
! [A: $tType,B: $tType,B4: A > B > $o,X: A,Y: B,A5: A > B > $o] :
( ( ~ ( B4 @ X @ Y )
=> ( A5 @ X @ Y ) )
=> ( sup_sup @ ( A > B > $o ) @ A5 @ B4 @ X @ Y ) ) ).
% sup2CI
thf(fact_469_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_470_inf1I,axiom,
! [A: $tType,A5: A > $o,X: A,B4: A > $o] :
( ( A5 @ X )
=> ( ( B4 @ X )
=> ( inf_inf @ ( A > $o ) @ A5 @ B4 @ X ) ) ) ).
% inf1I
thf(fact_471_inf2I,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,X: A,Y: B,B4: A > B > $o] :
( ( A5 @ X @ Y )
=> ( ( B4 @ X @ Y )
=> ( inf_inf @ ( A > B > $o ) @ A5 @ B4 @ X @ Y ) ) ) ).
% inf2I
thf(fact_472_sup1CI,axiom,
! [A: $tType,B4: A > $o,X: A,A5: A > $o] :
( ( ~ ( B4 @ X )
=> ( A5 @ X ) )
=> ( sup_sup @ ( A > $o ) @ A5 @ B4 @ X ) ) ).
% sup1CI
thf(fact_473_less__by__empty,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A )] :
( ( A5
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B4 ) ) ).
% less_by_empty
thf(fact_474_inf1E,axiom,
! [A: $tType,A5: A > $o,B4: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A5 @ B4 @ X )
=> ~ ( ( A5 @ X )
=> ~ ( B4 @ X ) ) ) ).
% inf1E
thf(fact_475_inf2E,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,B4: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A5 @ B4 @ X @ Y )
=> ~ ( ( A5 @ X @ Y )
=> ~ ( B4 @ X @ Y ) ) ) ).
% inf2E
thf(fact_476_sup1E,axiom,
! [A: $tType,A5: A > $o,B4: A > $o,X: A] :
( ( sup_sup @ ( A > $o ) @ A5 @ B4 @ X )
=> ( ~ ( A5 @ X )
=> ( B4 @ X ) ) ) ).
% sup1E
thf(fact_477_sup2E,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,B4: A > B > $o,X: A,Y: B] :
( ( sup_sup @ ( A > B > $o ) @ A5 @ B4 @ X @ Y )
=> ( ~ ( A5 @ X @ Y )
=> ( B4 @ X @ Y ) ) ) ).
% sup2E
thf(fact_478_inf1D1,axiom,
! [A: $tType,A5: A > $o,B4: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A5 @ B4 @ X )
=> ( A5 @ X ) ) ).
% inf1D1
thf(fact_479_inf1D2,axiom,
! [A: $tType,A5: A > $o,B4: A > $o,X: A] :
( ( inf_inf @ ( A > $o ) @ A5 @ B4 @ X )
=> ( B4 @ X ) ) ).
% inf1D2
thf(fact_480_inf2D1,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,B4: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A5 @ B4 @ X @ Y )
=> ( A5 @ X @ Y ) ) ).
% inf2D1
thf(fact_481_inf2D2,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,B4: A > B > $o,X: A,Y: B] :
( ( inf_inf @ ( A > B > $o ) @ A5 @ B4 @ X @ Y )
=> ( B4 @ X @ Y ) ) ).
% inf2D2
thf(fact_482_sup1I1,axiom,
! [A: $tType,A5: A > $o,X: A,B4: A > $o] :
( ( A5 @ X )
=> ( sup_sup @ ( A > $o ) @ A5 @ B4 @ X ) ) ).
% sup1I1
thf(fact_483_sup1I2,axiom,
! [A: $tType,B4: A > $o,X: A,A5: A > $o] :
( ( B4 @ X )
=> ( sup_sup @ ( A > $o ) @ A5 @ B4 @ X ) ) ).
% sup1I2
thf(fact_484_sup2I1,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,X: A,Y: B,B4: A > B > $o] :
( ( A5 @ X @ Y )
=> ( sup_sup @ ( A > B > $o ) @ A5 @ B4 @ X @ Y ) ) ).
% sup2I1
thf(fact_485_sup2I2,axiom,
! [A: $tType,B: $tType,B4: A > B > $o,X: A,Y: B,A5: A > B > $o] :
( ( B4 @ X @ Y )
=> ( sup_sup @ ( A > B > $o ) @ A5 @ B4 @ X @ Y ) ) ).
% sup2I2
thf(fact_486_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_487_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_488_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_489_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_490_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F3 @ Z2 @ Less_eq @ Less )
=> ( ( Z2
= ( F3 @ A3 @ B2 ) )
= ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_491_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F3: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F3 @ Z2 @ Less_eq @ Less )
=> ( ( ( F3 @ A3 @ B2 )
= Z2 )
= ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_492_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X @ Y ) ).
% bot2E
thf(fact_493_not__psubset__empty,axiom,
! [A: $tType,A5: set @ A] :
~ ( ord_less @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_494_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A8 @ B7 )
| ( A8 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_495_subset__psubset__trans,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_496_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
& ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_497_psubset__subset__trans,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_498_psubset__imp__subset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_499_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ B7 )
& ( A8 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_500_psubsetE,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% psubsetE
thf(fact_501_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G4 )
& ~ ( ord_less_eq @ ( A > B ) @ G4 @ F4 ) ) ) ) ) ).
% less_fun_def
thf(fact_502_less__assn__def,axiom,
( ( ord_less @ assn )
= ( ^ [A7: assn,B6: assn] :
( ( ord_less_eq @ assn @ A7 @ B6 )
& ( A7 != B6 ) ) ) ) ).
% less_assn_def
thf(fact_503_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y6: A] :
( ( ord_less @ B @ ( F3 @ Y6 ) @ ( F3 @ X3 ) )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_504_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y6: A] :
( ( ord_less @ B @ ( F3 @ Y6 ) @ ( F3 @ X3 ) )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_505_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S2: B,R2: set @ ( product_prod @ A @ B ),S4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S2 ) @ R2 )
=> ( ( S4 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S4 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_506_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less @ nat @ M @ N2 )
| ( ord_less @ nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_507_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_508_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_509_less__not__refl3,axiom,
! [S2: nat,T6: nat] :
( ( ord_less @ nat @ S2 @ T6 )
=> ( S2 != T6 ) ) ).
% less_not_refl3
thf(fact_510_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_511_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ M4 @ N5 )
=> ( P @ M4 ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_512_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M4: nat] :
( ( ord_less @ nat @ M4 @ N5 )
& ~ ( P @ M4 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_513_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_514_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y6: A] :
( ( ord_less @ nat @ ( V @ Y6 ) @ ( V @ X3 ) )
& ~ ( P @ Y6 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_515_le__refl,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% le_refl
thf(fact_516_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_517_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_518_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_519_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
| ( ord_less_eq @ nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_520_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_521_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y6 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_522_sngr__prec,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( precise @ A @ ( ref @ A )
@ ^ [X4: A,P6: ref @ A] : ( sngr_assn @ A @ P6 @ X4 ) ) ) ).
% sngr_prec
thf(fact_523_snga__prec,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( precise @ ( list @ A ) @ ( array @ A )
@ ^ [X4: list @ A,P6: array @ A] : ( snga_assn @ A @ P6 @ X4 ) ) ) ).
% snga_prec
thf(fact_524_preciseD_H,axiom,
! [B: $tType,A: $tType,R2: A > B > assn,A3: A,P3: B,F5: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),A4: A,F6: assn] :
( ( precise @ A @ B @ R2 )
=> ( ( rep_assn @ ( times_times @ assn @ ( R2 @ A3 @ P3 ) @ F5 ) @ H )
=> ( ( rep_assn @ ( times_times @ assn @ ( R2 @ A4 @ P3 ) @ F6 ) @ H )
=> ( A3 = A4 ) ) ) ) ).
% preciseD'
thf(fact_525_prop__restrict,axiom,
! [A: $tType,X: A,Z5: set @ A,X6: set @ A,P: A > $o] :
( ( member @ A @ X @ Z5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_526_Collect__restrict,axiom,
! [A: $tType,X6: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( P @ X4 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_527_precise__def,axiom,
! [B: $tType,A: $tType] :
( ( precise @ A @ B )
= ( ^ [R6: A > B > assn] :
! [A7: A,A10: A,H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P6: B,F7: assn,F8: assn] :
( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R6 @ A7 @ P6 ) @ F7 ) @ ( times_times @ assn @ ( R6 @ A10 @ P6 ) @ F8 ) ) @ H6 )
=> ( A7 = A10 ) ) ) ) ).
% precise_def
thf(fact_528_preciseI,axiom,
! [B: $tType,A: $tType,R2: A > B > assn] :
( ! [A6: A,A11: A,H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),P7: B,F9: assn,F10: assn] :
( ( rep_assn @ ( inf_inf @ assn @ ( times_times @ assn @ ( R2 @ A6 @ P7 ) @ F9 ) @ ( times_times @ assn @ ( R2 @ A11 @ P7 ) @ F10 ) ) @ H4 )
=> ( A6 = A11 ) )
=> ( precise @ A @ B @ R2 ) ) ).
% preciseI
thf(fact_529_reflclp__idemp,axiom,
! [A: $tType,P: A > A > $o] :
( ( sup_sup @ ( A > A > $o )
@ ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) ) ).
% reflclp_idemp
thf(fact_530_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ~ ( ord_less_eq @ A @ T6 @ X7 ) ) ) ).
% minf(8)
thf(fact_531_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ord_less_eq @ A @ X7 @ T6 ) ) ) ).
% minf(6)
thf(fact_532_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ord_less_eq @ A @ T6 @ X7 ) ) ) ).
% pinf(8)
thf(fact_533_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ~ ( ord_less_eq @ A @ X7 @ T6 ) ) ) ).
% pinf(6)
thf(fact_534_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B3: B,A4: B] :
( ( ~ ( ord_less_eq @ B @ B3 @ A4 ) )
= ( ord_less @ B @ A4 @ B3 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_535_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B2 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
& ( ord_less_eq @ A @ C3 @ B2 )
& ! [X7: A] :
( ( ( ord_less_eq @ A @ A3 @ X7 )
& ( ord_less @ A @ X7 @ C3 ) )
=> ( P @ X7 ) )
& ! [D5: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less @ A @ X3 @ D5 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D5 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_536_mod__h__bot__iff_I8_J,axiom,
! [C: $tType,R2: C > assn,H: heap_ext @ product_unit] :
( ( rep_assn @ ( ex_assn @ C @ R2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ? [X4: C] : ( rep_assn @ ( R2 @ X4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(8)
thf(fact_537_bijective__Empty,axiom,
! [B: $tType,A: $tType] : ( bijective @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% bijective_Empty
thf(fact_538_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_539_times__assn__raw_Opelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) )
=> ? [As13: set @ nat,As23: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As13 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As23 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
thf(fact_540_ex__assn__const,axiom,
! [A: $tType,C2: assn] :
( ( ex_assn @ A
@ ^ [X4: A] : C2 )
= C2 ) ).
% ex_assn_const
thf(fact_541_mod__ex__dist,axiom,
! [A: $tType,P: A > assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H )
= ( ? [X4: A] : ( rep_assn @ ( P @ X4 ) @ H ) ) ) ).
% mod_ex_dist
thf(fact_542_psubsetD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% psubsetD
thf(fact_543_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A8 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ).
% less_set_def
thf(fact_544_psubset__trans,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% psubset_trans
thf(fact_545_mod__exE,axiom,
! [A: $tType,P: A > assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( ex_assn @ A @ P ) @ H )
=> ~ ! [X3: A] :
~ ( rep_assn @ ( P @ X3 ) @ H ) ) ).
% mod_exE
thf(fact_546_mod__exI,axiom,
! [A: $tType,P: A > assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ? [X7: A] : ( rep_assn @ ( P @ X7 ) @ H )
=> ( rep_assn @ ( ex_assn @ A @ P ) @ H ) ) ).
% mod_exI
thf(fact_547_ex__one__point__gen,axiom,
! [A: $tType,P: A > assn,V2: A] :
( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),X3: A] :
( ( rep_assn @ ( P @ X3 ) @ H4 )
=> ( X3 = V2 ) )
=> ( ( ex_assn @ A @ P )
= ( P @ V2 ) ) ) ).
% ex_one_point_gen
thf(fact_548_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_549_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_550_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_551_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_552_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
| ~ ( ord_less_eq @ A @ A3 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_553_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(2)
thf(fact_554_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A3: A] :
? [B5: A] :
( ( ord_less @ A @ A3 @ B5 )
| ( ord_less @ A @ B5 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_555_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_556_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P8: A > $o,Q: A > $o,Q5: A > $o] :
( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P8 @ X3 ) ) )
=> ( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q5 @ X3 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P8 @ X7 )
& ( Q5 @ X7 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_557_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P8: A > $o,Q: A > $o,Q5: A > $o] :
( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P8 @ X3 ) ) )
=> ( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ Z6 @ X3 )
=> ( ( Q @ X3 )
= ( Q5 @ X3 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P8 @ X7 )
| ( Q5 @ X7 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_558_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( X7 != T6 ) ) ) ).
% pinf(3)
thf(fact_559_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( X7 != T6 ) ) ) ).
% pinf(4)
thf(fact_560_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ~ ( ord_less @ A @ X7 @ T6 ) ) ) ).
% pinf(5)
thf(fact_561_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ord_less @ A @ T6 @ X7 ) ) ) ).
% pinf(7)
thf(fact_562_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z3: C] :
! [X7: C] :
( ( ord_less @ C @ Z3 @ X7 )
=> ( F5 = F5 ) ) ) ).
% pinf(11)
thf(fact_563_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P8: A > $o,Q: A > $o,Q5: A > $o] :
( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P8 @ X3 ) ) )
=> ( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q5 @ X3 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P8 @ X7 )
& ( Q5 @ X7 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_564_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P8: A > $o,Q: A > $o,Q5: A > $o] :
( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P8 @ X3 ) ) )
=> ( ? [Z6: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z6 )
=> ( ( Q @ X3 )
= ( Q5 @ X3 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P8 @ X7 )
| ( Q5 @ X7 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_565_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( X7 != T6 ) ) ) ).
% minf(3)
thf(fact_566_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( X7 != T6 ) ) ) ).
% minf(4)
thf(fact_567_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ord_less @ A @ X7 @ T6 ) ) ) ).
% minf(5)
thf(fact_568_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T6: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ~ ( ord_less @ A @ T6 @ X7 ) ) ) ).
% minf(7)
thf(fact_569_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z3: C] :
! [X7: C] :
( ( ord_less @ C @ X7 @ Z3 )
=> ( F5 = F5 ) ) ) ).
% minf(11)
thf(fact_570_bijective__def,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B )] :
( ! [X4: A,Y4: B,Z7: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z7 ) @ R6 ) )
=> ( Y4 = Z7 ) )
& ! [X4: A,Y4: A,Z7: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z7 ) @ R6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y4 @ Z7 ) @ R6 ) )
=> ( X4 = Y4 ) ) ) ) ) ).
% bijective_def
thf(fact_571_times__assn__raw_Opelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( Y
= ( ? [As12: set @ nat,As22: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As12 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As22 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(1)
thf(fact_572_times__assn__raw_Opelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) )
=> ~ ? [As1: set @ nat,As2: set @ nat] :
( ( As4
= ( sup_sup @ ( set @ nat ) @ As1 @ As2 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As2 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
thf(fact_573_the__dflt__None__set,axiom,
! [A: $tType,X: set @ A] :
( ( the_default @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( dflt_None_set @ A @ X ) )
= X ) ).
% the_dflt_None_set
thf(fact_574_accp__subset,axiom,
! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
=> ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).
% accp_subset
thf(fact_575_accp__subset__induct,axiom,
! [A: $tType,D4: A > $o,R2: A > A > $o,X: A,P: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ D4 @ ( accp @ A @ R2 ) )
=> ( ! [X3: A,Z3: A] :
( ( D4 @ X3 )
=> ( ( R2 @ Z3 @ X3 )
=> ( D4 @ Z3 ) ) )
=> ( ( D4 @ X )
=> ( ! [X3: A] :
( ( D4 @ X3 )
=> ( ! [Z6: A] :
( ( R2 @ Z6 @ X3 )
=> ( P @ Z6 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_576_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_577_mod__h__bot__iff_I1_J,axiom,
! [B2: $o,H: heap_ext @ product_unit] :
( ( rep_assn @ ( pure_assn @ B2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= B2 ) ).
% mod_h_bot_iff(1)
thf(fact_578_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A3: B,B2: C] :
( ( uncurry @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( F3 @ A3 @ B2 ) ) ).
% uncurry_apply
thf(fact_579_hoare__triple__effect,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H: heap_ext @ product_unit,As: set @ nat] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ? [H5: heap_ext @ product_unit,R: A,T3: nat] :
( ( heap_Time_effect @ A @ C2 @ H @ H5 @ R @ T3 )
& ( rep_assn @ ( Q @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( hoare_new_addrs @ H @ As @ H5 ) ) ) ) ) ) ).
% hoare_triple_effect
thf(fact_580_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_581_wand__assnI,axiom,
! [H: heap_ext @ product_unit,As: set @ nat,Q: assn,R2: assn] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( ! [H5: heap_ext @ product_unit,As6: set @ nat] :
( ( ( inf_inf @ ( set @ nat ) @ As @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( relH @ As @ H @ H5 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As ) )
=> ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As6 ) )
=> ( rep_assn @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As @ As6 ) ) ) ) ) ) )
=> ( rep_assn @ ( wand_assn @ Q @ R2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ).
% wand_assnI
thf(fact_582_pure__assn__eq__conv,axiom,
! [P: $o,Q: $o] :
( ( ( pure_assn @ P )
= ( pure_assn @ Q ) )
= ( P = Q ) ) ).
% pure_assn_eq_conv
thf(fact_583_merge__pure__star,axiom,
! [A3: $o,B2: $o] :
( ( times_times @ assn @ ( pure_assn @ A3 ) @ ( pure_assn @ B2 ) )
= ( pure_assn
@ ( A3
& B2 ) ) ) ).
% merge_pure_star
thf(fact_584_merge__pure__or,axiom,
! [A3: $o,B2: $o] :
( ( sup_sup @ assn @ ( pure_assn @ A3 ) @ ( pure_assn @ B2 ) )
= ( pure_assn
@ ( A3
| B2 ) ) ) ).
% merge_pure_or
thf(fact_585_pure__assn__eq__false__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( bot_bot @ assn ) )
= ~ P ) ).
% pure_assn_eq_false_iff
thf(fact_586_pure__false,axiom,
( ( pure_assn @ $false )
= ( bot_bot @ assn ) ) ).
% pure_false
thf(fact_587_merge__pure__and,axiom,
! [A3: $o,B2: $o] :
( ( inf_inf @ assn @ ( pure_assn @ A3 ) @ ( pure_assn @ B2 ) )
= ( pure_assn
@ ( A3
& B2 ) ) ) ).
% merge_pure_and
thf(fact_588_in__range__empty,axiom,
! [H: heap_ext @ product_unit] : ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% in_range_empty
thf(fact_589_mod__pure__star__dist,axiom,
! [P: assn,B2: $o,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ H )
= ( ( rep_assn @ P @ H )
& B2 ) ) ).
% mod_pure_star_dist
thf(fact_590_in__range__dist__union,axiom,
! [H: heap_ext @ product_unit,As: set @ nat,As3: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( sup_sup @ ( set @ nat ) @ As @ As3 ) ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As3 ) ) ) ) ).
% in_range_dist_union
thf(fact_591_precise__extr__pure_I1_J,axiom,
! [B: $tType,A: $tType,P: $o,R2: A > B > assn] :
( ( precise @ A @ B
@ ^ [X4: A,Y4: B] : ( times_times @ assn @ ( pure_assn @ P ) @ ( R2 @ X4 @ Y4 ) ) )
= ( P
=> ( precise @ A @ B @ R2 ) ) ) ).
% precise_extr_pure(1)
thf(fact_592_precise__extr__pure_I2_J,axiom,
! [B: $tType,A: $tType,R2: A > B > assn,P: $o] :
( ( precise @ A @ B
@ ^ [X4: A,Y4: B] : ( times_times @ assn @ ( R2 @ X4 @ Y4 ) @ ( pure_assn @ P ) ) )
= ( P
=> ( precise @ A @ B @ R2 ) ) ) ).
% precise_extr_pure(2)
thf(fact_593_effect__LetI,axiom,
! [B: $tType,A: $tType,X: A,T6: A,F3: A > ( heap_Time_Heap @ B ),H: heap_ext @ product_unit,H3: heap_ext @ product_unit,R3: B,N2: nat] :
( ( X = T6 )
=> ( ( heap_Time_effect @ B @ ( F3 @ X ) @ H @ H3 @ R3 @ N2 )
=> ( heap_Time_effect @ B @ ( F3 @ T6 ) @ H @ H3 @ R3 @ N2 ) ) ) ).
% effect_LetI
thf(fact_594_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_595_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( A3 != Top )
= ( Less @ A3 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_596_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
= ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_597_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A3 ) ) ).
% ordering_top.extremum_strict
thf(fact_598_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( Less_eq @ A3 @ Top ) ) ).
% ordering_top.extremum
thf(fact_599_models__in__range,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> ( in_range @ H ) ) ).
% models_in_range
thf(fact_600_relH__in__rangeI_I2_J,axiom,
! [As: set @ nat,H: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( relH @ As @ H @ H3 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As ) ) ) ).
% relH_in_rangeI(2)
thf(fact_601_relH__in__rangeI_I1_J,axiom,
! [As: set @ nat,H: heap_ext @ product_unit,H3: heap_ext @ product_unit] :
( ( relH @ As @ H @ H3 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ).
% relH_in_rangeI(1)
thf(fact_602_relH__refl,axiom,
! [H: heap_ext @ product_unit,As: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( relH @ As @ H @ H ) ) ).
% relH_refl
thf(fact_603_in__range__subset,axiom,
! [As: set @ nat,As3: set @ nat,H: heap_ext @ product_unit] :
( ( ord_less_eq @ ( set @ nat ) @ As @ As3 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As3 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) ) ) ) ).
% in_range_subset
thf(fact_604_effectI,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H: heap_ext @ product_unit,R3: A,H3: heap_ext @ product_unit,N2: 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 ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N2 ) ) ) )
=> ( heap_Time_effect @ A @ C2 @ H @ H3 @ R3 @ N2 ) ) ).
% effectI
thf(fact_605_effect__def,axiom,
! [A: $tType] :
( ( heap_Time_effect @ A )
= ( ^ [C5: heap_Time_Heap @ A,H6: heap_ext @ product_unit,H7: heap_ext @ product_unit,R4: A,N: nat] :
( ( heap_Time_execute @ A @ C5 @ H6 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ N ) ) ) ) ) ) ).
% effect_def
thf(fact_606_in__range_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( in_range @ X )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ! [X3: nat] :
( ( member @ nat @ X3 @ As4 )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H4 ) ) ) ) ) ).
% in_range.elims(3)
thf(fact_607_in__range_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ X )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ~ ! [X7: nat] :
( ( member @ nat @ X7 @ As4 )
=> ( ord_less @ nat @ X7 @ ( lim @ product_unit @ H4 ) ) ) ) ) ).
% in_range.elims(2)
thf(fact_608_in__range_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( in_range @ X )
= Y )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( Y
= ( ~ ! [X4: nat] :
( ( member @ nat @ X4 @ As4 )
=> ( ord_less @ nat @ X4 @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ).
% in_range.elims(1)
thf(fact_609_in__range_Osimps,axiom,
! [H: heap_ext @ product_unit,As: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ As )
=> ( ord_less @ nat @ X4 @ ( lim @ product_unit @ H ) ) ) ) ) ).
% in_range.simps
thf(fact_610_Heap__eqI,axiom,
! [A: $tType,F3: heap_Time_Heap @ A,G3: heap_Time_Heap @ A] :
( ! [H4: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ F3 @ H4 )
= ( heap_Time_execute @ A @ G3 @ H4 ) )
=> ( F3 = G3 ) ) ).
% Heap_eqI
thf(fact_611_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_612_the__default_Osimps_I1_J,axiom,
! [A: $tType,Uu: A,X: A] :
( ( the_default @ A @ Uu @ ( some @ A @ X ) )
= X ) ).
% the_default.simps(1)
thf(fact_613_wand__raw_Oelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
& ! [H5: heap_ext @ product_unit,As6: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H4 @ H5 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As6 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As4 @ As6 ) ) ) ) ) ) ) ).
% wand_raw.elims(3)
thf(fact_614_wand__raw_Oelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
& ! [H8: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H4 @ H8 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As7 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ ( sup_sup @ ( set @ nat ) @ As4 @ As7 ) ) ) ) ) ) ) ).
% wand_raw.elims(2)
thf(fact_615_wand__raw_Oelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( Y
= ( ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
& ! [H7: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H4 @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As4 @ As8 ) ) ) ) ) ) ) ) ) ).
% wand_raw.elims(1)
thf(fact_616_wand__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H: heap_ext @ product_unit,As: set @ nat] :
( ( wand_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
& ! [H7: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As ) )
& ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
=> ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As @ As8 ) ) ) ) ) ) ).
% wand_raw.simps
thf(fact_617_in__measure,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F3 ) )
= ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ).
% in_measure
thf(fact_618_wand__raw_Opelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
& ! [H5: heap_ext @ product_unit,As6: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H4 @ H5 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ As6 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 @ ( sup_sup @ ( set @ nat ) @ As4 @ As6 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
thf(fact_619_wand__raw_Opelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
& ! [H8: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H4 @ H8 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ As7 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H8 @ ( sup_sup @ ( set @ nat ) @ As4 @ As7 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
thf(fact_620_wand__raw_Opelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( Y
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
& ! [H7: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As4 @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As4 @ H4 @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As4 ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As4 @ As8 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(1)
thf(fact_621_in__range_Opelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( in_range @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ! [X3: nat] :
( ( member @ nat @ X3 @ As4 )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ).
% in_range.pelims(3)
thf(fact_622_in__range_Opelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ~ ! [X7: nat] :
( ( member @ nat @ X7 @ As4 )
=> ( ord_less @ nat @ X7 @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ).
% in_range.pelims(2)
thf(fact_623_in__range_Opelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( in_range @ X )
= Y )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( Y
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ As4 )
=> ( ord_less @ nat @ X4 @ ( lim @ product_unit @ H4 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) ) ) ) ).
% in_range.pelims(1)
thf(fact_624_wand__assn__def,axiom,
( wand_assn
= ( ^ [P4: assn,Q3: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P4 ) @ ( rep_assn @ Q3 ) ) ) ) ) ).
% wand_assn_def
thf(fact_625_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ^ [A7: A,B6: A] :
( ( P @ A7 @ B6 )
| ( A7 = B6 ) ) ) ).
% eq_subset
thf(fact_626_sup__option__def,axiom,
! [A: $tType] :
( ( sup @ A )
=> ( ( sup_sup @ ( option @ A ) )
= ( ^ [X4: option @ A,Y4: option @ A] :
( case_option @ ( option @ A ) @ A @ Y4
@ ^ [X8: A] :
( case_option @ ( option @ A ) @ A @ X4
@ ^ [Z7: A] : ( some @ A @ ( sup_sup @ A @ X8 @ Z7 ) )
@ Y4 )
@ X4 ) ) ) ) ).
% sup_option_def
thf(fact_627_dflt__None__set__def,axiom,
! [A: $tType] :
( ( dflt_None_set @ A )
= ( ^ [S5: set @ A] :
( if @ ( option @ ( set @ A ) )
@ ( S5
= ( bot_bot @ ( set @ A ) ) )
@ ( none @ ( set @ A ) )
@ ( some @ ( set @ A ) @ S5 ) ) ) ) ).
% dflt_None_set_def
thf(fact_628_times__assn__def,axiom,
( ( times_times @ assn )
= ( ^ [P4: assn,Q3: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P4 ) @ ( rep_assn @ Q3 ) ) ) ) ) ).
% times_assn_def
thf(fact_629_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A8: set @ A] :
( A8
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_630_hoare__triple__success,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H: heap_ext @ product_unit,As: set @ nat] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( heap_Time_success @ A @ C2 @ H ) ) ) ).
% hoare_triple_success
thf(fact_631_subset__Collect__iff,axiom,
! [A: $tType,B4: set @ A,A5: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_632_subset__CollectI,axiom,
! [A: $tType,B4: set @ A,A5: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_633_conj__subset__def,axiom,
! [A: $tType,A5: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A5
@ ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_634_not__None__eq,axiom,
! [A: $tType,X: option @ A] :
( ( X
!= ( none @ A ) )
= ( ? [Y4: A] :
( X
= ( some @ A @ Y4 ) ) ) ) ).
% not_None_eq
thf(fact_635_not__Some__eq,axiom,
! [A: $tType,X: option @ A] :
( ( ! [Y4: A] :
( X
!= ( some @ A @ Y4 ) ) )
= ( X
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_636_Rep__assn__inverse,axiom,
! [X: assn] :
( ( abs_assn @ ( rep_assn @ X ) )
= X ) ).
% Rep_assn_inverse
thf(fact_637_less__eq__option__None__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% less_eq_option_None_code
thf(fact_638_less__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
~ ( ord_less @ ( option @ A ) @ X @ ( none @ A ) ) ) ).
% less_option_None
thf(fact_639_sup__None__2,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [X: option @ A] :
( ( sup_sup @ ( option @ A ) @ X @ ( none @ A ) )
= X ) ) ).
% sup_None_2
thf(fact_640_sup__None__1,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [Y: option @ A] :
( ( sup_sup @ ( option @ A ) @ ( none @ A ) @ Y )
= Y ) ) ).
% sup_None_1
thf(fact_641_inf__None__2,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [X: option @ A] :
( ( inf_inf @ ( option @ A ) @ X @ ( none @ A ) )
= ( none @ A ) ) ) ).
% inf_None_2
thf(fact_642_inf__None__1,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [Y: option @ A] :
( ( inf_inf @ ( option @ A ) @ ( none @ A ) @ Y )
= ( none @ A ) ) ) ).
% inf_None_1
thf(fact_643_less__eq__option__Some__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( none @ A ) ) ) ).
% less_eq_option_Some_None
thf(fact_644_less__option__None__Some__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% less_option_None_Some_code
thf(fact_645_the__dflt__None__empty,axiom,
! [A: $tType] :
( ( dflt_None_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( none @ ( set @ A ) ) ) ).
% the_dflt_None_empty
thf(fact_646_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > B] :
( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(4)
thf(fact_647_bot__option__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( bot_bot @ ( option @ A ) )
= ( none @ A ) ) ) ).
% bot_option_def
thf(fact_648_success__LetI,axiom,
! [A: $tType,B: $tType,X: A,T6: A,F3: A > ( heap_Time_Heap @ B ),H: heap_ext @ product_unit] :
( ( X = T6 )
=> ( ( heap_Time_success @ B @ ( F3 @ X ) @ H )
=> ( heap_Time_success @ B @ ( F3 @ T6 ) @ H ) ) ) ).
% success_LetI
thf(fact_649_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F1: B,F22: A > B,Option: option @ A] :
( ( H @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( case_option @ C @ A @ ( H @ F1 )
@ ^ [X4: A] : ( H @ ( F22 @ X4 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_650_option_Odistinct_I1_J,axiom,
! [A: $tType,X2: A] :
( ( none @ A )
!= ( some @ A @ X2 ) ) ).
% option.distinct(1)
thf(fact_651_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X2: A] :
( ( Option
= ( some @ A @ X2 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_652_option_Oexhaust,axiom,
! [A: $tType,Y: option @ A] :
( ( Y
!= ( none @ A ) )
=> ~ ! [X22: A] :
( Y
!= ( some @ A @ X22 ) ) ) ).
% option.exhaust
thf(fact_653_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P5: ( option @ A ) > $o] :
? [X5: option @ A] : ( P5 @ X5 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
| ? [X4: A] : ( P4 @ ( some @ A @ X4 ) ) ) ) ) ).
% split_option_ex
thf(fact_654_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P5: ( option @ A ) > $o] :
! [X5: option @ A] : ( P5 @ X5 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
& ! [X4: A] : ( P4 @ ( some @ A @ X4 ) ) ) ) ) ).
% split_option_all
thf(fact_655_combine__options__cases,axiom,
! [A: $tType,B: $tType,X: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
( ( ( X
= ( none @ A ) )
=> ( P @ X @ Y ) )
=> ( ( ( Y
= ( none @ B ) )
=> ( P @ X @ Y ) )
=> ( ! [A6: A,B5: B] :
( ( X
= ( some @ A @ A6 ) )
=> ( ( Y
= ( some @ B @ B5 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_656_less__eq__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% less_eq_option_None
thf(fact_657_less__eq__option__None__is__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ X @ ( none @ A ) )
=> ( X
= ( none @ A ) ) ) ) ).
% less_eq_option_None_is_None
thf(fact_658_Abs__assn__eqI_I1_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H4 )
= ( rep_assn @ Pr @ H4 ) )
=> ( ( abs_assn @ P )
= Pr ) ) ).
% Abs_assn_eqI(1)
thf(fact_659_Abs__assn__eqI_I2_J,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Pr: assn] :
( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( P @ H4 )
= ( rep_assn @ Pr @ H4 ) )
=> ( Pr
= ( abs_assn @ P ) ) ) ).
% Abs_assn_eqI(2)
thf(fact_660_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > B,X2: A] :
( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X2 ) )
= ( F22 @ X2 ) ) ).
% option.simps(5)
thf(fact_661_the__default_Osimps_I2_J,axiom,
! [A: $tType,X: A] :
( ( the_default @ A @ X @ ( none @ A ) )
= X ) ).
% the_default.simps(2)
thf(fact_662_inf__option__def,axiom,
! [A: $tType] :
( ( inf @ A )
=> ( ( inf_inf @ ( option @ A ) )
= ( ^ [X4: option @ A,Y4: option @ A] :
( case_option @ ( option @ A ) @ A @ ( none @ A )
@ ^ [Z7: A] :
( case_option @ ( option @ A ) @ A @ ( none @ A )
@ ^ [Aa2: A] : ( some @ A @ ( inf_inf @ A @ Z7 @ Aa2 ) )
@ Y4 )
@ X4 ) ) ) ) ).
% inf_option_def
thf(fact_663_bot__assn__def,axiom,
( ( bot_bot @ assn )
= ( abs_assn
@ ^ [Uu2: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] : $false ) ) ).
% bot_assn_def
thf(fact_664_less__option__None__is__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
( ( ord_less @ ( option @ A ) @ ( none @ A ) @ X )
=> ? [Z3: A] :
( X
= ( some @ A @ Z3 ) ) ) ) ).
% less_option_None_is_Some
thf(fact_665_less__option__None__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% less_option_None_Some
thf(fact_666_sup__assn__def,axiom,
( ( sup_sup @ assn )
= ( ^ [P4: assn,Q3: assn] :
( abs_assn
@ ^ [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P4 @ H6 )
| ( rep_assn @ Q3 @ H6 ) ) ) ) ) ).
% sup_assn_def
thf(fact_667_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R2: $o,X: A,Y: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R2 )
=> ( R2
& ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_668_inf__assn__def,axiom,
( ( inf_inf @ assn )
= ( ^ [P4: assn,Q3: assn] :
( abs_assn
@ ^ [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P4 @ H6 )
& ( rep_assn @ Q3 @ H6 ) ) ) ) ) ).
% inf_assn_def
thf(fact_669_successE,axiom,
! [A: $tType,F3: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ( heap_Time_success @ A @ F3 @ H )
=> ~ ! [R: A,H5: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F3 @ H )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ H5 ) ) ) ) ).
% successE
thf(fact_670_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_671_success__bind__executeI,axiom,
! [A: $tType,B: $tType,F3: heap_Time_Heap @ A,H: heap_ext @ product_unit,X: A,H3: heap_ext @ product_unit,N2: nat,G3: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F3 @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N2 ) ) ) )
=> ( ( heap_Time_success @ B @ ( G3 @ X ) @ H3 )
=> ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F3 @ G3 ) @ H ) ) ) ).
% success_bind_executeI
thf(fact_672_sngr__assn__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( sngr_assn @ A )
= ( ^ [R4: ref @ A,X4: A] : ( abs_assn @ ( sngr_assn_raw @ A @ R4 @ X4 ) ) ) ) ) ).
% sngr_assn_def
thf(fact_673_snga__assn__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( snga_assn @ A )
= ( ^ [R4: array @ A,A7: list @ A] : ( abs_assn @ ( snga_assn_raw @ A @ R4 @ A7 ) ) ) ) ) ).
% snga_assn_def
thf(fact_674_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_675_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_676_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > $o,X: option @ A,R2: B > $o,F3: B,G3: A > B] :
( ( case_option @ $o @ A @ P @ Q @ X )
=> ( ( P
=> ( R2 @ F3 ) )
=> ( ! [Q6: A] :
( ( Q @ Q6 )
=> ( R2 @ ( G3 @ Q6 ) ) )
=> ( R2 @ ( case_option @ B @ A @ F3 @ G3 @ X ) ) ) ) ) ).
% disjE_realizer2
thf(fact_677_pure__assn__def,axiom,
( pure_assn
= ( ^ [B6: $o] : ( abs_assn @ ( pure_assn_raw @ ( heap_ext @ product_unit ) @ nat @ B6 ) ) ) ) ).
% pure_assn_def
thf(fact_678_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_679_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B2: C] :
( ( produc5280177257484947105e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( C2 @ A3 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_680_Heap__Time__Monad_Obind__bind,axiom,
! [C: $tType,B: $tType,A: $tType,F3: heap_Time_Heap @ A,G3: A > ( heap_Time_Heap @ C ),K: C > ( heap_Time_Heap @ B )] :
( ( heap_Time_bind @ C @ B @ ( heap_Time_bind @ A @ C @ F3 @ G3 ) @ K )
= ( heap_Time_bind @ A @ B @ F3
@ ^ [X4: A] : ( heap_Time_bind @ C @ B @ ( G3 @ X4 ) @ K ) ) ) ).
% Heap_Time_Monad.bind_bind
thf(fact_681_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V2: option @ ( product_prod @ A @ B )] :
( ( ! [X4: A,Y4: B] :
( V2
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) )
= ( V2
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_682_map__to__set__inverse,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( set_to_map @ A @ B @ ( map_to_set @ A @ B @ M ) )
= M ) ).
% map_to_set_inverse
thf(fact_683_execute__bind_I2_J,axiom,
! [A: $tType,B: $tType,F3: heap_Time_Heap @ A,H: heap_ext @ product_unit,G3: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F3 @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F3 @ G3 ) @ H )
= ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_bind(2)
thf(fact_684_distrib__if__bind,axiom,
! [A: $tType,B: $tType,B2: $o,C2: heap_Time_Heap @ B,D3: heap_Time_Heap @ B,F3: B > ( heap_Time_Heap @ A )] :
( ( B2
=> ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D3 ) @ F3 )
= ( heap_Time_bind @ B @ A @ C2 @ F3 ) ) )
& ( ~ B2
=> ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D3 ) @ F3 )
= ( heap_Time_bind @ B @ A @ D3 @ F3 ) ) ) ) ).
% distrib_if_bind
thf(fact_685_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu2: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_686_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu2: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_687_success__def,axiom,
! [A: $tType] :
( ( heap_Time_success @ A )
= ( ^ [F4: heap_Time_Heap @ A,H6: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ F4 @ H6 )
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% success_def
thf(fact_688_successI,axiom,
! [A: $tType,F3: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ( ( heap_Time_execute @ A @ F3 @ H )
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( heap_Time_success @ A @ F3 @ H ) ) ).
% successI
thf(fact_689_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X )
=> ( ( ( X
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y3: A] :
( ( X
= ( some @ A @ Y3 ) )
=> ~ ( Q @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_690_map__to__set__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( ( map_to_set @ A @ B @ M )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( M
= ( ^ [X4: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(1)
thf(fact_691_map__to__set__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
= ( map_to_set @ A @ B @ M ) )
= ( M
= ( ^ [X4: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(2)
thf(fact_692_less__eq__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less_eq @ ( option @ A ) )
= ( ^ [X4: option @ A,Y4: option @ A] :
( case_option @ $o @ A @ $true
@ ^ [Z7: A] : ( case_option @ $o @ A @ $false @ ( ord_less_eq @ A @ Z7 ) @ Y4 )
@ X4 ) ) ) ) ).
% less_eq_option_def
thf(fact_693_less__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ ( option @ A ) )
= ( ^ [X4: option @ A] :
( case_option @ $o @ A @ $false
@ ^ [Y4: A] :
( case_option @ $o @ A @ $true
@ ^ [Z7: A] : ( ord_less @ A @ Z7 @ Y4 )
@ X4 ) ) ) ) ) ).
% less_option_def
thf(fact_694_Heap__cases,axiom,
! [A: $tType,F3: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ! [X3: A,H5: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F3 @ H )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ H5 ) ) )
=> ( ( heap_Time_execute @ A @ F3 @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% Heap_cases
thf(fact_695_timeFrame_Ocases,axiom,
! [A: $tType,X: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ! [N5: nat,R: A,H4: heap_ext @ product_unit,N4: nat] :
( X
!= ( 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 ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N4 ) ) ) ) )
=> ~ ! [N5: nat] :
( X
!= ( 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_696_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_697_pure__assn__raw_Oelims_I3_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ~ ( pure_assn_raw @ A @ B @ X @ Xa )
=> ~ ! [H4: A,As4: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As4 ) )
=> ( ( As4
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ).
% pure_assn_raw.elims(3)
thf(fact_698_pure__assn__raw_Oelims_I2_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ( pure_assn_raw @ A @ B @ X @ Xa )
=> ~ ! [H4: A,As4: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As4 ) )
=> ~ ( ( As4
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ).
% pure_assn_raw.elims(2)
thf(fact_699_pure__assn__raw_Oelims_I1_J,axiom,
! [A: $tType,B: $tType,X: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
( ( ( pure_assn_raw @ A @ B @ X @ Xa )
= Y )
=> ~ ! [H4: A,As4: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As4 ) )
=> ( Y
= ( ~ ( ( As4
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.elims(1)
thf(fact_700_pure__assn__raw_Osimps,axiom,
! [A: $tType,B: $tType,B2: $o,H: A,As: set @ B] :
( ( pure_assn_raw @ A @ B @ B2 @ ( product_Pair @ A @ ( set @ B ) @ H @ As ) )
= ( ( As
= ( bot_bot @ ( set @ B ) ) )
& B2 ) ) ).
% pure_assn_raw.simps
thf(fact_701_pure__assn__raw_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
( ( ( pure_assn_raw @ A @ B @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
=> ~ ! [H4: A,As4: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As4 ) )
=> ( ( Y
= ( ( As4
= ( bot_bot @ ( set @ B ) ) )
& X ) )
=> ~ ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H4 @ As4 ) ) ) ) ) ) ) ).
% pure_assn_raw.pelims(1)
thf(fact_702_pure__assn__raw_Opelims_I2_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ( pure_assn_raw @ A @ B @ X @ Xa )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
=> ~ ! [H4: A,As4: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H4 @ As4 ) ) )
=> ~ ( ( As4
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
thf(fact_703_pure__assn__raw_Opelims_I3_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ~ ( pure_assn_raw @ A @ B @ X @ Xa )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
=> ~ ! [H4: A,As4: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H4 @ As4 ) ) )
=> ( ( As4
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
thf(fact_704_execute__bind_I1_J,axiom,
! [A: $tType,B: $tType,F3: heap_Time_Heap @ A,H: heap_ext @ product_unit,X: A,H3: heap_ext @ product_unit,N2: nat,G3: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F3 @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N2 ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F3 @ G3 ) @ H )
= ( heap_Time_timeFrame @ B @ N2 @ ( heap_Time_execute @ B @ ( G3 @ X ) @ H3 ) ) ) ) ).
% execute_bind(1)
thf(fact_705_execute__bind__eq__SomeI,axiom,
! [A: $tType,B: $tType,F3: heap_Time_Heap @ A,H: heap_ext @ product_unit,X: A,H3: heap_ext @ product_unit,N2: nat,G3: A > ( heap_Time_Heap @ B ),Y: B,H9: heap_ext @ product_unit,N6: nat] :
( ( ( heap_Time_execute @ A @ F3 @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N2 ) ) ) )
=> ( ( ( heap_Time_execute @ B @ ( G3 @ X ) @ H3 )
= ( 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 @ H9 @ N6 ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F3 @ G3 ) @ H )
= ( 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 @ H9 @ ( plus_plus @ nat @ N2 @ N6 ) ) ) ) ) ) ) ).
% execute_bind_eq_SomeI
thf(fact_706_set__to__map__def,axiom,
! [A: $tType,B: $tType] :
( ( set_to_map @ B @ A )
= ( ^ [S5: set @ ( product_prod @ B @ A ),K3: B] :
( eps_Opt @ A
@ ^ [V3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K3 @ V3 ) @ S5 ) ) ) ) ).
% set_to_map_def
thf(fact_707_execute__guard_I2_J,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H: heap_ext @ product_unit,F3: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( P @ H )
=> ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F3 ) @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F3 @ H ) ) ) ) ).
% execute_guard(2)
thf(fact_708_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_709_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_710_mlex__leq,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R2 ) ) ) ) ).
% mlex_leq
thf(fact_711_curryI,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( F3 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( product_curry @ A @ B @ $o @ F3 @ A3 @ B2 ) ) ).
% curryI
thf(fact_712_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_713_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_714_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F4: ( product_prod @ B @ C ) > A,A7: B,B6: C] : ( F4 @ ( product_Pair @ B @ C @ A7 @ B6 ) ) ) ) ).
% curry_conv
thf(fact_715_some__opt__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( eps_Opt @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X ) )
= ( some @ A @ X ) ) ).
% some_opt_sym_eq_trivial
thf(fact_716_some__opt__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( eps_Opt @ A
@ ^ [Y4: A] : Y4 = X )
= ( some @ A @ X ) ) ).
% some_opt_eq_trivial
thf(fact_717_some__opt__false__trivial,axiom,
! [A: $tType] :
( ( eps_Opt @ A
@ ^ [Uu2: A] : $false )
= ( none @ A ) ) ).
% some_opt_false_trivial
thf(fact_718_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C2: C] :
( ( product_curry @ A @ B @ C
@ ^ [X4: product_prod @ A @ B] : C2 )
= ( ^ [X4: A,Y4: B] : C2 ) ) ).
% curry_K
thf(fact_719_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
=> ( ord_less @ nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_720_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_721_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_722_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_723_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_724_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_725_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_726_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_727_add__leE,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 )
=> ~ ( ( ord_less_eq @ nat @ M @ N2 )
=> ~ ( ord_less_eq @ nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_728_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M ) ) ).
% le_add1
thf(fact_729_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq @ nat @ N2 @ ( plus_plus @ nat @ M @ N2 ) ) ).
% le_add2
thf(fact_730_add__leD1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% add_leD1
thf(fact_731_add__leD2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N2 )
=> ( ord_less_eq @ nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_732_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N5: nat] :
( L
= ( plus_plus @ nat @ K @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_733_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_734_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_735_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_736_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_737_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_738_curryE,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F3 @ A3 @ B2 )
=> ( F3 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% curryE
thf(fact_739_curryD,axiom,
! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F3 @ A3 @ B2 )
=> ( F3 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% curryD
thf(fact_740_timeFrame_Osimps_I1_J,axiom,
! [A: $tType,N2: nat,R3: A,H: heap_ext @ product_unit,N6: nat] :
( ( heap_Time_timeFrame @ A @ N2 @ ( 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 @ N6 ) ) ) )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( plus_plus @ nat @ N2 @ N6 ) ) ) ) ) ).
% timeFrame.simps(1)
thf(fact_741_timeFrame_Oelims,axiom,
! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( ( heap_Time_timeFrame @ A @ X @ Xa )
= Y )
=> ( ! [R: A,H4: 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 ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N4 ) ) ) )
=> ( Y
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( plus_plus @ nat @ X @ 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_742_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N5: nat] :
( ( ord_less @ nat @ M5 @ N5 )
=> ( ord_less @ nat @ ( F3 @ M5 ) @ ( F3 @ N5 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F3 @ M ) @ K ) @ ( F3 @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_743_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F3: A > nat,N2: nat] :
( ( P @ K )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ? [Y6: A] :
( ( P @ Y6 )
& ~ ( ord_less_eq @ nat @ ( F3 @ Y6 ) @ ( F3 @ X3 ) ) ) )
=> ? [Y3: A] :
( ( P @ Y3 )
& ~ ( ord_less @ nat @ ( F3 @ Y3 ) @ ( plus_plus @ nat @ ( F3 @ K ) @ N2 ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_744_Eps__Opt__eq__Some,axiom,
! [A: $tType,P: A > $o,X: A] :
( ! [X9: A] :
( ( P @ X )
=> ( ( P @ X9 )
=> ( X9 = X ) ) )
=> ( ( ( eps_Opt @ A @ P )
= ( some @ A @ X ) )
= ( P @ X ) ) ) ).
% Eps_Opt_eq_Some
thf(fact_745_Eps__Opt__eq__Some__implies,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( ( eps_Opt @ A @ P )
= ( some @ A @ X ) )
=> ( P @ X ) ) ).
% Eps_Opt_eq_Some_implies
thf(fact_746_execute__guard_I1_J,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H: heap_ext @ product_unit,F3: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ~ ( P @ H )
=> ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F3 ) @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_guard(1)
thf(fact_747_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C5: ( product_prod @ A @ B ) > C,X4: A,Y4: B] : ( C5 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) ) ).
% curry_def
thf(fact_748_mlex__less,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R2 ) ) ) ).
% mlex_less
thf(fact_749_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R2 ) )
= ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% mlex_iff
thf(fact_750_timeFrame_Opelims,axiom,
! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( ( heap_Time_timeFrame @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ Xa ) )
=> ( ! [R: A,H4: 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 ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N4 ) ) ) )
=> ( ( Y
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( plus_plus @ nat @ X @ 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 ) ) ) @ X @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ 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 ) ) ) @ X @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ) ) ) ).
% timeFrame.pelims
thf(fact_751_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_752_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_753_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_754_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_755_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_756_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_757_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_758_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_759_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( A3 != B2 )
& ( C2 != D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A3 @ D3 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_760_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I2 ) @ ( times_times @ nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_761_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_762_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_763_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_764_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_765_add__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N2 ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_766_add__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_767_mlex__bound,axiom,
! [A3: nat,A5: nat,B2: nat,N7: nat] :
( ( ord_less @ nat @ A3 @ A5 )
=> ( ( ord_less @ nat @ B2 @ N7 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N7 ) @ B2 ) @ ( times_times @ nat @ A5 @ N7 ) ) ) ) ).
% mlex_bound
thf(fact_768_mlex__fst__decrI,axiom,
! [A3: nat,A4: nat,B2: nat,N7: nat,B3: nat] :
( ( ord_less @ nat @ A3 @ A4 )
=> ( ( ord_less @ nat @ B2 @ N7 )
=> ( ( ord_less @ nat @ B3 @ N7 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N7 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A4 @ N7 ) @ B3 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_769_mlex__snd__decrI,axiom,
! [A3: nat,A4: nat,B2: nat,B3: nat,N7: nat] :
( ( A3 = A4 )
=> ( ( ord_less @ nat @ B2 @ B3 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N7 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A4 @ N7 ) @ B3 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_770_mlex__leI,axiom,
! [A3: nat,A4: nat,B2: nat,B3: nat,N7: nat] :
( ( ord_less_eq @ nat @ A3 @ A4 )
=> ( ( ord_less_eq @ nat @ B2 @ B3 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N7 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A4 @ N7 ) @ B3 ) ) ) ) ).
% mlex_leI
thf(fact_771_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_772_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A7: A,B6: A] : ( times_times @ A @ B6 @ A7 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_773_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_774_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_775_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_776_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B6: A] :
? [C5: A] :
( B6
= ( plus_plus @ A @ A7 @ C5 ) ) ) ) ) ).
% le_iff_add
thf(fact_777_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_778_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ! [C3: A] :
( B2
!= ( plus_plus @ A @ A3 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_779_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_780_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_781_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_782_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( I2 = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_783_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_784_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_785_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_786_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_787_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_788_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_789_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_790_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( I2 = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_791_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_792_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W2: A,Y: A,X: A,Z2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y ) @ ( times_times @ A @ X @ Z2 ) )
= ( plus_plus @ A @ ( times_times @ A @ W2 @ Z2 ) @ ( times_times @ A @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
thf(fact_793_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_794_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_795_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_796_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_797_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_798_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,E3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E3 ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_799_execute__raise,axiom,
! [A: $tType,S2: list @ char] :
( ( heap_Time_execute @ A @ ( heap_Time_raise @ A @ S2 ) )
= ( ^ [Uu2: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_raise
thf(fact_800_guard__def,axiom,
! [A: $tType] :
( ( heap_Time_guard @ A )
= ( ^ [P4: ( heap_ext @ product_unit ) > $o,F4: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( heap_Time_Heap2 @ A
@ ^ [H6: heap_ext @ product_unit] : ( if @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( P4 @ H6 ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F4 @ H6 ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% guard_def
thf(fact_801_execute__assert_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,H: heap_ext @ product_unit] :
( ~ ( P @ X )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_assert(2)
thf(fact_802_ent__pure__post__iff,axiom,
! [P: assn,Q: assn,B2: $o] :
( ( entails @ P @ ( times_times @ assn @ Q @ ( pure_assn @ B2 ) ) )
= ( ! [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H6 )
=> B2 )
& ( entails @ P @ Q ) ) ) ).
% ent_pure_post_iff
thf(fact_803_Heap__execute,axiom,
! [A: $tType,F3: heap_Time_Heap @ A] :
( ( heap_Time_Heap2 @ A @ ( heap_Time_execute @ A @ F3 ) )
= F3 ) ).
% Heap_execute
thf(fact_804_triv__exI,axiom,
! [A: $tType,Q: A > assn,X: A] : ( entails @ ( Q @ X ) @ ( ex_assn @ A @ Q ) ) ).
% triv_exI
thf(fact_805_ent__pure__pre__iff,axiom,
! [P: assn,B2: $o,Q: assn] :
( ( entails @ ( times_times @ assn @ P @ ( pure_assn @ B2 ) ) @ Q )
= ( B2
=> ( entails @ P @ Q ) ) ) ).
% ent_pure_pre_iff
thf(fact_806_ent__false__iff,axiom,
! [P: assn] :
( ( entails @ P @ ( bot_bot @ assn ) )
= ( ! [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H6 ) ) ) ).
% ent_false_iff
thf(fact_807_ent__trans,axiom,
! [P: assn,Q: assn,R2: assn] :
( ( entails @ P @ Q )
=> ( ( entails @ Q @ R2 )
=> ( entails @ P @ R2 ) ) ) ).
% ent_trans
thf(fact_808_ent__refl,axiom,
! [P: assn] : ( entails @ P @ P ) ).
% ent_refl
thf(fact_809_ent__iffI,axiom,
! [A5: assn,B4: assn] :
( ( entails @ A5 @ B4 )
=> ( ( entails @ B4 @ A5 )
=> ( A5 = B4 ) ) ) ).
% ent_iffI
thf(fact_810_raise__def,axiom,
! [A: $tType] :
( ( heap_Time_raise @ A )
= ( ^ [S6: list @ char] :
( heap_Time_Heap2 @ A
@ ^ [Uu2: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% raise_def
thf(fact_811_entails__def,axiom,
( entails
= ( ^ [P4: assn,Q3: assn] :
! [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P4 @ H6 )
=> ( rep_assn @ Q3 @ H6 ) ) ) ) ).
% entails_def
thf(fact_812_entailsI,axiom,
! [P: assn,Q: assn] :
( ! [H4: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H4 )
=> ( rep_assn @ Q @ H4 ) )
=> ( entails @ P @ Q ) ) ).
% entailsI
thf(fact_813_entailsD,axiom,
! [P: assn,Q: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( entails @ P @ Q )
=> ( ( rep_assn @ P @ H )
=> ( rep_assn @ Q @ H ) ) ) ).
% entailsD
thf(fact_814_ent__fwd,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Q: assn] :
( ( rep_assn @ P @ H )
=> ( ( entails @ P @ Q )
=> ( rep_assn @ Q @ H ) ) ) ).
% ent_fwd
thf(fact_815_ent__star__mono,axiom,
! [P: assn,P8: assn,Q: assn,Q5: assn] :
( ( entails @ P @ P8 )
=> ( ( entails @ Q @ Q5 )
=> ( entails @ ( times_times @ assn @ P @ Q ) @ ( times_times @ assn @ P8 @ Q5 ) ) ) ) ).
% ent_star_mono
thf(fact_816_ent__disjE,axiom,
! [A5: assn,C4: assn,B4: assn] :
( ( entails @ A5 @ C4 )
=> ( ( entails @ B4 @ C4 )
=> ( entails @ ( sup_sup @ assn @ A5 @ B4 ) @ C4 ) ) ) ).
% ent_disjE
thf(fact_817_ent__disjI1,axiom,
! [P: assn,Q: assn,R2: assn] :
( ( entails @ ( sup_sup @ assn @ P @ Q ) @ R2 )
=> ( entails @ P @ R2 ) ) ).
% ent_disjI1
thf(fact_818_ent__disjI2,axiom,
! [P: assn,Q: assn,R2: assn] :
( ( entails @ ( sup_sup @ assn @ P @ Q ) @ R2 )
=> ( entails @ Q @ R2 ) ) ).
% ent_disjI2
thf(fact_819_ent__disjI1_H,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( entails @ A5 @ B4 )
=> ( entails @ A5 @ ( sup_sup @ assn @ B4 @ C4 ) ) ) ).
% ent_disjI1'
thf(fact_820_ent__disjI2_H,axiom,
! [A5: assn,C4: assn,B4: assn] :
( ( entails @ A5 @ C4 )
=> ( entails @ A5 @ ( sup_sup @ assn @ B4 @ C4 ) ) ) ).
% ent_disjI2'
thf(fact_821_ent__disjI1__direct,axiom,
! [A5: assn,B4: assn] : ( entails @ A5 @ ( sup_sup @ assn @ A5 @ B4 ) ) ).
% ent_disjI1_direct
thf(fact_822_ent__disjI2__direct,axiom,
! [B4: assn,A5: assn] : ( entails @ B4 @ ( sup_sup @ assn @ A5 @ B4 ) ) ).
% ent_disjI2_direct
thf(fact_823_execute_Osimps,axiom,
! [A: $tType,F3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_execute @ A @ ( heap_Time_Heap2 @ A @ F3 ) )
= F3 ) ).
% execute.simps
thf(fact_824_ent__false,axiom,
! [P: assn] : ( entails @ ( bot_bot @ assn ) @ P ) ).
% ent_false
thf(fact_825_ent__conjI,axiom,
! [A5: assn,B4: assn,C4: assn] :
( ( entails @ A5 @ B4 )
=> ( ( entails @ A5 @ C4 )
=> ( entails @ A5 @ ( inf_inf @ assn @ B4 @ C4 ) ) ) ) ).
% ent_conjI
thf(fact_826_ent__conjE1,axiom,
! [A5: assn,C4: assn,B4: assn] :
( ( entails @ A5 @ C4 )
=> ( entails @ ( inf_inf @ assn @ A5 @ B4 ) @ C4 ) ) ).
% ent_conjE1
thf(fact_827_ent__conjE2,axiom,
! [B4: assn,C4: assn,A5: assn] :
( ( entails @ B4 @ C4 )
=> ( entails @ ( inf_inf @ assn @ A5 @ B4 ) @ C4 ) ) ).
% ent_conjE2
thf(fact_828_ent__ex__preI,axiom,
! [A: $tType,P: A > assn,Q: assn] :
( ! [X3: A] : ( entails @ ( P @ X3 ) @ Q )
=> ( entails @ ( ex_assn @ A @ P ) @ Q ) ) ).
% ent_ex_preI
thf(fact_829_ent__ex__postI,axiom,
! [A: $tType,P: assn,Q: A > assn,X: A] :
( ( entails @ P @ ( Q @ X ) )
=> ( entails @ P @ ( ex_assn @ A @ Q ) ) ) ).
% ent_ex_postI
thf(fact_830_ent__wandI,axiom,
! [Q: assn,P: assn,R2: assn] :
( ( entails @ ( times_times @ assn @ Q @ P ) @ R2 )
=> ( entails @ P @ ( wand_assn @ Q @ R2 ) ) ) ).
% ent_wandI
thf(fact_831_ent__mp,axiom,
! [P: assn,Q: assn] : ( entails @ ( times_times @ assn @ P @ ( wand_assn @ P @ Q ) ) @ Q ) ).
% ent_mp
thf(fact_832_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_833_execute__assert_I1_J,axiom,
! [A: $tType,P: A > $o,X: A,H: heap_ext @ product_unit] :
( ( P @ X )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_assert(1)
thf(fact_834_Heap__lub__empty,axiom,
! [A: $tType] :
( ( heap_Time_Heap_lub @ A @ ( bot_bot @ ( set @ ( heap_Time_Heap @ A ) ) ) )
= ( heap_Time_Heap2 @ A
@ ^ [X4: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% Heap_lub_empty
thf(fact_835_ent__pure__post__iff__sng,axiom,
! [P: assn,B2: $o] :
( ( entails @ P @ ( pure_assn @ B2 ) )
= ( ! [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H6 )
=> B2 )
& ( entails @ P @ ( one_one @ assn ) ) ) ) ).
% ent_pure_post_iff_sng
thf(fact_836_pairself_Opelims,axiom,
! [B: $tType,A: $tType,X: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ Xa ) )
=> ~ ! [A6: A,B5: A] :
( ( Xa
= ( product_Pair @ A @ A @ A6 @ B5 ) )
=> ( ( Y
= ( product_Pair @ B @ B @ ( X @ A6 ) @ ( X @ B5 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ ( product_Pair @ A @ A @ A6 @ B5 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_837_one__assn__raw_Osimps,axiom,
! [H: heap_ext @ product_unit,As: set @ nat] :
( ( one_assn_raw @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
= ( As
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% one_assn_raw.simps
thf(fact_838_one__assn__raw_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( Y
= ( As4
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% one_assn_raw.elims(1)
thf(fact_839_one__assn__raw_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( one_assn_raw @ X )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( As4
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(2)
thf(fact_840_one__assn__raw_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( one_assn_raw @ X )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( As4
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(3)
thf(fact_841_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_842_combine__options__def,axiom,
! [A: $tType] :
( ( combine_options @ A )
= ( ^ [F4: A > A > A,X4: option @ A,Y4: option @ A] :
( case_option @ ( option @ A ) @ A @ Y4
@ ^ [Z7: A] :
( case_option @ ( option @ A ) @ A @ ( some @ A @ Z7 )
@ ^ [Aa2: A] : ( some @ A @ ( F4 @ Z7 @ Aa2 ) )
@ Y4 )
@ X4 ) ) ) ).
% combine_options_def
thf(fact_843_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_844_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_845_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_846_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_847_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_848_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( times_times @ A @ C2 @ A3 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_849_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( times_times @ A @ A3 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_850_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_851_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_852_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_853_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_854_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_855_le0,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% le0
thf(fact_856_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% bot_nat_0.extremum
thf(fact_857_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N2
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_858_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_859_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_860_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_861_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_862_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_863_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times @ nat @ M @ N2 )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N2
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_864_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N2 ) )
= ( ( M
= ( one_one @ nat ) )
& ( N2
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_865_combine__options__simps_I3_J,axiom,
! [A: $tType,F3: A > A > A,A3: A,B2: A] :
( ( combine_options @ A @ F3 @ ( some @ A @ A3 ) @ ( some @ A @ B2 ) )
= ( some @ A @ ( F3 @ A3 @ B2 ) ) ) ).
% combine_options_simps(3)
thf(fact_866_combine__options__simps_I2_J,axiom,
! [A: $tType,F3: A > A > A,X: option @ A] :
( ( combine_options @ A @ F3 @ X @ ( none @ A ) )
= X ) ).
% combine_options_simps(2)
thf(fact_867_combine__options__simps_I1_J,axiom,
! [A: $tType,F3: A > A > A,Y: option @ A] :
( ( combine_options @ A @ F3 @ ( none @ A ) @ Y )
= Y ) ).
% combine_options_simps(1)
thf(fact_868_pure__true,axiom,
( ( pure_assn @ $true )
= ( one_one @ assn ) ) ).
% pure_true
thf(fact_869_pure__assn__eq__emp__iff,axiom,
! [P: $o] :
( ( ( pure_assn @ P )
= ( one_one @ assn ) )
= P ) ).
% pure_assn_eq_emp_iff
thf(fact_870_assn__basic__inequalities_I3_J,axiom,
( ( bot_bot @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(3)
thf(fact_871_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_872_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_873_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_874_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_875_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_876_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_877_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_878_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_879_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_880_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_881_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_882_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_883_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_884_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A3: A] :
( ( ( times_times @ A @ C2 @ A3 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_885_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_886_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A,C2: A] :
( ( ( times_times @ A @ A3 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_887_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% add_gr_0
thf(fact_888_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_889_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_890_less__one,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( one_one @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_891_ent__pure__pre__iff__sng,axiom,
! [B2: $o,Q: assn] :
( ( entails @ ( pure_assn @ B2 ) @ Q )
= ( B2
=> ( entails @ ( one_one @ assn ) @ Q ) ) ) ).
% ent_pure_pre_iff_sng
thf(fact_892_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_893_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_894_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_895_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_896_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_897_combine__options__assoc,axiom,
! [A: $tType,F3: A > A > A,X: option @ A,Y: option @ A,Z2: option @ A] :
( ! [X3: A,Y3: A,Z3: A] :
( ( F3 @ ( F3 @ X3 @ Y3 ) @ Z3 )
= ( F3 @ X3 @ ( F3 @ Y3 @ Z3 ) ) )
=> ( ( combine_options @ A @ F3 @ ( combine_options @ A @ F3 @ X @ Y ) @ Z2 )
= ( combine_options @ A @ F3 @ X @ ( combine_options @ A @ F3 @ Y @ Z2 ) ) ) ) ).
% combine_options_assoc
thf(fact_898_combine__options__commute,axiom,
! [A: $tType,F3: A > A > A,X: option @ A,Y: option @ A] :
( ! [X3: A,Y3: A] :
( ( F3 @ X3 @ Y3 )
= ( F3 @ Y3 @ X3 ) )
=> ( ( combine_options @ A @ F3 @ X @ Y )
= ( combine_options @ A @ F3 @ Y @ X ) ) ) ).
% combine_options_commute
thf(fact_899_combine__options__left__commute,axiom,
! [A: $tType,F3: A > A > A,Y: option @ A,X: option @ A,Z2: option @ A] :
( ! [X3: A,Y3: A] :
( ( F3 @ X3 @ Y3 )
= ( F3 @ Y3 @ X3 ) )
=> ( ! [X3: A,Y3: A,Z3: A] :
( ( F3 @ ( F3 @ X3 @ Y3 ) @ Z3 )
= ( F3 @ X3 @ ( F3 @ Y3 @ Z3 ) ) )
=> ( ( combine_options @ A @ F3 @ Y @ ( combine_options @ A @ F3 @ X @ Z2 ) )
= ( combine_options @ A @ F3 @ X @ ( combine_options @ A @ F3 @ Y @ Z2 ) ) ) ) ) ).
% combine_options_left_commute
thf(fact_900_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_901_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X )
= ( X
!= ( zero_zero @ nat ) ) ) ).
% nat_geq_1_eq_neqz
thf(fact_902_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times @ nat @ M @ N2 ) )
=> ( ( N2
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_903_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F3: ( A > B ) > C,G3: C] :
( ( F3
= ( ^ [X4: A > B] : G3 ) )
=> ( ( F3
@ ^ [X4: A] : ( zero_zero @ B ) )
= G3 ) ) ) ).
% fun_cong_unused_0
thf(fact_904_one__assn__def,axiom,
( ( one_one @ assn )
= ( abs_assn @ one_assn_raw ) ) ).
% one_assn_def
thf(fact_905_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ A3 ) ) ) ) ).
% mult_left_le
thf(fact_906_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_907_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_908_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_909_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_910_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_911_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_912_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.comm_neutral
thf(fact_913_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
= ( N2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_914_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M: A,N2: A] :
( ( ord_less @ A @ M @ N2 )
=> ( N2
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_915_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_916_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N2: A] :
( ( N2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% gr_zeroI
thf(fact_917_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_918_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_919_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_920_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A3 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_921_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_922_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y6: A] :
( ( ord_less @ nat @ ( V @ Y6 ) @ ( V @ X3 ) )
& ~ ( P @ Y6 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_923_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M4: nat] :
( ( ord_less @ nat @ M4 @ N5 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_924_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( N2
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_925_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_926_not__less0,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_927_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_928_gr0I,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% gr0I
thf(fact_929_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_930_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times @ nat @ N2 @ ( one_one @ nat ) )
= N2 ) ).
% nat_mult_1_right
thf(fact_931_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_932_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_933_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_934_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_935_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_936_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_937_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus @ nat @ M @ N2 )
= M )
=> ( N2
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_938_mult__0,axiom,
! [N2: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_939_assn__one__left,axiom,
! [P: assn] :
( ( times_times @ assn @ ( one_one @ assn ) @ P )
= P ) ).
% assn_one_left
thf(fact_940_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_941_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_942_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_943_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_944_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_945_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_946_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_947_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_948_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A3: A,Y: A,U: A,V2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V2 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_949_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X4: A] : X4 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_950_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H6: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_951_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A3: A,Y: A,U: A,V2: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ( ord_less @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V2 @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_952_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_953_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_954_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: A,N2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N2 ) ) ) ) ) ).
% less_1_mult
thf(fact_955_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_956_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_957_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_958_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_959_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_960_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_961_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_962_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_963_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_964_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_965_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_966_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_967_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_968_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_969_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_970_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_971_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_972_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_973_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_974_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_975_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_976_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_977_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ A3 ) ) ) ).
% zero_le_square
thf(fact_978_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_979_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_980_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_981_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ! [C3: A] :
( ( B2
= ( plus_plus @ A @ A3 @ C3 ) )
=> ( C3
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_982_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_983_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_984_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_985_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_986_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_987_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_988_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_989_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_990_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_991_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_992_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_993_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_994_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_995_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_996_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_997_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_998_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_999_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_1000_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_1001_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_1002_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_1003_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_1004_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R3: A,A3: A,B2: A,C2: A,D3: A] :
( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B2 )
& ( C2 != D3 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R3 @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R3 @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1005_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N2 )
& ! [I: nat] :
( ( ord_less @ nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1006_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ( plus_plus @ nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1007_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ I2 @ K ) @ ( times_times @ nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1008_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ K @ I2 ) @ ( times_times @ nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1009_pairself_Oelims,axiom,
! [B: $tType,A: $tType,X: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X @ Xa )
= Y )
=> ~ ! [A6: A,B5: A] :
( ( Xa
= ( product_Pair @ A @ A @ A6 @ B5 ) )
=> ( Y
!= ( product_Pair @ B @ B @ ( X @ A6 ) @ ( X @ B5 ) ) ) ) ) ).
% pairself.elims
thf(fact_1010_pairself_Osimps,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: A,B2: A] :
( ( pairself @ A @ B @ F3 @ ( product_Pair @ A @ A @ A3 @ B2 ) )
= ( product_Pair @ B @ B @ ( F3 @ A3 ) @ ( F3 @ B2 ) ) ) ).
% pairself.simps
thf(fact_1011_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X4: nat,Y4: nat] : ( ord_less_eq @ nat @ Y4 @ X4 )
@ ^ [X4: nat,Y4: nat] : ( ord_less @ nat @ Y4 @ X4 )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1012_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1013_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1014_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_1015_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_1016_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_1017_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_1018_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1019_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1020_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1021_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1022_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1023_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1024_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1025_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1026_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1027_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1028_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1029_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1030_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1031_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1032_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1033_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1034_mod__emp__simp,axiom,
! [H: heap_ext @ product_unit] : ( rep_assn @ ( one_one @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_emp_simp
thf(fact_1035_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1036_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1037_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_1038_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1039_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A7 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_1040_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1041_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1042_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ X ) @ ( times_times @ A @ Z2 @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1043_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1044_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X7: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X7 ) ) ).
% linordered_field_no_lb
thf(fact_1045_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X7: A] :
? [X_1: A] : ( ord_less @ A @ X7 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_1046_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_1047_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1048_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N2 ) )
= ( M = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1049_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1050_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E2 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_epsilon
thf(fact_1051_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q4: A,R3: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q4 @ R3 ) )
= ( R3
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_1052_tap__def,axiom,
! [A: $tType] :
( ( heap_Time_tap @ A )
= ( ^ [F4: ( heap_ext @ product_unit ) > A] :
( heap_Time_Heap2 @ A
@ ^ [H6: 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 ) @ ( F4 @ H6 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% tap_def
thf(fact_1053_execute__tap,axiom,
! [A: $tType,F3: ( heap_ext @ product_unit ) > A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( heap_Time_tap @ A @ F3 ) @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F3 @ H ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ).
% execute_tap
thf(fact_1054_one__assn__raw_Opelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( one_assn_raw @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( As4
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% one_assn_raw.pelims(3)
thf(fact_1055_one__assn__raw_Opelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( one_assn_raw @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( As4
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% one_assn_raw.pelims(2)
thf(fact_1056_one__assn__raw_Opelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( Y
= ( As4
= ( bot_bot @ ( set @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) ) ) ) ).
% one_assn_raw.pelims(1)
thf(fact_1057_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_1058_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_1059_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M6: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M6 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X7: nat] :
( ( P @ X7 )
=> ( ord_less_eq @ nat @ X7 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1060_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F2: nat > A > A,A6: nat,B5: nat,Acc: A] :
( X
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B5 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_1061_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_1062_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_1063_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_1064_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A1 @ ( product_Pair @ nat @ A @ A22 @ A32 ) ) ) )
=> ( ! [F2: 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 ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B5 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B5 @ A6 )
=> ( P @ F2 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B5 @ ( F2 @ A6 @ Acc ) ) )
=> ( P @ F2 @ A6 @ B5 @ Acc ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_1065_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N2 ) )
= ( ord_less_eq @ nat @ N2 @ M ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_1066_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F4: nat > A > A,A7: nat,B6: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B6 @ A7 ) @ Acc2 @ ( set_fo6178422350223883121st_nat @ A @ F4 @ ( plus_plus @ nat @ A7 @ ( one_one @ nat ) ) @ B6 @ ( F4 @ A7 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_1067_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_1068_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N2: nat] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_1069_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_1070_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1071_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1072_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1073_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% times_divide_eq_right
thf(fact_1074_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_1075_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_1076_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ).
% times_divide_eq_left
thf(fact_1077_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1078_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1079_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1080_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1081_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1082_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1083_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1084_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1085_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ( power_power @ A @ A3 @ M )
= ( power_power @ A @ A3 @ N2 ) )
= ( M = N2 ) ) ) ) ).
% power_inject_exp
thf(fact_1086_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_1087_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_1088_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_1089_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1090_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1091_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1092_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1093_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_1094_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1095_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1096_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_1097_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N2: nat] :
( ( ( power_power @ A @ A3 @ N2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% power_eq_0_iff
thf(fact_1098_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1099_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N2 ) )
= ( ord_less @ nat @ N2 @ M ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1100_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_1101_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N2: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N2 ) @ Y )
= ( times_times @ A @ Y @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_1102_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,N2: nat] :
( ( power_power @ A @ ( times_times @ A @ A3 @ B2 ) @ N2 )
= ( times_times @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ).
% power_mult_distrib
thf(fact_1103_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N2: nat] :
( ( times_times @ A @ ( power_power @ A @ A3 @ N2 ) @ A3 )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% power_commutes
thf(fact_1104_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_1105_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z2: A,W2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ W2 ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_1106_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z2: A,W2: A] :
( ( times_times @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z2 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ Y @ W2 ) ) ) ) ).
% times_divide_times_eq
thf(fact_1107_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N2: nat] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ).
% power_mono
thf(fact_1108_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% zero_le_power
thf(fact_1109_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% zero_less_power
thf(fact_1110_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% one_le_power
thf(fact_1111_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_1112_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_right_mono
thf(fact_1113_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1114_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1115_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1116_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1117_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1118_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A3 @ C2 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1119_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_neg_neg
thf(fact_1120_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_1121_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_1122_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_pos_pos
thf(fact_1123_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_1124_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) )
& ( C2
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_1125_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1126_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1127_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1128_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N2: nat] :
( ( ( times_times @ A @ X @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N2 ) @ ( power_power @ A @ Y @ N2 ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_1129_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y )
= ( divide_divide @ A @ W2 @ Z2 ) )
= ( ( times_times @ A @ X @ Z2 )
= ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1130_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ C2 )
= A3 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1131_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1132_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A3 @ C2 ) )
=> ( ( divide_divide @ A @ B2 @ C2 )
= A3 ) ) ) ) ).
% divide_eq_imp
thf(fact_1133_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C2 )
= B2 )
=> ( A3
= ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1134_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C2 )
= A3 )
= ( B2
= ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1135_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( A3
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( times_times @ A @ A3 @ C2 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1136_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: nat,N2: nat] :
( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% power_add
thf(fact_1137_nat__power__less__imp__less,axiom,
! [I2: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I2 )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I2 @ M ) @ ( power_power @ nat @ I2 @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1138_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_1139_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_1140_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,W2: A,Z2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_le
thf(fact_1141_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less
thf(fact_1142_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less @ A @ W2 @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_1143_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_1144_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_1145_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1146_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1147_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_1148_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_1149_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_positive
thf(fact_1150_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1151_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N2 ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1152_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1153_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1154_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1155_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1156_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1157_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1158_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1159_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z2 @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1160_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z2 @ Y ) @ X )
=> ( ord_less @ A @ Z2 @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1161_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1162_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1163_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_1164_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_1165_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1166_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1167_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1168_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X: A,Z2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z2 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_1169_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_1170_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1171_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z2 ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1172_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_1173_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N7: nat,A3: A] :
( ( ord_less @ nat @ N2 @ N7 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ A3 @ N7 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_1174_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N7: nat,A3: A] :
( ( ord_less_eq @ nat @ N2 @ N7 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ A3 @ N7 ) ) ) ) ) ).
% power_increasing
thf(fact_1175_zero__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N2 )
= ( zero_zero @ A ) ) ) ) ).
% zero_power
thf(fact_1176_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_1177_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_1178_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( divide_divide @ nat @ M @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1179_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_1180_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F3: nat > A > A,A3: nat,B2: nat,Acc3: 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 ) @ A3 @ ( product_Pair @ nat @ A @ B2 @ Acc3 ) ) ) )
=> ( ( ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A3 @ B2 @ Acc3 )
= Acc3 ) )
& ( ~ ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A3 @ B2 @ Acc3 )
= ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B2 @ ( F3 @ A3 @ Acc3 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_1181_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N2 ) ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ).
% power_Suc_less
thf(fact_1182_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1183_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ Y ) @ X )
=> ( ord_less_eq @ A @ Z2 @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1184_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z2 @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1185_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1186_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1187_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1188_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1189_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1190_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1191_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1192_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_1193_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_1194_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N7: nat,A3: A] :
( ( ord_less @ nat @ N2 @ N7 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N7 ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1195_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,N7: nat,A3: A] :
( ( ord_less_eq @ nat @ N2 @ N7 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N7 ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1196_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,A3: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A3 @ N2 )
= ( power_power @ A @ B2 @ N2 ) )
= ( A3 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1197_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat,B2: A] :
( ( ( power_power @ A @ A3 @ N2 )
= ( power_power @ A @ B2 @ N2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( A3 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1198_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N2 ) )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_1199_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ).
% self_le_power
thf(fact_1200_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ).
% one_less_power
thf(fact_1201_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_1202_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_1203_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_1204_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_1205_div__mult__self1__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N2 @ M ) @ N2 )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1206_div__mult__self__is__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N2 ) @ N2 )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1207_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_1208_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_1209_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1210_div__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( divide_divide @ nat @ M @ N2 )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_1211_div__le__mono,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_1212_div__le__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N2 ) @ M ) ).
% div_le_dividend
thf(fact_1213_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N2: nat] :
( ( ( divide_divide @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M @ N2 )
| ( N2
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1214_less__mult__imp__div__less,axiom,
! [M: nat,I2: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( times_times @ nat @ I2 @ N2 ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N2 ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_1215_times__div__less__eq__dividend,axiom,
! [N2: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1216_div__times__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1217_div__greater__zero__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N2 ) )
= ( ( ord_less_eq @ nat @ N2 @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1218_div__le__mono2,axiom,
! [M: nat,N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N2 ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1219_div__less__iff__less__mult,axiom,
! [Q4: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q4 ) @ N2 )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1220_div__eq__dividend__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ( divide_divide @ nat @ M @ N2 )
= M )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_1221_div__less__dividend,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N2 ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1222_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N2 @ Q4 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q4 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1223_split__div,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide @ nat @ M @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N2 )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I4 ) @ J3 ) )
=> ( P @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_1224_dividend__less__div__times,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N2 ) @ N2 ) ) ) ) ).
% dividend_less_div_times
thf(fact_1225_dividend__less__times__div,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ N2 @ ( divide_divide @ nat @ M @ N2 ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1226_verit__le__mono__div,axiom,
! [A5: nat,B4: nat,N2: nat] :
( ( ord_less @ nat @ A5 @ B4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A5 @ N2 )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B4 @ N2 )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B4 @ N2 ) ) ) ) ).
% verit_le_mono_div
thf(fact_1227_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1228_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,C2: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_1229_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,A3: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_1230_split__div_H,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( divide_divide @ nat @ M @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q7: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q7 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q7 ) ) )
& ( P @ Q7 ) ) ) ) ).
% split_div'
thf(fact_1231_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N2: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ A3 )
= ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% power_minus_mult
thf(fact_1232_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P6: A,M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1233_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,N2: nat,M: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N2 ) )
= ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1234_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1235_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A8: A > B,B7: A > B,X4: A] : ( minus_minus @ B @ ( A8 @ X4 ) @ ( B7 @ X4 ) ) ) ) ) ).
% minus_apply
thf(fact_1236_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1237_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_1238_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat,N2: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( M = N2 ) ) ) ).
% of_nat_eq_iff
thf(fact_1239_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N2: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less_eq @ num @ M @ N2 ) ) ) ).
% numeral_le_iff
thf(fact_1240_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N2: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less @ num @ M @ N2 ) ) ) ).
% numeral_less_iff
thf(fact_1241_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_1242_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M: num,N2: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ).
% numeral_times_numeral
thf(fact_1243_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N2: num] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N2 ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N2 ) ) ) ) ) ).
% power_add_numeral
thf(fact_1244_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N2: num,B2: A] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N2 ) ) @ B2 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N2 ) ) ) @ B2 ) ) ) ).
% power_add_numeral2
thf(fact_1245_lessI,axiom,
! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1246_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1247_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_1248_Suc__le__mono,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N2 @ M ) ) ).
% Suc_le_mono
thf(fact_1249_add__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( plus_plus @ nat @ M @ ( suc @ N2 ) )
= ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1250_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_1251_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_1252_Suc__diff__diff,axiom,
! [M: nat,N2: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1253_diff__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( minus_minus @ nat @ M @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1254_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq @ nat @ I2 @ N2 )
=> ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1255_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K )
= ( minus_minus @ nat @ I2 @ ( plus_plus @ nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1256_mod__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( modulo_modulo @ nat @ M @ N2 )
= M ) ) ).
% mod_less
thf(fact_1257_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% zero_comp_diff_simps(1)
thf(fact_1258_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_1259_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% zero_comp_diff_simps(2)
thf(fact_1260_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_1261_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
= A3 ) ) ) ).
% le_add_diff_inverse
thf(fact_1262_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% le_add_diff_inverse2
thf(fact_1263_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V2: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ).
% distrib_left_numeral
thf(fact_1264_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A3: A,B2: A,V2: num] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_1265_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V2: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1266_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A3: A,B2: A,V2: num] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1267_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A3 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_1268_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_1269_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_1270_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( ( zero_zero @ nat )
= N2 ) ) ) ).
% of_nat_0_eq_iff
thf(fact_1271_of__nat__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_0
thf(fact_1272_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_1273_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_1274_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_1275_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_1276_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ) ).
% of_nat_less_iff
thf(fact_1277_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% of_nat_le_iff
thf(fact_1278_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1279_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_1280_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N2: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% of_nat_add
thf(fact_1281_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_1282_mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times @ nat @ M @ N2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1283_one__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M @ N2 ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1284_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N2: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% of_nat_mult
thf(fact_1285_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_1286_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1287_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( semiring_1_of_nat @ A @ N2 )
= ( one_one @ A ) )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_1288_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_1289_of__nat__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% of_nat_1
thf(fact_1290_mult__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( times_times @ nat @ M @ ( suc @ N2 ) )
= ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_1291_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1292_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
= ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1293_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1294_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( suc @ N2 ) @ ( one_one @ nat ) )
= N2 ) ).
% diff_Suc_1
thf(fact_1295_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1296_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1297_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1298_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1299_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1300_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1301_of__nat__le__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_le_0_iff
thf(fact_1302_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) ) ) ).
% of_nat_Suc
thf(fact_1303_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1304_one__le__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_1305_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ I2 @ ( suc @ ( minus_minus @ nat @ J @ K ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1306_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I2 )
= ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1307_Suc__diff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ M ) )
= ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_diff
thf(fact_1308_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% of_nat_0_less_iff
thf(fact_1309_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1310_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1311_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1312_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1313_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1314_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I2: num,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N2 ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N2 ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_1315_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: num,N2: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N2 ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_1316_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I2: num,N2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N2 ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N2 ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_1317_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: num,N2: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N2 ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_1318_mod__induct,axiom,
! [P: nat > $o,N2: nat,P3: nat,M: nat] :
( ( P @ N2 )
=> ( ( ord_less @ nat @ N2 @ P3 )
=> ( ( ord_less @ nat @ M @ P3 )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ P3 )
=> ( ( P @ N5 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N5 ) @ P3 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_1319_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M3: nat,N: nat] : ( if @ nat @ ( ord_less @ nat @ M3 @ N ) @ M3 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M3 @ N ) @ N ) ) ) ) ).
% mod_if
thf(fact_1320_mod__Suc__le__divisor,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N2 ) ) @ N2 ) ).
% mod_Suc_le_divisor
thf(fact_1321_le__mod__geq,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( modulo_modulo @ nat @ M @ N2 )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ).
% le_mod_geq
thf(fact_1322_Suc__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ N2 @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N2 ) ) )
= ( minus_minus @ nat @ M @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1323_diff__less__Suc,axiom,
! [M: nat,N2: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1324_Suc__diff__le,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N2 )
= ( suc @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1325_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A8: A > B,B7: A > B,X4: A] : ( minus_minus @ B @ ( A8 @ X4 ) @ ( B7 @ X4 ) ) ) ) ) ).
% fun_diff_def
thf(fact_1326_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N5: nat] :
( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) )
=> ( P @ ( minus_minus @ nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1327_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1328_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_1329_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1330_of__nat__diff,axiom,
! [A: $tType] :
( ( semiring_1_cancel @ A )
=> ! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N2 ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% of_nat_diff
thf(fact_1331_mod__geq,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less @ nat @ M @ N2 )
=> ( ( modulo_modulo @ nat @ M @ N2 )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ).
% mod_geq
thf(fact_1332_Suc__to__right,axiom,
! [N2: nat,M: nat] :
( ( ( suc @ N2 )
= M )
=> ( N2
= ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_to_right
thf(fact_1333_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N2 ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1334_of__nat__neq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ N2 ) )
!= ( zero_zero @ A ) ) ) ).
% of_nat_neq_0
thf(fact_1335_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_1336_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_1337_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_1338_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_1339_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( divide_divide @ int @ ( power_power @ int @ K @ M ) @ K )
= ( power_power @ int @ K @ ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1340_diff__Suc__less,axiom,
! [N2: nat,I2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ I2 ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1341_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_eq
thf(fact_1342_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,A4: A,B2: A,B3: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ A4 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B3 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B3 ) @ C2 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_1343_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_mult2
thf(fact_1344_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( times_times @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% mult_mod_right
thf(fact_1345_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_left_eq
thf(fact_1346_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_right_eq
thf(fact_1347_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ D3 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_1348_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_1349_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_1350_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
= ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1351_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% mult_of_nat_commute
thf(fact_1352_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ D3 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_1353_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
= ( ord_less @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_1354_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_1355_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_1356_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_1357_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A3 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) ) ) ) ).
% left_diff_distrib'
thf(fact_1358_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_1359_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_1360_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ! [X7: A,K4: A] :
( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K4 @ D4 ) ) )
& ( Q @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1361_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ! [X7: A,K4: A] :
( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K4 @ D4 ) ) )
| ( Q @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1362_mod__less__eq__dividend,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ M ) ).
% mod_less_eq_dividend
thf(fact_1363_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A7: nat,B6: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B6 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1364_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A7: nat,B6: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B6 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1365_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N2 @ M )
= ( zero_zero @ nat ) )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1366_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_1367_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1368_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1369_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_1370_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1371_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1372_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_1373_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N5: nat] :
( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_1374_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_1375_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N5: nat] :
( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_1376_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1377_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_1378_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1379_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_1380_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_1381_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1382_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N2 )
=> ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_1383_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1384_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less @ nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_1385_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less @ nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_1386_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_1387_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ N2 )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1388_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
= ( ( ord_less @ nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_1389_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less @ nat @ M @ N2 ) )
= ( ord_less @ nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1390_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N2 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1391_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N2 ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less @ nat @ N2 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1392_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_1393_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_1394_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1395_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1396_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1397_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1398_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1399_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1400_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1401_diff__le__mono,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1402_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_1403_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_eq @ nat @ A3 @ C2 )
=> ( ( ord_less_eq @ nat @ B2 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A3 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_1404_diff__le__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1405_diff__add__inverse2,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ N2 )
= M ) ).
% diff_add_inverse2
thf(fact_1406_diff__add__inverse,axiom,
! [N2: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ N2 )
= M ) ).
% diff_add_inverse
thf(fact_1407_diff__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) )
= ( minus_minus @ nat @ M @ N2 ) ) ).
% diff_cancel2
thf(fact_1408_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( minus_minus @ nat @ M @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1409_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_1410_le__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N2 )
=> ( M
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1411_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1412_Suc__le__D,axiom,
! [N2: nat,M8: nat] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ M8 )
=> ? [M5: nat] :
( M8
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1413_le__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_eq @ nat @ M @ N2 )
| ( M
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1414_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1415_not__less__eq__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N2 ) )
= ( ord_less_eq @ nat @ ( suc @ N2 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1416_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ! [M4: nat] :
( ( ord_less_eq @ nat @ ( suc @ M4 ) @ N5 )
=> ( P @ M4 ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1417_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( P @ M )
=> ( ! [N5: nat] :
( ( ord_less_eq @ nat @ M @ N5 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1418_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ! [N5: nat] : ( R2 @ N5 @ ( suc @ N5 ) )
=> ( R2 @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1419_add__Suc__shift,axiom,
! [M: nat,N2: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N2 )
= ( plus_plus @ nat @ M @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1420_add__Suc,axiom,
! [M: nat,N2: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N2 )
= ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ).
% add_Suc
thf(fact_1421_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A3: nat] :
( ( A5
= ( plus_plus @ nat @ K @ A3 ) )
=> ( ( suc @ A5 )
= ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1422_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1423_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1424_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M )
= ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( M = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_1425_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,C2: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_1426_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,A3: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_1427_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M: nat,N2: nat] :
( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_1428_nz__le__conv__less,axiom,
! [K: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ K @ ( suc @ ( zero_zero @ nat ) ) ) @ M ) ) ) ).
% nz_le_conv_less
thf(fact_1429_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( N2
= ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_1430_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N2 )
= ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1431_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M3: nat,N: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ N
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1432_Suc__n__minus__m__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ M ) )
= ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_n_minus_m_eq
thf(fact_1433_div__geq,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( ord_less @ nat @ M @ N2 )
=> ( ( divide_divide @ nat @ M @ N2 )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% div_geq
thf(fact_1434_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M3: nat,N: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M3 @ N )
| ( N
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M3 @ N ) @ N ) ) ) ) ) ).
% div_if
thf(fact_1435_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A7: nat,B6: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B6 ) ) ) ) ).
% nat_less_as_int
thf(fact_1436_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A7: nat,B6: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B6 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1437_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ A3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1438_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1439_Suc__times__mod__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N2 ) ) @ M )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_1440_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% of_nat_0_le_iff
thf(fact_1441_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_1442_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
=> ~ ! [D2: A] :
( B2
!= ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% mod_eqE
thf(fact_1443_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B6: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A7 @ B6 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1444_le__div__geq,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( divide_divide @ nat @ M @ N2 )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% le_div_geq
thf(fact_1445_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A,B6: A] : ( ord_less @ A @ ( minus_minus @ A @ A7 @ B6 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1446_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: A,K: A,N2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ N2 )
=> ( ord_less_eq @ A @ I2 @ ( minus_minus @ A @ N2 @ K ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_1447_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: A,K: A,N2: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ N2 )
=> ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K ) @ N2 )
=> ( ( ord_less_eq @ A @ N2 @ ( plus_plus @ A @ J @ K ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N2 @ K ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1448_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_1449_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_1450_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1451_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% le_add_diff
thf(fact_1452_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1453_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1454_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1455_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1456_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1457_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1458_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1459_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A3 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A3 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1460_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_1461_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ) ).
% of_nat_less_imp_less
thf(fact_1462_of__nat__mono,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [I2: nat,J: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% of_nat_mono
thf(fact_1463_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( minus_minus @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% diff_shunt_var
thf(fact_1464_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A] :
( ~ ( ord_less @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
= A3 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1465_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( ord_less @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_1466_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_1467_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M: nat,N2: nat] :
( ( divide_divide @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N2 ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% div_mult2_eq'
thf(fact_1468_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E3 ) @ C2 )
= D3 ) ) ) ).
% eq_add_iff1
thf(fact_1469_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E3 ) @ D3 ) ) ) ) ).
% eq_add_iff2
thf(fact_1470_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ A @ X @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_1471_mod__less__divisor,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ N2 ) ) ).
% mod_less_divisor
thf(fact_1472_zero__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% zero_le_numeral
thf(fact_1473_not__numeral__le__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_le_zero
thf(fact_1474_zero__less__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% zero_less_numeral
thf(fact_1475_not__numeral__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_less_zero
thf(fact_1476_one__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% one_le_numeral
thf(fact_1477_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_1478_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N2: nat,N6: nat] :
( ! [N5: nat] : ( ord_less @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
=> ( ( ord_less @ nat @ N2 @ N6 )
=> ( ord_less @ A @ ( F3 @ N2 ) @ ( F3 @ N6 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1479_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N2: nat,M: nat] :
( ! [N5: nat] : ( ord_less @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
=> ( ( ord_less @ A @ ( F3 @ N2 ) @ ( F3 @ M ) )
= ( ord_less @ nat @ N2 @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1480_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N2: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
=> ( ( ord_less_eq @ nat @ N2 @ N6 )
=> ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( F3 @ N6 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1481_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N2: nat,N6: nat] :
( ! [N5: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N5 ) ) @ ( F3 @ N5 ) )
=> ( ( ord_less_eq @ nat @ N2 @ N6 )
=> ( ord_less_eq @ A @ ( F3 @ N6 ) @ ( F3 @ N2 ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1482_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_1483_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N2: nat] :
( ( power_power @ A @ A3 @ ( suc @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ N2 ) @ A3 ) ) ) ).
% power_Suc2
thf(fact_1484_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A,N2: nat] :
( ( power_power @ A @ A3 @ ( suc @ N2 ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% power_Suc
thf(fact_1485_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1486_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
= ( ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1487_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1488_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1489_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1490_diff__add__0,axiom,
! [N2: nat,M: nat] :
( ( minus_minus @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_1491_less__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( ord_less @ nat @ M @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1492_diff__less__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ C2 @ A3 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1493_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1494_add__diff__inverse__nat,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less @ nat @ M @ N2 )
=> ( ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ M @ N2 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1495_nat__compl__induct_H,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N5: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N5 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N5 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_compl_induct'
thf(fact_1496_nat__compl__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N5: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N5 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N5 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_compl_induct
thf(fact_1497_add__is__1,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus @ nat @ M @ N2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_1498_one__is__add,axiom,
! [M: nat,N2: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M @ N2 ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_1499_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
= ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1500_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less_eq @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1501_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
= ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1502_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K )
= ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1503_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ( minus_minus @ nat @ J @ I2 )
= K )
= ( J
= ( plus_plus @ nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1504_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_1505_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
= ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_1506_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N5: nat] :
( ( ord_less_eq @ nat @ I2 @ N5 )
=> ( ( ord_less @ nat @ N5 @ J )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1507_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N5: nat] :
( ( ord_less_eq @ nat @ I2 @ N5 )
=> ( ( ord_less @ nat @ N5 @ J )
=> ( ( P @ ( suc @ N5 ) )
=> ( P @ N5 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1508_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
=> ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_1509_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1510_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_1511_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N: nat] : ( ord_less_eq @ nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1512_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_1513_nat__in__between__eq_I2_J,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B2 )
& ( ord_less @ nat @ B2 @ ( suc @ A3 ) ) )
= ( B2 = A3 ) ) ).
% nat_in_between_eq(2)
thf(fact_1514_nat__in__between__eq_I1_J,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less @ nat @ A3 @ B2 )
& ( ord_less_eq @ nat @ B2 @ ( suc @ A3 ) ) )
= ( B2
= ( suc @ A3 ) ) ) ).
% nat_in_between_eq(1)
thf(fact_1515_less__natE,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ~ ! [Q6: nat] :
( N2
!= ( suc @ ( plus_plus @ nat @ M @ Q6 ) ) ) ) ).
% less_natE
thf(fact_1516_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1517_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1518_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N: nat] :
? [K3: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1519_less__imp__Suc__add,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ? [K2: nat] :
( N2
= ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1520_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1521_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_1522_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1523_mult__Suc,axiom,
! [M: nat,N2: nat] :
( ( times_times @ nat @ ( suc @ M ) @ N2 )
= ( plus_plus @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ).
% mult_Suc
thf(fact_1524_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_1525_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1526_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_1527_Suc__div__le__mono,axiom,
! [M: nat,N2: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N2 ) @ ( divide_divide @ nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_div_le_mono
thf(fact_1528_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= A3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1529_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1530_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1531_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1532_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( A3
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_1533_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% div_mult_mod_eq
thf(fact_1534_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= A3 ) ) ).
% mod_div_mult_eq
thf(fact_1535_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= A3 ) ) ).
% mod_mult_div_eq
thf(fact_1536_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A3: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% mult_div_mod_eq
thf(fact_1537_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_1538_le__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E3 ) @ C2 ) @ D3 ) ) ) ).
% le_add_iff1
thf(fact_1539_le__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E3 ) @ D3 ) ) ) ) ).
% le_add_iff2
thf(fact_1540_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E3 ) @ C2 ) @ D3 ) ) ) ).
% less_add_iff1
thf(fact_1541_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E3: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E3 ) @ D3 ) ) ) ) ).
% less_add_iff2
thf(fact_1542_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_1543_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1544_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1545_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1546_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z2 ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1547_mod__le__divisor,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N2 ) @ N2 ) ) ).
% mod_le_divisor
thf(fact_1548_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat,B2: A] :
( ( ( power_power @ A @ A3 @ ( suc @ N2 ) )
= ( power_power @ A @ B2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( A3 = B2 ) ) ) ) ) ).
% power_inject_base
thf(fact_1549_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat,B2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N2 ) ) @ ( power_power @ A @ B2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_1550_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1551_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1552_mod__eq__nat1E,axiom,
! [M: nat,Q4: nat,N2: nat] :
( ( ( modulo_modulo @ nat @ M @ Q4 )
= ( modulo_modulo @ nat @ N2 @ Q4 ) )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ~ ! [S7: nat] :
( M
!= ( plus_plus @ nat @ N2 @ ( times_times @ nat @ Q4 @ S7 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1553_mod__eq__nat2E,axiom,
! [M: nat,Q4: nat,N2: nat] :
( ( ( modulo_modulo @ nat @ M @ Q4 )
= ( modulo_modulo @ nat @ N2 @ Q4 ) )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ~ ! [S7: nat] :
( N2
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q4 @ S7 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1554_nat__mod__eq__lemma,axiom,
! [X: nat,N2: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N2 )
= ( modulo_modulo @ nat @ Y @ N2 ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
=> ? [Q6: nat] :
( X
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N2 @ Q6 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_1555_div__less__mono,axiom,
! [A5: nat,B4: nat,N2: nat] :
( ( ord_less @ nat @ A5 @ B4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ( modulo_modulo @ nat @ A5 @ N2 )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B4 @ N2 )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A5 @ N2 ) @ ( divide_divide @ nat @ B4 @ N2 ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1556_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N2 ) ) ) ) ) ).
% power_gt1
thf(fact_1557_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ( ( ord_less @ nat @ A3 @ B2 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D6: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D6 ) )
=> ( P @ D6 ) ) ) ) ).
% nat_diff_split
thf(fact_1558_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A3 @ B2 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D6: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D6 ) )
& ~ ( P @ D6 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1559_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I2 )
= ( ord_less @ nat @ J @ ( plus_plus @ nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1560_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N2 )
& ! [I: nat] :
( ( ord_less_eq @ nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1561_n__less__n__mult__m,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N2 @ ( times_times @ nat @ N2 @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1562_n__less__m__mult__n,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N2 @ ( times_times @ nat @ M @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1563_one__less__mult,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_1564_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1565_nat__eq__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1566_nat__eq__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1567_nat__le__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1568_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1569_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1570_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1571_power__gt__expt,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ord_less @ nat @ K @ ( power_power @ nat @ N2 @ K ) ) ) ).
% power_gt_expt
thf(fact_1572_nat__one__le__power,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I2 )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I2 @ N2 ) ) ) ).
% nat_one_le_power
thf(fact_1573_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1574_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1575_split__mod,axiom,
! [P: nat > $o,M: nat,N2: nat] :
( ( P @ ( modulo_modulo @ nat @ M @ N2 ) )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ( P @ M ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N2 )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N2 @ I4 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_1576_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1577_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1578_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N2 ) ) @ A3 ) ) ) ) ).
% power_Suc_le_self
thf(fact_1579_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_1580_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,N2: nat,M: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N2 ) )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_diff
thf(fact_1581_nat__less__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1582_nat__less__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N2 ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1583_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M3: nat,N: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) @ N ) ) ) ) ) ).
% mult_eq_if
thf(fact_1584_div__nat__eqI,axiom,
! [N2: nat,Q4: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N2 @ Q4 ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N2 @ ( suc @ Q4 ) ) )
=> ( ( divide_divide @ nat @ M @ N2 )
= Q4 ) ) ) ).
% div_nat_eqI
thf(fact_1585_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_1586_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V2: A,R3: A,S2: A] :
( ( ord_less_eq @ A @ U @ V2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R3 )
=> ( ( ord_less_eq @ A @ R3 @ S2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R3 @ ( minus_minus @ A @ V2 @ U ) ) @ S2 ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_1587_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1588_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_1589_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N5: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N5 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1590_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I4: A] : ( plus_plus @ A @ I4 @ ( one_one @ A ) )
@ N
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_1591_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [M5: nat] : ( P @ M5 @ ( zero_zero @ nat ) )
=> ( ! [M5: nat,N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( P @ N5 @ ( modulo_modulo @ nat @ M5 @ N5 ) )
=> ( P @ M5 @ N5 ) ) )
=> ( P @ M @ N2 ) ) ) ).
% gcd_nat_induct
thf(fact_1592_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X2: A] :
( ( size_option @ A @ X @ ( some @ A @ X2 ) )
= ( plus_plus @ nat @ ( X @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_1593_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A7: A,N: nat] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_1594_option_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_option @ A @ X @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_1595_Diff__cancel,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_1596_empty__Diff,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_1597_Diff__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Diff_empty
thf(fact_1598_Un__Diff__cancel2,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) @ A5 )
= ( sup_sup @ ( set @ A ) @ B4 @ A5 ) ) ).
% Un_Diff_cancel2
thf(fact_1599_Un__Diff__cancel,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_1600_Diff__eq__empty__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_1601_Diff__disjoint,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_1602_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_1603_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_1604_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_1605_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_1606_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_1607_verit__le__mono__div__int,axiom,
! [A5: int,B4: int,N2: int] :
( ( ord_less @ int @ A5 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A5 @ N2 )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B4 @ N2 )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B4 @ N2 ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1608_zdiv__mono__strict,axiom,
! [A5: int,B4: int,N2: int] :
( ( ord_less @ int @ A5 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ( modulo_modulo @ int @ A5 @ N2 )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B4 @ N2 )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A5 @ N2 ) @ ( divide_divide @ int @ B4 @ N2 ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1609_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_1610_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N2: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N2 @ K ) @ ( modulo_modulo @ int @ N2 @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_1611_zmod__zmult2__eq,axiom,
! [C2: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
= ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_1612_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_1613_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L ) @ ( zero_zero @ int ) ) ) ).
% neg_mod_sign
thf(fact_1614_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( modulo_modulo @ int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_1615_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less @ int @ L @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_1616_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
= ( plus_plus @ int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_1617_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ M @ K ) @ M ) ) ).
% zmod_le_nonneg_dividend
thf(fact_1618_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_1619_neg__mod__conj,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% neg_mod_conj
thf(fact_1620_pos__mod__conj,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ B2 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ).
% pos_mod_conj
thf(fact_1621_zmod__trivial__iff,axiom,
! [I2: int,K: int] :
( ( ( modulo_modulo @ int @ I2 @ K )
= I2 )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
& ( ord_less @ int @ I2 @ K ) )
| ( ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I2 ) ) ) ) ).
% zmod_trivial_iff
thf(fact_1622_plusinfinity,axiom,
! [D3: int,P8: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int,K2: int] :
( ( P8 @ X3 )
= ( P8 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D3 ) ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less @ int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P8 @ X3 ) ) )
=> ( ? [X_12: int] : ( P8 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1623_minusinfinity,axiom,
! [D3: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D3 ) ) ) )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1624_decr__mult__lemma,axiom,
! [D3: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X7: int] :
( ( P @ X7 )
=> ( P @ ( minus_minus @ int @ X7 @ ( times_times @ int @ K @ D3 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1625_int__mod__pos__eq,axiom,
! [A3: int,B2: int,Q4: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ( modulo_modulo @ int @ A3 @ B2 )
= R3 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_1626_int__mod__neg__eq,axiom,
! [A3: int,B2: int,Q4: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R3 )
=> ( ( modulo_modulo @ int @ A3 @ B2 )
= R3 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_1627_split__zmod,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N2 @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N2 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_1628_q__pos__lemma,axiom,
! [B3: int,Q8: int,R7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q8 ) @ R7 ) )
=> ( ( ord_less @ int @ R7 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q8 ) ) ) ) ).
% q_pos_lemma
thf(fact_1629_zdiv__mono2__lemma,axiom,
! [B2: int,Q4: int,R3: int,B3: int,Q8: int,R7: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R3 )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q8 ) @ R7 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q8 ) @ R7 ) )
=> ( ( ord_less @ int @ R7 @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ Q4 @ Q8 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1630_incr__mult__lemma,axiom,
! [D3: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X7: int] :
( ( P @ X7 )
=> ( P @ ( plus_plus @ int @ X7 @ ( times_times @ int @ K @ D3 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1631_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q4: int,R3: int,B3: int,Q8: int,R7: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R3 )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q8 ) @ R7 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q8 ) @ R7 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ Q8 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1632_unique__quotient__lemma,axiom,
! [B2: int,Q8: int,R7: int,Q4: int,R3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q8 ) @ R7 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R7 )
=> ( ( ord_less @ int @ R7 @ B2 )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ord_less_eq @ int @ Q8 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1633_unique__quotient__lemma__neg,axiom,
! [B2: int,Q8: int,R7: int,Q4: int,R3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q8 ) @ R7 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R3 )
=> ( ( ord_less @ int @ B2 @ R7 )
=> ( ord_less_eq @ int @ Q4 @ Q8 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1634_imp__le__cong,axiom,
! [X: int,X10: int,P: $o,P8: $o] :
( ( X = X10 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X10 )
=> ( P = P8 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X10 )
=> P8 ) ) ) ) ).
% imp_le_cong
thf(fact_1635_conj__le__cong,axiom,
! [X: int,X10: int,P: $o,P8: $o] :
( ( X = X10 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X10 )
=> ( P = P8 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X10 )
& P8 ) ) ) ) ).
% conj_le_cong
thf(fact_1636_verit__la__generic,axiom,
! [A3: int,X: int] :
( ( ord_less_eq @ int @ A3 @ X )
| ( A3 = X )
| ( ord_less_eq @ int @ X @ A3 ) ) ).
% verit_la_generic
thf(fact_1637_split__zdiv,axiom,
! [P: int > $o,N2: int,K: int] :
( ( P @ ( divide_divide @ int @ N2 @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N2
= ( plus_plus @ int @ ( times_times @ int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_1638_int__div__neg__eq,axiom,
! [A3: int,B2: int,Q4: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R3 )
=> ( ( divide_divide @ int @ A3 @ B2 )
= Q4 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1639_int__div__pos__eq,axiom,
! [A3: int,B2: int,Q4: int,R3: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R3 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
=> ( ( ord_less @ int @ R3 @ B2 )
=> ( ( divide_divide @ int @ A3 @ B2 )
= Q4 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1640_zdiv__zmult2__eq,axiom,
! [C2: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( divide_divide @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1641_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ( ord_less_eq @ int @ B2 @ A3 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1642_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1643_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1644_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I2 @ K ) )
= ( ord_less_eq @ int @ K @ I2 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1645_div__nonpos__pos__le0,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1646_div__nonneg__neg__le0,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1647_div__positive__int,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ L @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ).
% div_positive_int
thf(fact_1648_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) )
= ( ( K
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
| ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_1649_zdiv__mono2__neg,axiom,
! [A3: int,B3: int,B2: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B3 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1650_zdiv__mono1__neg,axiom,
! [A3: int,A4: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ A4 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A4 @ B2 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1651_zdiv__eq__0__iff,axiom,
! [I2: int,K: int] :
( ( ( divide_divide @ int @ I2 @ K )
= ( zero_zero @ int ) )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
& ( ord_less @ int @ I2 @ K ) )
| ( ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I2 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1652_zdiv__mono2,axiom,
! [A3: int,B3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A3 @ B3 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1653_zdiv__mono1,axiom,
! [A3: int,A4: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ A4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A4 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1654_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1655_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1656_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1657_div__neg__pos__less0,axiom,
! [A3: int,B2: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_1658_Diff__mono,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,D4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_1659_Diff__subset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ A5 ) ).
% Diff_subset
thf(fact_1660_double__diff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B4 @ ( minus_minus @ ( set @ A ) @ C4 @ A5 ) )
= A5 ) ) ) ).
% double_diff
thf(fact_1661_int__ops_I6_J,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1662_Diff__Int__distrib2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1663_Diff__Int__distrib,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ C4 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C4 @ A5 ) @ ( inf_inf @ ( set @ A ) @ C4 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_1664_Diff__Diff__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_1665_Diff__Int2,axiom,
! [A: $tType,A5: set @ A,C4: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ C4 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_1666_Int__Diff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Int_Diff
thf(fact_1667_set__diff__diff__left,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( minus_minus @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% set_diff_diff_left
thf(fact_1668_Un__Diff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ C4 ) @ ( minus_minus @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Un_Diff
thf(fact_1669_psubset__imp__ex__mem,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1670_subset__minus__empty,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_1671_disjoint__alt__simp2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B4 )
!= A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp2
thf(fact_1672_disjoint__alt__simp1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B4 )
= A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp1
thf(fact_1673_Int__Diff__disjoint,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_1674_Diff__triv,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ B4 )
= A5 ) ) ).
% Diff_triv
thf(fact_1675_Diff__partition,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_1676_Diff__subset__conv,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_1677_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N2 ) ) ) ) ).
% pochhammer_pos
thf(fact_1678_Un__Diff__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= A5 ) ).
% Un_Diff_Int
thf(fact_1679_Int__Diff__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= A5 ) ).
% Int_Diff_Un
thf(fact_1680_Diff__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ ( minus_minus @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Diff_Int
thf(fact_1681_Diff__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ ( minus_minus @ ( set @ A ) @ A5 @ C4 ) ) ) ).
% Diff_Un
thf(fact_1682_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat,N2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ N2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_1683_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N2: nat,M: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N2 )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_1684_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X @ Y )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_1685_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,I2: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I2 )
= I2 ) ) ).
% of_nat_aux.simps(1)
thf(fact_1686_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N2: nat,I2: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N2 ) @ I2 )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N2 @ ( Inc @ I2 ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_1687_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N2 ) ) ) ) ).
% pochhammer_nonneg
thf(fact_1688_disjoint__alt__simp3,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp3
thf(fact_1689_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N2 ) )
= ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N2 ) ) ) ) ).
% pochhammer_rec
thf(fact_1690_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N2 ) )
= ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) ) ) ) ).
% pochhammer_rec'
thf(fact_1691_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N2 ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_1692_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N2: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N2 @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N2 ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_1693_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N2: nat,Z2: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_1694_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N5: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N5 ) ) ) ).
% real_arch_simple
thf(fact_1695_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N5: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N5 ) ) ) ).
% reals_Archimedean2
thf(fact_1696_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ord_less @ int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1697_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1698_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( K
= ( semiring_1_of_nat @ int @ N5 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1699_pos__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N5: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N5 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ).
% pos_int_cases
thf(fact_1700_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K: nat] :
( ( ord_less @ int @ I2 @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I2 ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1701_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N5: nat] :
( K
!= ( semiring_1_of_nat @ int @ N5 ) ) ) ).
% nonneg_int_cases
thf(fact_1702_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N5: nat] :
( K
= ( semiring_1_of_nat @ int @ N5 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1703_zle__int,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% zle_int
thf(fact_1704_Diff__idemp,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ).
% Diff_idemp
thf(fact_1705_Diff__iff,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( ( member @ A @ C2 @ A5 )
& ~ ( member @ A @ C2 @ B4 ) ) ) ).
% Diff_iff
thf(fact_1706_DiffI,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A5 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1707_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A8 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_1708_DiffD2,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
=> ~ ( member @ A @ C2 @ B4 ) ) ).
% DiffD2
thf(fact_1709_DiffD1,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
=> ( member @ A @ C2 @ A5 ) ) ).
% DiffD1
thf(fact_1710_DiffE,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
=> ~ ( ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% DiffE
thf(fact_1711_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A8 )
& ~ ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1712_int__ge__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_1713_int__le__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_1714_int__induct,axiom,
! [P: int > $o,K: int,I2: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_induct
thf(fact_1715_int__gr__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less @ int @ K @ I2 )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_1716_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1717_int__less__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less @ int @ I2 @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_1718_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_1719_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_1720_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_1721_zmult__zless__mono2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less @ int @ I2 @ J )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ K @ I2 ) @ ( times_times @ int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1722_pos__zmult__eq__1__iff,axiom,
! [M: int,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ( times_times @ int @ M @ N2 )
= ( one_one @ int ) )
= ( ( M
= ( one_one @ int ) )
& ( N2
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1723_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z2 )
= ( ord_less @ int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1724_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less @ int @ W2 @ Z2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1725_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W3: int,Z7: int] :
? [N: nat] :
( Z7
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1726_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1727_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1728_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z7: int] :
? [N: nat] :
( Z7
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1729_nat0__intermed__int__val,axiom,
! [N2: nat,F3: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N2 ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N2 )
& ( ( F3 @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1730_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q4: int,R3: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q4 ) @ R3 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R3 )
& ( ord_less @ int @ R3 @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R3 )
& ( ord_less_eq @ int @ R3 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q4
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_1731_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% pochhammer_minus'
thf(fact_1732_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_1733_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y ) )
= ( X = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_1734_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A8: A > B,X4: A] : ( uminus_uminus @ B @ ( A8 @ X4 ) ) ) ) ) ).
% uminus_apply
thf(fact_1735_Compl__anti__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% Compl_anti_mono
thf(fact_1736_Compl__subset__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1737_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_1738_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_1739_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_1740_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_1741_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_1742_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A3 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_1743_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_1744_Compl__disjoint2,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_1745_Compl__disjoint,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_1746_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% abs_mult_self_eq
thf(fact_1747_inter__compl__diff__conv,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ).
% inter_compl_diff_conv
thf(fact_1748_Diff__Compl,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ).
% Diff_Compl
thf(fact_1749_abs__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat] :
( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N2 ) )
= ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% abs_of_nat
thf(fact_1750_Compl__Diff__eq,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ B4 ) ) ).
% Compl_Diff_eq
thf(fact_1751_negative__zle,axiom,
! [N2: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zle
thf(fact_1752_mod__not__dist,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( uminus_uminus @ assn @ P ) @ H )
= ( ( in_range @ H )
& ~ ( rep_assn @ P @ H ) ) ) ).
% mod_not_dist
thf(fact_1753_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_1754_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_1755_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_le_0_iff_le
thf(fact_1756_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_1757_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_0_iff_less
thf(fact_1758_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_1759_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_pos
thf(fact_1760_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_1761_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_1762_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_1763_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_nonneg
thf(fact_1764_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% abs_le_self_iff
thf(fact_1765_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_1766_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_1767_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_right
thf(fact_1768_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_left
thf(fact_1769_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_1770_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_1771_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( inf_inf @ A @ X @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_1772_boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( sup_sup @ A @ X @ Y ) )
= ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_1773_boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( inf_inf @ A @ X @ Y ) )
= ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_1774_negative__zless,axiom,
! [N2: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zless
thf(fact_1775_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_nonpos
thf(fact_1776_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_1777_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_1778_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat,A3: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ A3 ) )
= A3 ) ) ).
% left_minus_one_mult_self
thf(fact_1779_minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) )
= ( one_one @ A ) ) ) ).
% minus_one_mult_self
thf(fact_1780_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_1781_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_1782_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_1783_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N2: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1784_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N2: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1785_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N2: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N2 ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1786_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less @ int @ ( abs_abs @ int @ Z2 ) @ ( one_one @ int ) )
= ( Z2
= ( zero_zero @ int ) ) ) ).
% zabs_less_one_iff
thf(fact_1787_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( ord_less_eq @ num @ N2 @ M ) ) ) ).
% neg_numeral_le_iff
thf(fact_1788_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( ord_less @ num @ N2 @ M ) ) ) ).
% neg_numeral_less_iff
thf(fact_1789_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1790_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1791_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1792_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1793_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1794_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1795_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N2 ) )
= ( ( A3
!= ( zero_zero @ A ) )
| ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_1796_small__lazy_H_Ocases,axiom,
! [X: product_prod @ int @ int] :
~ ! [D2: int,I3: int] :
( X
!= ( product_Pair @ int @ int @ D2 @ I3 ) ) ).
% small_lazy'.cases
thf(fact_1797_exhaustive__int_H_Ocases,axiom,
! [X: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F2: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I3: int] :
( X
!= ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F2 @ ( product_Pair @ int @ int @ D2 @ I3 ) ) ) ).
% exhaustive_int'.cases
thf(fact_1798_full__exhaustive__int_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F2: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I3: int] :
( X
!= ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F2 @ ( product_Pair @ int @ int @ D2 @ I3 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_1799_unique__remainder,axiom,
! [A3: int,B2: int,Q4: int,R3: int,Q8: int,R7: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q8 @ R7 ) )
=> ( R3 = R7 ) ) ) ).
% unique_remainder
thf(fact_1800_unique__quotient,axiom,
! [A3: int,B2: int,Q4: int,R3: int,Q8: int,R7: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q8 @ R7 ) )
=> ( Q4 = Q8 ) ) ) ).
% unique_quotient
thf(fact_1801_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A8: A > B,X4: A] : ( uminus_uminus @ B @ ( A8 @ X4 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_1802_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_minus_self
thf(fact_1803_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_1804_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_1805_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_1806_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less @ A @ A3 @ B2 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_less_iff
thf(fact_1807_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( abs_abs @ A @ A3 )
= B2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
& ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_1808_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( abs_abs @ A @ B2 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ( B2 = A3 )
| ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_1809_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_1810_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A7: A] : ( if @ A @ ( ord_less @ A @ A7 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A7 ) @ A7 ) ) ) ) ).
% abs_if
thf(fact_1811_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A7: A] : ( if @ A @ ( ord_less @ A @ A7 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A7 ) @ A7 ) ) ) ) ).
% abs_if_raw
thf(fact_1812_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_neg
thf(fact_1813_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I4: int] : ( if @ int @ ( ord_less @ int @ I4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1814_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_1815_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_self
thf(fact_1816_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_mult
thf(fact_1817_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_1818_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_1819_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_1820_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_1821_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_1822_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_1823_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_less_iff
thf(fact_1824_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% less_minus_iff
thf(fact_1825_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1826_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_1827_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_1828_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ A3 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_1829_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_1830_minus__assn__def,axiom,
( ( minus_minus @ assn )
= ( ^ [A7: assn,B6: assn] : ( inf_inf @ assn @ A7 @ ( uminus_uminus @ assn @ B6 ) ) ) ) ).
% minus_assn_def
thf(fact_1831_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_1832_eucl__rel__int__by0,axiom,
! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ).
% eucl_rel_int_by0
thf(fact_1833_div__int__unique,axiom,
! [K: int,L: int,Q4: int,R3: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( divide_divide @ int @ K @ L )
= Q4 ) ) ).
% div_int_unique
thf(fact_1834_zminus1__lemma,axiom,
! [A3: int,B2: int,Q4: int,R3: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R3
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q4 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q4 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R3 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_1835_mod__int__unique,axiom,
! [K: int,L: int,Q4: int,R3: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( ( modulo_modulo @ int @ K @ L )
= R3 ) ) ).
% mod_int_unique
thf(fact_1836_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_zero
thf(fact_1837_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_pos
thf(fact_1838_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_1839_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_1840_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_1841_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_1842_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1843_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D3 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D3 ) ) ) ) ) ).
% abs_mult_less
thf(fact_1844_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% neg_numeral_le_numeral
thf(fact_1845_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_1846_le__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% le_minus_one_simps(2)
thf(fact_1847_le__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% le_minus_one_simps(4)
thf(fact_1848_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% neg_numeral_less_numeral
thf(fact_1849_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_1850_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_1851_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_1852_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W2: num,X: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ ( uminus_uminus @ A @ X ) )
= ( times_times @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_1853_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X: A] :
( ( ( times_times @ A @ X @ X )
= ( one_one @ A ) )
= ( ( X
= ( one_one @ A ) )
| ( X
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_1854_diff__eq,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( minus_minus @ A )
= ( ^ [X4: A,Y4: A] : ( inf_inf @ A @ X4 @ ( uminus_uminus @ A @ Y4 ) ) ) ) ) ).
% diff_eq
thf(fact_1855_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( inf_inf @ A @ X @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_1856_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ A3 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_1857_subset__Compl__self__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_1858_Compl__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) ) ) ).
% Compl_Int
thf(fact_1859_Compl__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) ) ) ).
% Compl_Un
thf(fact_1860_not__int__zless__negative,axiom,
! [N2: nat,M: nat] :
~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1861_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E2 ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_1862_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
| ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_1863_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X ) ) ) ) ) ).
% abs_mult_pos
thf(fact_1864_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C2 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1865_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_1866_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R3 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R3 ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R3 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1867_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R3 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1868_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y )
= ( abs_abs @ A @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% abs_div_pos
thf(fact_1869_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N2 ) ) ) ).
% zero_le_power_abs
thf(fact_1870_not__zero__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] :
~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_1871_neg__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_le_zero
thf(fact_1872_le__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% le_minus_one_simps(1)
thf(fact_1873_le__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% le_minus_one_simps(3)
thf(fact_1874_not__zero__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] :
~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1875_neg__numeral__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_less_zero
thf(fact_1876_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_1877_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_1878_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1879_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_le_neg_one
thf(fact_1880_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% neg_numeral_le_neg_one
thf(fact_1881_neg__one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% neg_one_le_numeral
thf(fact_1882_neg__numeral__le__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_le_one
thf(fact_1883_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1884_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1885_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_less_neg_one
thf(fact_1886_neg__one__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M ) ) ) ).
% neg_one_less_numeral
thf(fact_1887_neg__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_less_one
thf(fact_1888_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( ( times_times @ A @ C2 @ B2 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1889_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= C2 )
= ( ( uminus_uminus @ A @ A3 )
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1890_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
= A3 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A3 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1891_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1892_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% inf_shunt
thf(fact_1893_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P3: A,Q4: A,R3: A] :
( ( ord_less_eq @ A @ P3 @ ( sup_sup @ A @ Q4 @ R3 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P3 @ ( uminus_uminus @ A @ Q4 ) ) @ R3 ) ) ) ).
% sup_neg_inf
thf(fact_1894_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y ) ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% shunt2
thf(fact_1895_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z2 ) ) ) ) ).
% shunt1
thf(fact_1896_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N2: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% power_minus
thf(fact_1897_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q4: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q4 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_1898_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ B4 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_1899_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L ) ) @ ( abs_abs @ int @ L ) ) ) ).
% abs_mod_less
thf(fact_1900_int__cases4,axiom,
! [M: int] :
( ! [N5: nat] :
( M
!= ( semiring_1_of_nat @ int @ N5 ) )
=> ~ ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( M
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ) ).
% int_cases4
thf(fact_1901_int__zle__neg,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) )
= ( ( N2
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_1902_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( zero_zero @ int ) ) ).
% negative_zle_0
thf(fact_1903_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N5: nat] :
( K
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1904_eucl__rel__int,axiom,
! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% eucl_rel_int
thf(fact_1905_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_1906_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_1907_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_1908_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_1909_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_1910_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_1911_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_1912_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_1913_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_1914_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1915_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1916_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_1917_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_1918_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_1919_uminus__assn__def,axiom,
( ( uminus_uminus @ assn )
= ( ^ [P4: assn] :
( abs_assn
@ ^ [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ H6 )
& ~ ( rep_assn @ P4 @ H6 ) ) ) ) ) ).
% uminus_assn_def
thf(fact_1920_int__cases3,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ! [N5: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N5 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 ) )
=> ~ ! [N5: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N5 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ).
% int_cases3
thf(fact_1921_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N2: nat,K: nat] :
( ( ord_less @ nat @ N2 @ K )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_1922_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N2: nat,K: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N2 @ K ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_1923_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N2 )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N2 )
& ( A3
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_1924_not__zle__0__negative,axiom,
! [N2: nat] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ).
% not_zle_0_negative
thf(fact_1925_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ K )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_1926_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N5: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N5 ) ) ) ) ) ).
% negD
thf(fact_1927_negative__zless__0,axiom,
! [N2: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) @ ( zero_zero @ int ) ) ).
% negative_zless_0
thf(fact_1928_verit__less__mono__div__int2,axiom,
! [A5: int,B4: int,N2: int] :
( ( ord_less_eq @ int @ A5 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N2 ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B4 @ N2 ) @ ( divide_divide @ int @ A5 @ N2 ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1929_div__eq__minus1,axiom,
! [B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).
% div_eq_minus1
thf(fact_1930_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,N2: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% power_minus'
thf(fact_1931_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_1932_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_1933_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_1934_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_1935_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_1936_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_1937_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_1938_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_1939_neg__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N5: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N5 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ).
% neg_int_cases
thf(fact_1940_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
= ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_1941_zmod__minus1,axiom,
! [B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
= ( minus_minus @ int @ B2 @ ( one_one @ int ) ) ) ) ).
% zmod_minus1
thf(fact_1942_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_1943_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_1944_nat__intermed__int__val,axiom,
! [M: nat,N2: nat,F3: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less @ nat @ I3 @ N2 ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ int @ ( F3 @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N2 ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less_eq @ nat @ I3 @ N2 )
& ( ( F3 @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1945_incr__lemma,axiom,
! [D3: int,Z2: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) ) ) ).
% incr_lemma
thf(fact_1946_decr__lemma,axiom,
! [D3: int,X: int,Z2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_1947_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R3 ) @ K ) )
= ( times_times @ A @ R3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R3 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_1948_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K @ L )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1949_nat__ivt__aux,axiom,
! [N2: nat,F3: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F3 @ N2 ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N2 )
& ( ( F3 @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1950_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_minus
thf(fact_1951_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N2 @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1952_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I3: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I3 @ J2 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I3 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_1953_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_1954_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_1955_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_1956_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_1957_ComplI,axiom,
! [A: $tType,C2: A,A5: set @ A] :
( ~ ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) ) ).
% ComplI
thf(fact_1958_Compl__iff,axiom,
! [A: $tType,C2: A,A5: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( ~ ( member @ A @ C2 @ A5 ) ) ) ).
% Compl_iff
thf(fact_1959_Compl__eq__Compl__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A5 )
= ( uminus_uminus @ ( set @ A ) @ B4 ) )
= ( A5 = B4 ) ) ).
% Compl_eq_Compl_iff
thf(fact_1960_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_1961_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_less_ceiling
thf(fact_1962_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% ceiling_le_numeral
thf(fact_1963_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_1964_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% one_le_ceiling
thf(fact_1965_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X ) ) ) ).
% numeral_less_ceiling
thf(fact_1966_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_1967_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_less_ceiling
thf(fact_1968_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_1969_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% zero_le_ceiling
thf(fact_1970_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_1971_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_1972_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_1973_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_1974_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_1975_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A8: set @ A] :
( collect @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A8 ) ) ) ) ).
% Compl_eq
thf(fact_1976_ComplD,axiom,
! [A: $tType,C2: A,A5: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
=> ~ ( member @ A @ C2 @ A5 ) ) ).
% ComplD
thf(fact_1977_double__complement,axiom,
! [A: $tType,A5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= A5 ) ).
% double_complement
thf(fact_1978_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A8: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A8 ) ) ) ) ) ).
% uminus_set_def
thf(fact_1979_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y ) )
=> ( X != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_1980_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_1981_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% ceiling_less_cancel
thf(fact_1982_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_1983_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% ceiling_add_le
thf(fact_1984_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A3 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A3 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_1985_option_Osize_I4_J,axiom,
! [A: $tType,X2: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X2 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_1986_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N2: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N2 ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N2 ) @ A3 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_1987_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N2 ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N2 ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_1988_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N2: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N2 ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N2 ) @ A3 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_1989_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N2 ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N2 ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_1990_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_1991_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_1992_nat__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N2 ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_1993_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ W2 @ Z2 ) ) ) ).
% of_int_le_iff
thf(fact_1994_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_1995_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_1996_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z2 )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_1997_nat__0__iff,axiom,
! [I2: int] :
( ( ( nat2 @ I2 )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_1998_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_1999_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_2000_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_2001_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% of_int_0_le_iff
thf(fact_2002_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_2003_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_2004_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,Z2: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N2 ) @ Z2 ) ) ) ).
% of_int_numeral_le_iff
thf(fact_2005_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int,N2: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less_eq @ int @ Z2 @ ( numeral_numeral @ int @ N2 ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_2006_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: num,Z2: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N2 ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N2 ) @ Z2 ) ) ) ).
% of_int_numeral_less_iff
thf(fact_2007_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int,N2: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less @ int @ Z2 @ ( numeral_numeral @ int @ N2 ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_2008_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_2009_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% of_int_1_le_iff
thf(fact_2010_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_2011_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_2012_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_le_floor
thf(fact_2013_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_2014_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V2 ) @ X ) ) ) ).
% numeral_le_floor
thf(fact_2015_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% zero_less_floor
thf(fact_2016_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_2017_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_2018_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% floor_less_numeral
thf(fact_2019_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_le_floor
thf(fact_2020_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_2021_of__nat__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ A @ ( nat2 @ Z2 ) )
= ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_nat_nat
thf(fact_2022_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_2023_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_2024_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_2025_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_2026_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_2027_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_2028_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_2029_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_2030_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_2031_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N2 ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_2032_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N2: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N2 ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) @ A3 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_2033_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N2 ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_2034_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N2: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N2 ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) @ A3 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_2035_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N2: nat,A3: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N2 ) @ ( nat2 @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) @ A3 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_2036_nat__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N2: nat] :
( ( ord_less @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N2 ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_2037_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N2: nat,A3: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N2 ) @ ( nat2 @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N2 ) @ A3 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_2038_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X ) ) ).
% of_int_floor_le
thf(fact_2039_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less_eq @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ) ).
% le_floor_iff
thf(fact_2040_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
= ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% floor_less_iff
thf(fact_2041_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_le_of_int
thf(fact_2042_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_less_of_int
thf(fact_2043_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ).
% ex_of_int_less
thf(fact_2044_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: int,Y: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% mult_of_int_commute
thf(fact_2045_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_2046_of__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R3 )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R3 ) ) ) @ R3 ) ) ) ).
% of_nat_floor
thf(fact_2047_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z2 ) ) ) ) ).
% floor_unique
thf(fact_2048_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A3 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_2049_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T6: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T6 ) )
= ( ! [I4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I4 ) @ T6 )
& ( ord_less @ A @ T6 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% floor_split
thf(fact_2050_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_2051_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_2052_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_2053_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_2054_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% floor_less_cancel
thf(fact_2055_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ X @ Y )
=> ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_2056_eq__nat__nat__iff,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z8 ) )
= ( Z2 = Z8 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_2057_all__nat,axiom,
( ( ^ [P5: nat > $o] :
! [X5: nat] : ( P5 @ X5 ) )
= ( ^ [P4: nat > $o] :
! [X4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
=> ( P4 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_2058_ex__nat,axiom,
( ( ^ [P5: nat > $o] :
? [X5: nat] : ( P5 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [X4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
& ( P4 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_2059_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% le_of_int_ceiling
thf(fact_2060_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% floor_le_ceiling
thf(fact_2061_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P3 @ Q4 ) ) ) @ Q4 ) @ P3 ) ) ) ).
% floor_divide_lower
thf(fact_2062_of__int__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K3: int] : ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K3 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K3 ) ) ) ) ) ) ).
% of_int_of_nat
thf(fact_2063_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ P3 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P3 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) ) ) ) ).
% floor_divide_upper
thf(fact_2064_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_2065_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% le_floor_add
thf(fact_2066_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( ( ord_less @ nat @ M @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_2067_nat__le__iff,axiom,
! [X: int,N2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N2 )
= ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_2068_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_2069_int__eq__iff,axiom,
! [M: nat,Z2: int] :
( ( ( semiring_1_of_nat @ int @ M )
= Z2 )
= ( ( M
= ( nat2 @ Z2 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_2070_of__nat__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R3: A] : ( ord_less_eq @ A @ R3 @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R3 ) ) ) ) ) ).
% of_nat_ceiling
thf(fact_2071_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% ceiling_le_iff
thf(fact_2072_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less @ int @ Z2 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ) ).
% less_ceiling_iff
thf(fact_2073_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A7: nat,B6: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B6 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_2074_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A7: nat,B6: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B6 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_2075_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A7: nat,B6: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B6 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_2076_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A7: nat,B6: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B6 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_2077_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A7: nat,B6: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B6 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_2078_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_nonneg
thf(fact_2079_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_2080_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X )
=> ( ( N2
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_2081_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N2 ) ) @ X )
=> ( ( N2
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_2082_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_2083_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
& ! [Y6: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y6 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y6 @ ( one_one @ int ) ) ) ) )
=> ( Y6 = X3 ) ) ) ) ).
% floor_exists1
thf(fact_2084_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_2085_nat__le__eq__zle,axiom,
! [W2: int,Z2: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq @ int @ W2 @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_2086_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_2087_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_2088_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,X: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N2 ) @ X ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_2089_split__nat,axiom,
! [P: nat > $o,I2: int] :
( ( P @ ( nat2 @ I2 ) )
= ( ! [N: nat] :
( ( I2
= ( semiring_1_of_nat @ int @ N ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ I2 @ ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_2090_le__nat__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ nat @ N2 @ ( nat2 @ K ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_2091_nat__add__distrib,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( nat2 @ ( plus_plus @ int @ Z2 @ Z8 ) )
= ( plus_plus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_2092_nat__mult__distrib,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( nat2 @ ( times_times @ int @ Z2 @ Z8 ) )
= ( times_times @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ).
% nat_mult_distrib
thf(fact_2093_nat__diff__distrib,axiom,
! [Z8: int,Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z8 )
=> ( ( ord_less_eq @ int @ Z8 @ Z2 )
=> ( ( nat2 @ ( minus_minus @ int @ Z2 @ Z8 ) )
= ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_2094_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( minus_minus @ int @ X @ Y ) )
= ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_2095_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_2096_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_2097_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_2098_nat__mod__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( modulo_modulo @ int @ X @ Y ) )
= ( modulo_modulo @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_2099_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_2100_nat__power__eq,axiom,
! [Z2: int,N2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( nat2 @ ( power_power @ int @ Z2 @ N2 ) )
= ( power_power @ nat @ ( nat2 @ Z2 ) @ N2 ) ) ) ).
% nat_power_eq
thf(fact_2101_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A3 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_2102_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% ceiling_correct
thf(fact_2103_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z2 ) ) ) ) ).
% ceiling_unique
thf(fact_2104_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archimedean_ceiling @ A @ X )
= A3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_2105_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T6: A] :
( ( P @ ( archimedean_ceiling @ A @ T6 ) )
= ( ! [I4: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) @ T6 )
& ( ord_less_eq @ A @ T6 @ ( ring_1_of_int @ A @ I4 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% ceiling_split
thf(fact_2106_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_2107_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less_eq @ int @ Z2 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_2108_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_2109_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M )
= ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_2110_nat__mult__distrib__neg,axiom,
! [Z2: int,Z8: int] :
( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z2 @ Z8 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z8 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_2111_nat__abs__int__diff,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ B2 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ A3 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_2112_diff__nat__eq__if,axiom,
! [Z8: int,Z2: int] :
( ( ( ord_less @ int @ Z8 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
= ( nat2 @ Z2 ) ) )
& ( ~ ( ord_less @ int @ Z8 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z2 @ Z8 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z2 @ Z8 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_2113_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ P3 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P3 @ Q4 ) ) ) @ Q4 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2114_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P3 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) @ P3 ) ) ) ).
% ceiling_divide_lower
thf(fact_2115_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_2116_diff__size__le__size__Diff,axiom,
! [A: $tType,M6: multiset @ A,M9: multiset @ A] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M6 ) @ ( size_size @ ( multiset @ A ) @ M9 ) ) @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M6 @ M9 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_2117_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A33: product_prod @ int @ int] :
( ? [K3: int] :
( ( A12 = K3 )
& ( A23
= ( zero_zero @ int ) )
& ( A33
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L2: int,K3: int,Q7: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A33
= ( product_Pair @ int @ int @ Q7 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q7 @ L2 ) ) )
| ? [R4: int,L2: int,K3: int,Q7: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A33
= ( product_Pair @ int @ int @ Q7 @ R4 ) )
& ( ( sgn_sgn @ int @ R4 )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R4 ) @ ( abs_abs @ int @ L2 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q7 @ L2 ) @ R4 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_2118_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod @ int @ int] :
( ( eucl_rel_int @ A1 @ A22 @ A32 )
=> ( ( ( A22
= ( zero_zero @ int ) )
=> ( A32
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q6: int] :
( ( A32
= ( product_Pair @ int @ int @ Q6 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q6 @ A22 ) ) ) )
=> ~ ! [R: int,Q6: int] :
( ( A32
= ( product_Pair @ int @ int @ Q6 @ R ) )
=> ( ( ( sgn_sgn @ int @ R )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q6 @ A22 ) @ R ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_2119_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_2120_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A3 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A3 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_2121_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_2122_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_greater
thf(fact_2123_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ A3 @ ( sgn_sgn @ A @ B2 ) )
= ( times_times @ A @ A3 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% divide_sgn
thf(fact_2124_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_2125_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) )
= ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_2126_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_2127_union__diff__assoc,axiom,
! [A: $tType,C4: multiset @ A,B4: multiset @ A,A5: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C4 @ B4 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) @ C4 )
= ( plus_plus @ ( multiset @ A ) @ A5 @ ( minus_minus @ ( multiset @ A ) @ B4 @ C4 ) ) ) ) ).
% union_diff_assoc
thf(fact_2128_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_mult
thf(fact_2129_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_2130_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K3: A] : ( times_times @ A @ K3 @ ( sgn_sgn @ A @ K3 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_2131_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= A3 ) ) ).
% abs_mult_sgn
thf(fact_2132_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= A3 ) ) ).
% sgn_mult_abs
thf(fact_2133_mult__sgn__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( abs_abs @ A @ X ) )
= X ) ) ).
% mult_sgn_abs
thf(fact_2134_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) ) ) ).
% frac_ge_0
thf(fact_2135_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_2136_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: A] :
( ( ( archimedean_frac @ A @ X )
= A3 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_2137_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_1_pos
thf(fact_2138_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I4: int] :
( if @ int
@ ( I4
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I4 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_2139_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_2140_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_2141_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_2142_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_2143_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_2144_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_2145_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= X )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_2146_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_2147_eucl__rel__int__remainderI,axiom,
! [R3: int,L: int,K: int,Q4: int] :
( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ L ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q4 @ L ) @ R3 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R3 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_2148_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A3 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A3 @ B2 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_2149_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_2150_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_2151_pochhammer__same,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ N2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% pochhammer_same
thf(fact_2152_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( semiring_char_0_fact @ A @ N2 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_2153_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M3 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_2154_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_2155_cpmi,axiom,
! [D4: int,P: int > $o,P8: int > $o,B4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z6 )
=> ( ( P @ X3 )
= ( P8 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B4 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P8 @ X3 )
= ( P8 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D4 ) ) ) )
=> ( ( ? [X11: int] : ( P @ X11 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P8 @ X4 ) )
| ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y4: int] :
( ( member @ int @ Y4 @ B4 )
& ( P @ ( plus_plus @ int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_2156_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H: A,L3: A,H3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H )
= ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H = H3 ) )
| ( ~ ( ord_less_eq @ A @ L @ H )
& ~ ( ord_less_eq @ A @ L3 @ H3 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_2157_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I2 )
& ( ord_less_eq @ A @ I2 @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_2158_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_2159_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_2160_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_2161_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_2162_slice__complete,axiom,
! [A: $tType,Xs: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs )
= Xs ) ).
% slice_complete
thf(fact_2163_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% fact_Suc
thf(fact_2164_union__le__mono1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B4: multiset @ A,D4: multiset @ A,C4: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B4 @ D4 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B4 @ C4 ) @ ( plus_plus @ ( multiset @ A ) @ D4 @ C4 ) ) ) ) ).
% union_le_mono1
thf(fact_2165_union__le__mono2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B4: multiset @ A,D4: multiset @ A,C4: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B4 @ D4 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ B4 ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ D4 ) ) ) ) ).
% union_le_mono2
thf(fact_2166_union__less__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: multiset @ A,C4: multiset @ A,B4: multiset @ A,D4: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ A5 @ C4 )
=> ( ( ord_less @ ( multiset @ A ) @ B4 @ D4 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ D4 ) ) ) ) ) ).
% union_less_mono
thf(fact_2167_mset__distrib,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,M6: multiset @ A,N7: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A5 @ B4 )
= ( plus_plus @ ( multiset @ A ) @ M6 @ N7 ) )
=> ~ ! [Am: multiset @ A,An: multiset @ A] :
( ( A5
= ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
=> ! [Bm: multiset @ A,Bn: multiset @ A] :
( ( B4
= ( plus_plus @ ( multiset @ A ) @ Bm @ Bn ) )
=> ( ( M6
= ( plus_plus @ ( multiset @ A ) @ Am @ Bm ) )
=> ( N7
!= ( plus_plus @ ( multiset @ A ) @ An @ Bn ) ) ) ) ) ) ).
% mset_distrib
thf(fact_2168_fact__mono__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ).
% fact_mono_nat
thf(fact_2169_fact__ge__self,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( semiring_char_0_fact @ nat @ N2 ) ) ).
% fact_ge_self
thf(fact_2170_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_2171_fact__less__mono__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ).
% fact_less_mono_nat
thf(fact_2172_fact__ge__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).
% fact_ge_zero
thf(fact_2173_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ K )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_2174_fact__not__neg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] :
~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% fact_not_neg
thf(fact_2175_fact__gt__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).
% fact_gt_zero
thf(fact_2176_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ).
% fact_ge_1
thf(fact_2177_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% fact_mono
thf(fact_2178_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A7 ) @ K3 ) ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_2179_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_2180_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_2181_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_2182_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,A3: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_2183_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 )
& ( ( ord_less @ A @ C2 @ A3 )
| ( ord_less @ A @ B2 @ D3 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_2184_fact__ge__Suc__0__nat,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N2 ) ) ).
% fact_ge_Suc_0_nat
thf(fact_2185_fact__less__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ) ).
% fact_less_mono
thf(fact_2186_fact__mod,axiom,
! [A: $tType] :
( ( ( linordered_semidom @ A )
& ( semidom_modulo @ A ) )
=> ! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M ) )
= ( zero_zero @ A ) ) ) ) ).
% fact_mod
thf(fact_2187_fact__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N2 @ N2 ) ) ) ) ).
% fact_le_power
thf(fact_2188_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_2189_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_2190_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M: nat,A3: A] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_2191_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K3: nat] :
( if @ A
@ ( K3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ L2 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_2192_fact__div__fact__le__pow,axiom,
! [R3: nat,N2: nat] :
( ( ord_less_eq @ nat @ R3 @ N2 )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ R3 ) ) ) @ ( power_power @ nat @ N2 @ R3 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_2193_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_2194_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_2195_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A7 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_2196_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( one_one @ A ) ) @ K ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N2 ) ) ) ) ).
% gbinomial_index_swap
thf(fact_2197_periodic__finite__ex,axiom,
! [D3: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D3 ) ) ) )
=> ( ( ? [X11: int] : ( P @ X11 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P @ X4 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_2198_aset_I7_J,axiom,
! [D4: int,A5: set @ int,T6: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T6 @ X7 )
=> ( ord_less @ int @ T6 @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ).
% aset(7)
thf(fact_2199_aset_I5_J,axiom,
! [D4: int,T6: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T6 @ A5 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X7 @ T6 )
=> ( ord_less @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T6 ) ) ) ) ) ).
% aset(5)
thf(fact_2200_aset_I4_J,axiom,
! [D4: int,T6: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T6 @ A5 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X7 != T6 )
=> ( ( plus_plus @ int @ X7 @ D4 )
!= T6 ) ) ) ) ) ).
% aset(4)
thf(fact_2201_aset_I3_J,axiom,
! [D4: int,T6: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T6 @ ( one_one @ int ) ) @ A5 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X7 = T6 )
=> ( ( plus_plus @ int @ X7 @ D4 )
= T6 ) ) ) ) ) ).
% aset(3)
thf(fact_2202_bset_I7_J,axiom,
! [D4: int,T6: int,B4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T6 @ B4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T6 @ X7 )
=> ( ord_less @ int @ T6 @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ).
% bset(7)
thf(fact_2203_bset_I5_J,axiom,
! [D4: int,B4: set @ int,T6: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X7 @ T6 )
=> ( ord_less @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T6 ) ) ) ) ).
% bset(5)
thf(fact_2204_bset_I4_J,axiom,
! [D4: int,T6: int,B4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T6 @ B4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X7 != T6 )
=> ( ( minus_minus @ int @ X7 @ D4 )
!= T6 ) ) ) ) ) ).
% bset(4)
thf(fact_2205_bset_I3_J,axiom,
! [D4: int,T6: int,B4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T6 @ ( one_one @ int ) ) @ B4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X7 = T6 )
=> ( ( minus_minus @ int @ X7 @ D4 )
= T6 ) ) ) ) ) ).
% bset(3)
thf(fact_2206_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_2207_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_2208_aset_I8_J,axiom,
! [D4: int,A5: set @ int,T6: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T6 @ X7 )
=> ( ord_less_eq @ int @ T6 @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ).
% aset(8)
thf(fact_2209_aset_I6_J,axiom,
! [D4: int,T6: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T6 @ ( one_one @ int ) ) @ A5 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X7 @ T6 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T6 ) ) ) ) ) ).
% aset(6)
thf(fact_2210_bset_I8_J,axiom,
! [D4: int,T6: int,B4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T6 @ ( one_one @ int ) ) @ B4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T6 @ X7 )
=> ( ord_less_eq @ int @ T6 @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ).
% bset(8)
thf(fact_2211_bset_I6_J,axiom,
! [D4: int,B4: set @ int,T6: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B4 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X7 @ T6 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T6 ) ) ) ) ).
% bset(6)
thf(fact_2212_cppi,axiom,
! [D4: int,P: int > $o,P8: int > $o,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z6: int] :
! [X3: int] :
( ( ord_less @ int @ Z6 @ X3 )
=> ( ( P @ X3 )
= ( P8 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P8 @ X3 )
= ( P8 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D4 ) ) ) )
=> ( ( ? [X11: int] : ( P @ X11 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P8 @ X4 ) )
| ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y4: int] :
( ( member @ int @ Y4 @ A5 )
& ( P @ ( minus_minus @ int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_2213_slice__nth,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A,I2: nat] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ To @ From ) )
=> ( ( nth @ A @ ( slice @ A @ From @ To @ Xs ) @ I2 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ From @ I2 ) ) ) ) ) ) ).
% slice_nth
thf(fact_2214_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_2215_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_2216_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_2217_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_2218_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_2219_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_2220_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,X: B,A5: set @ B] :
( ( B2
= ( F3 @ X ) )
=> ( ( member @ B @ X @ A5 )
=> ( member @ A @ B2 @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ).
% image_eqI
thf(fact_2221_image__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : X4
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_2222_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_2223_empty__is__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image2 @ B @ A @ F3 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_2224_image__is__empty,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( ( image2 @ B @ A @ F3 @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_2225_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_2226_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I2: A,J: A] :
( ( image2 @ A @ A
@ ^ [N: A] : ( plus_plus @ A @ N @ K )
@ ( set_or1337092689740270186AtMost @ A @ I2 @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_2227_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( image2 @ A @ A
@ ^ [T4: A] : ( minus_minus @ A @ T4 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_2228_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F3: B > A,G3: B > A,S: set @ B] :
( ( image2 @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ S )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ S @ ( collect @ B @ P ) ) )
@ ( image2 @ B @ A @ G3
@ ( inf_inf @ ( set @ B ) @ S
@ ( collect @ B
@ ^ [X4: B] :
~ ( P @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_2229_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A3 ) @ ( times_times @ A @ D3 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_2230_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A
@ ^ [C5: A] : ( divide_divide @ A @ C5 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_2231_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,B2: B,F3: A > B] :
( ( member @ A @ X @ A5 )
=> ( ( B2
= ( F3 @ X ) )
=> ( member @ B @ B2 @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_2232_ball__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( P @ X3 ) )
=> ! [X7: B] :
( ( member @ B @ X7 @ A5 )
=> ( P @ ( F3 @ X7 ) ) ) ) ).
% ball_imageD
thf(fact_2233_image__cong,axiom,
! [B: $tType,A: $tType,M6: set @ A,N7: set @ A,F3: A > B,G3: A > B] :
( ( M6 = N7 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N7 )
=> ( ( F3 @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image2 @ A @ B @ F3 @ M6 )
= ( image2 @ A @ B @ G3 @ N7 ) ) ) ) ).
% image_cong
thf(fact_2234_bex__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
( ? [X7: A] :
( ( member @ A @ X7 @ ( image2 @ B @ A @ F3 @ A5 ) )
& ( P @ X7 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( P @ ( F3 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_2235_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F3: B > A,A5: set @ B] :
( ( member @ A @ Z2 @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( Z2
= ( F3 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_2236_imageI,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F3: A > B] :
( ( member @ A @ X @ A5 )
=> ( member @ B @ ( F3 @ X ) @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ).
% imageI
thf(fact_2237_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,A5: set @ B] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ~ ! [X3: B] :
( ( B2
= ( F3 @ X3 ) )
=> ~ ( member @ B @ X3 @ A5 ) ) ) ).
% imageE
thf(fact_2238_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G3: C > B,A5: set @ C] :
( ( image2 @ B @ A @ F3 @ ( image2 @ C @ B @ G3 @ A5 ) )
= ( image2 @ C @ A
@ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
@ A5 ) ) ).
% image_image
thf(fact_2239_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image2 @ B @ A @ F3 @ A5 ) )
& ( P @ X4 ) ) )
= ( image2 @ B @ A @ F3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( P @ ( F3 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_2240_prod__decode__aux_Ocases,axiom,
! [X: product_prod @ nat @ nat] :
~ ! [K2: nat,M5: nat] :
( X
!= ( product_Pair @ nat @ nat @ K2 @ M5 ) ) ).
% prod_decode_aux.cases
thf(fact_2241_mset__le__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M6: multiset @ A,N7: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M6 @ N7 )
=> ~ ( ord_less @ ( multiset @ A ) @ N7 @ M6 ) ) ) ).
% mset_le_asym
thf(fact_2242_mset__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K5: multiset @ A,M6: multiset @ A,N7: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ K5 @ M6 )
=> ( ( ord_less @ ( multiset @ A ) @ M6 @ N7 )
=> ( ord_less @ ( multiset @ A ) @ K5 @ N7 ) ) ) ) ).
% mset_le_trans
thf(fact_2243_mset__le__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M6: multiset @ A] :
~ ( ord_less @ ( multiset @ A ) @ M6 @ M6 ) ) ).
% mset_le_irrefl
thf(fact_2244_mset__le__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M6: multiset @ A,N7: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M6 @ N7 )
=> ~ ( ord_less @ ( multiset @ A ) @ N7 @ M6 ) ) ) ).
% mset_le_not_sym
thf(fact_2245_mset__le__not__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M6: multiset @ A] :
~ ( ord_less @ ( multiset @ A ) @ M6 @ M6 ) ) ).
% mset_le_not_refl
thf(fact_2246_less__eq__multiset__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less_eq @ ( multiset @ A ) )
= ( ^ [M10: multiset @ A,N8: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ M10 @ N8 )
| ( M10 = N8 ) ) ) ) ) ).
% less_eq_multiset_def
thf(fact_2247_subset__image__iff,axiom,
! [A: $tType,B: $tType,B4: set @ A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A5 )
& ( B4
= ( image2 @ B @ A @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_2248_image__subset__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A5 ) @ B4 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( member @ A @ ( F3 @ X4 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_2249_subset__imageE,axiom,
! [A: $tType,B: $tType,B4: set @ A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ~ ! [C6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C6 @ A5 )
=> ( B4
!= ( image2 @ B @ A @ F3 @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_2250_image__subsetI,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: A > B,B4: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ B @ ( F3 @ X3 ) @ B4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ B4 ) ) ).
% image_subsetI
thf(fact_2251_image__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ ( image2 @ A @ B @ F3 @ B4 ) ) ) ).
% image_mono
thf(fact_2252_image__Un,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,B4: set @ B] :
( ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A5 ) @ ( image2 @ B @ A @ F3 @ B4 ) ) ) ).
% image_Un
thf(fact_2253_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: A > B,B4: set @ B] :
( ! [X3: A] :
( ( P @ X3 )
=> ( member @ B @ ( F3 @ X3 ) @ B4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( collect @ A @ P ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_2254_obtain__list__from__elements,axiom,
! [A: $tType,N2: nat,P: A > nat > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ? [Li: A] : ( P @ Li @ I3 ) )
=> ~ ! [L4: list @ A] :
( ( ( size_size @ ( list @ A ) @ L4 )
= N2 )
=> ~ ! [I: nat] :
( ( ord_less @ nat @ I @ N2 )
=> ( P @ ( nth @ A @ L4 @ I ) @ I ) ) ) ) ).
% obtain_list_from_elements
thf(fact_2255_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z4: list @ A] : Y5 = Z4 )
= ( ^ [Xs3: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I4 )
= ( nth @ A @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_2256_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ? [X11: A] : ( P @ I4 @ X11 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ( P @ I4 @ ( nth @ A @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_2257_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys3 @ I3 ) ) )
=> ( Xs = Ys3 ) ) ) ).
% nth_equalityI
thf(fact_2258_image__Int__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,B4: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) @ ( inf_inf @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A5 ) @ ( image2 @ B @ A @ F3 @ B4 ) ) ) ).
% image_Int_subset
thf(fact_2259_image__diff__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,B4: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A5 ) @ ( image2 @ B @ A @ F3 @ B4 ) ) @ ( image2 @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) ) ).
% image_diff_subset
thf(fact_2260_ex__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
& ( P @ M3 ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_2261_all__nat__less,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
=> ( P @ M3 ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_2262_None__notin__image__Some,axiom,
! [A: $tType,A5: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) ) ).
% None_notin_image_Some
thf(fact_2263_nth__rotate1,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_2264_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_2265_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M3: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M3 @ K3 ) @ ( product_Pair @ nat @ nat @ M3 @ ( minus_minus @ nat @ K3 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M3 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_2266_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_2267_product__nth,axiom,
! [A: $tType,B: $tType,N2: nat,Xs: list @ A,Ys3: list @ B] :
( ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys3 ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys3 ) @ N2 )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys3 ) ) ) @ ( nth @ B @ Ys3 @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys3 ) ) ) ) ) ) ).
% product_nth
thf(fact_2268_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T6: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ ( uminus_uminus @ ( set @ A ) @ T6 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ T6 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_2269_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S2: set @ A,T6: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ ( minus_minus @ ( set @ A ) @ S2 @ T6 ) )
= ( minus_minus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ S2 )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ T6 ) ) ) ) ).
% translation_subtract_diff
thf(fact_2270_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S2: set @ A,T6: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ ( inf_inf @ ( set @ A ) @ S2 @ T6 ) )
= ( inf_inf @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ S2 )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ T6 ) ) ) ) ).
% translation_subtract_Int
thf(fact_2271_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N2: nat,M: nat,X: A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_2272_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F3: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( semiring_1_of_nat @ A @ ( F3 @ X4 ) )
@ A5 ) ) ) ).
% of_nat_sum
thf(fact_2273_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F3: B > int,A5: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F3 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( ring_1_of_int @ A @ ( F3 @ X4 ) )
@ A5 ) ) ) ).
% of_int_sum
thf(fact_2274_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N2: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N2 ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G3 @ ( suc @ N2 ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_2275_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F3: B > A,A3: A,A5: set @ B] :
( ( modulo_modulo @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( modulo_modulo @ A @ ( F3 @ I4 ) @ A3 )
@ A5 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ A3 ) ) ) ).
% mod_sum_eq
thf(fact_2276_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_2277_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_2278_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N2: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_2279_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G3 @ ( suc @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_2280_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( G3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_2281_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G3 @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G3 @ M )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_2282_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N2: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( minus_minus @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_2283_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,I5: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I4 ) )
@ I5 )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I5 ) ) ) ) ).
% sum_power_add
thf(fact_2284_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,A3: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A7: nat] : ( plus_plus @ A @ ( F3 @ A7 ) )
@ A3
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_2285_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G3: nat > A,P3: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N2 @ P3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P3 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_2286_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N2: nat,F3: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( minus_minus @ A @ ( F3 @ M ) @ ( F3 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_2287_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N2: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) )
= ( minus_minus @ A @ ( F3 @ N2 ) @ ( F3 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_2288_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N2: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_2289_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_2290_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,N2: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N2 ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_2291_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S2: set @ A,T6: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( inf_inf @ ( set @ A ) @ S2 @ T6 ) )
= ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S2 ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T6 ) ) ) ) ).
% translation_Int
thf(fact_2292_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A5: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F3 @ I4 ) )
@ A5 ) ) ) ).
% sum_abs_ge_zero
thf(fact_2293_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A5: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A5 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F3 @ I4 ) )
@ A5 ) ) ) ).
% sum_abs
thf(fact_2294_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_2295_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I5 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I3 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( times_times @ B @ ( A3 @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_2296_abs__sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A5: set @ A] :
( ( abs_abs @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A7: A] : ( abs_abs @ B @ ( F3 @ A7 ) )
@ A5 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A7: A] : ( abs_abs @ B @ ( F3 @ A7 ) )
@ A5 ) ) ) ).
% abs_sum_abs
thf(fact_2297_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu2: B] : ( zero_zero @ A )
@ A5 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_2298_int__sum,axiom,
! [B: $tType,F3: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X4: B] : ( semiring_1_of_nat @ int @ ( F3 @ X4 ) )
@ A5 ) ) ).
% int_sum
thf(fact_2299_sum__subtractf__nat,axiom,
! [A: $tType,A5: set @ A,G3: A > nat,F3: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ nat @ ( G3 @ X3 ) @ ( F3 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( minus_minus @ nat @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ A5 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G3 @ A5 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_2300_sum__SucD,axiom,
! [A: $tType,F3: A > nat,A5: set @ A,N2: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 )
= ( suc @ N2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_2301_sum_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > C > A,B4: set @ C,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G3 @ I4 ) @ B4 )
@ A5 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [J3: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [I4: B] : ( G3 @ I4 @ J3 )
@ A5 )
@ B4 ) ) ) ).
% sum.swap
thf(fact_2302_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K5: set @ B,F3: B > A,G3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ K5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ K5 ) ) ) ) ).
% sum_mono
thf(fact_2303_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,H: B > A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( G3 @ X4 ) @ ( H @ X4 ) )
@ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H @ A5 ) ) ) ) ).
% sum.distrib
thf(fact_2304_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F3: A > B,A5: set @ A,G3: C > B,B4: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G3 @ B4 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F3 @ I4 ) @ ( G3 @ J3 ) )
@ B4 )
@ A5 ) ) ) ).
% sum_product
thf(fact_2305_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F3: B > A,A5: set @ B,R3: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ R3 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N: B] : ( times_times @ A @ ( F3 @ N ) @ R3 )
@ A5 ) ) ) ).
% sum_distrib_right
thf(fact_2306_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R3: A,F3: B > A,A5: set @ B] :
( ( times_times @ A @ R3 @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N: B] : ( times_times @ A @ R3 @ ( F3 @ N ) )
@ A5 ) ) ) ).
% sum_distrib_left
thf(fact_2307_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F3: B > A,G3: B > A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ A5 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ).
% sum_subtractf
thf(fact_2308_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F3: B > A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( uminus_uminus @ A @ ( F3 @ X4 ) )
@ A5 )
= ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) ) ) ) ).
% sum_negf
thf(fact_2309_sum__divide__distrib,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F3: B > A,A5: set @ B,R3: A] :
( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ R3 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N: B] : ( divide_divide @ A @ ( F3 @ N ) @ R3 )
@ A5 ) ) ) ).
% sum_divide_distrib
thf(fact_2310_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) ) ) ) ).
% sum_nonneg
thf(fact_2311_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_2312_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A3 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_2313_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N2 ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_2314_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M3: nat,N: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M3 @ N ) @ ( modulo_modulo @ nat @ M3 @ N ) ) ) ) ).
% divmod_nat_def
thf(fact_2315_sorted__in__between,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I2: nat,J: nat,L: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ I2 )
=> ( ( ord_less @ nat @ I2 @ 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 @ I2 ) @ X )
=> ( ( ord_less @ A @ X @ ( nth @ A @ L @ J ) )
=> ~ ! [K2: nat] :
( ( ord_less_eq @ nat @ I2 @ K2 )
=> ( ( ord_less @ nat @ K2 @ J )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ K2 ) @ X )
=> ~ ( ord_less @ A @ X @ ( nth @ A @ L @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
thf(fact_2316_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_2317_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_2318_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K: A] :
( ( member @ A @ I2 @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I2 @ K ) ) ) ).
% atMost_iff
thf(fact_2319_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu2: B] : ( one_one @ A )
@ A5 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_2320_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F3: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F3 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( semiring_1_of_nat @ A @ ( F3 @ X4 ) )
@ A5 ) ) ) ).
% of_nat_prod
thf(fact_2321_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F3: B > int,A5: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F3 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( ring_1_of_int @ A @ ( F3 @ X4 ) )
@ A5 ) ) ) ).
% of_int_prod
thf(fact_2322_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_2323_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_2324_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H: A,H3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H ) @ ( set_ord_atMost @ A @ H3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ H @ H3 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_2325_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N2 ) ) @ ( G3 @ ( suc @ N2 ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_2326_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N2 ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N2 ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G3 @ ( suc @ N2 ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_2327_prod_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > C > A,B4: set @ C,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G3 @ I4 ) @ B4 )
@ A5 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [J3: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( G3 @ I4 @ J3 )
@ A5 )
@ B4 ) ) ) ).
% prod.swap
thf(fact_2328_sorted__wrt__true,axiom,
! [A: $tType,Xs: list @ A] :
( sorted_wrt @ A
@ ^ [Uu2: A,Uv: A] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_2329_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [H: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H ) ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_2330_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A,H: B > A,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( G3 @ X4 ) @ ( H @ X4 ) )
@ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ H @ A5 ) ) ) ) ).
% prod.distrib
thf(fact_2331_prod__dividef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F3: B > A,G3: B > A,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ A5 )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ).
% prod_dividef
thf(fact_2332_prod__power__distrib,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ B )
=> ! [F3: A > B,A5: set @ A,N2: nat] :
( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A5 ) @ N2 )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N2 )
@ A5 ) ) ) ).
% prod_power_distrib
thf(fact_2333_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F3: B > A,A3: A,A5: set @ B] :
( ( modulo_modulo @ A
@ ( groups7121269368397514597t_prod @ B @ A
@ ^ [I4: B] : ( modulo_modulo @ A @ ( F3 @ I4 ) @ A3 )
@ A5 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ A3 ) ) ) ).
% mod_prod_eq
thf(fact_2334_abs__prod,axiom,
! [A: $tType,B: $tType] :
( ( linordered_field @ A )
=> ! [F3: B > A,A5: set @ B] :
( ( abs_abs @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( abs_abs @ A @ ( F3 @ X4 ) )
@ A5 ) ) ) ).
% abs_prod
thf(fact_2335_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_2336_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_2337_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_2338_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ).
% prod_nonneg
thf(fact_2339_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ).
% prod_mono
thf(fact_2340_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ).
% prod_pos
thf(fact_2341_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ).
% prod_ge_1
thf(fact_2342_sorted__wrt__less__idx,axiom,
! [Ns: list @ nat,I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Ns ) )
=> ( ord_less_eq @ nat @ I2 @ ( nth @ nat @ Ns @ I2 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_2343_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H: A,L3: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_Iic_le_Icc
thf(fact_2344_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_2345_power__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: A,F3: B > nat,A5: set @ B] :
( ( power_power @ A @ C2 @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A7: B] : ( power_power @ A @ C2 @ ( F3 @ A7 ) )
@ A5 ) ) ) ).
% power_sum
thf(fact_2346_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_2347_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_2348_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P4: A > A > $o,Xs3: list @ A] :
! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P4 @ ( nth @ A @ Xs3 @ I4 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_2349_sorted__wrt__nth__less,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,I2: nat,J: nat] :
( ( sorted_wrt @ A @ P @ Xs )
=> ( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_2350_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_2351_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N2 @ M ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_2352_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P3: nat,K: nat,G3: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P3 )
=> ( ( ord_less_eq @ nat @ K @ P3 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G3 @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P3 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G3 @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_2353_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_2354_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_2355_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G3 @ ( suc @ N2 ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_2356_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G3 @ ( suc @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_2357_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G3 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_2358_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( G3 @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G3 @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_2359_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_2360_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,I2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) )
@ ( set_ord_atMost @ nat @ I2 ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ ( suc @ I2 ) ) ) ) ) ).
% sum_telescope
thf(fact_2361_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) ) ) ) ).
% fact_prod
thf(fact_2362_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F3: nat > A,A3: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A7: nat] : ( times_times @ A @ ( F3 @ A7 ) )
@ A3
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_2363_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_2364_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I2: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_2365_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_2366_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G3: nat > A,P3: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N2 @ P3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ P3 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_2367_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N2 ) ) @ ( one_one @ A ) ) @ N2 ) ) ) ).
% gbinomial_parallel_sum
thf(fact_2368_fact__eq__fact__times,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( semiring_char_0_fact @ nat @ M )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N2 )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_2369_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N2 ) ) ) ) ) ).
% sum_gp_basic
thf(fact_2370_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_2371_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N2: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_2372_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A7: A,N: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_2373_fact__div__fact,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_2374_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_2375_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_2376_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P3: nat,K: nat,G3: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P3 )
=> ( ( ord_less_eq @ nat @ K @ P3 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G3 @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P3 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G3 @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_2377_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_2378_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( N2
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_2379_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N2: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_2380_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A5: set @ nat,C2: nat > A,D3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A5 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D3 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2381_nth__enumerate__eq,axiom,
! [A: $tType,M: nat,Xs: list @ A,N2: nat] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) @ M )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N2 @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_2382_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_2383_semiring__norm_I78_J,axiom,
! [M: num,N2: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less @ num @ M @ N2 ) ) ).
% semiring_norm(78)
thf(fact_2384_semiring__norm_I71_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_eq @ num @ M @ N2 ) ) ).
% semiring_norm(71)
thf(fact_2385_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_2386_semiring__norm_I68_J,axiom,
! [N2: num] : ( ord_less_eq @ num @ one2 @ N2 ) ).
% semiring_norm(68)
thf(fact_2387_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I2 )
& ( ord_less @ A @ I2 @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_2388_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_2389_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_2390_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_2391_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_2392_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_2393_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I2: A,J: A,M: A,N2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N2 ) )
= ( ( ord_less_eq @ A @ J @ I2 )
| ( ( ord_less_eq @ A @ M @ I2 )
& ( ord_less_eq @ A @ J @ N2 ) ) ) ) ) ).
% ivl_subset
thf(fact_2394_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I2: A,N2: A,M: A] :
( ( ord_less_eq @ A @ I2 @ N2 )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ M ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ N2 ) )
= ( set_or7035219750837199246ssThan @ A @ N2 @ M ) ) ) ) ).
% ivl_diff
thf(fact_2395_binomial__eq__0__iff,axiom,
! [N2: nat,K: nat] :
( ( ( binomial @ N2 @ K )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N2 @ K ) ) ).
% binomial_eq_0_iff
thf(fact_2396_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_2397_semiring__norm_I76_J,axiom,
! [N2: num] : ( ord_less @ num @ one2 @ ( bit0 @ N2 ) ) ).
% semiring_norm(76)
thf(fact_2398_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_2399_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_2400_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I2: A,J: A] :
( ( image2 @ A @ A
@ ^ [N: A] : ( plus_plus @ A @ N @ K )
@ ( set_or7035219750837199246ssThan @ A @ I2 @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_2401_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_2402_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_2403_zero__less__binomial__iff,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) )
= ( ord_less_eq @ nat @ K @ N2 ) ) ).
% zero_less_binomial_iff
thf(fact_2404_prod__pos__nat__iff,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X4 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_2405_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) )
= ( M != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_2406_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( M != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_2407_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N2 @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_2408_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N2: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less @ num @ one2 @ N2 ) ) ) ).
% one_less_numeral_iff
thf(fact_2409_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% power2_less_eq_zero_iff
thf(fact_2410_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X = Y ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_2411_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_2412_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N2: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G3 @ N2 ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_2413_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G3 @ N2 ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_2414_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A5: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A5 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C2 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_2415_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% one_less_floor
thf(fact_2416_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_2417_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ nat ) ) ) ).
% mod2_gr_0
thf(fact_2418_int__prod,axiom,
! [B: $tType,F3: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F3 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X4: B] : ( semiring_1_of_nat @ int @ ( F3 @ X4 ) )
@ A5 ) ) ).
% int_prod
thf(fact_2419_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S: set @ A,S4: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [S6: set @ A] :
( ( minus_minus @ ( set @ A ) @ S6 @ S )
= ( minus_minus @ ( set @ A ) @ S4 @ S ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_2420_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_2421_binomial__le__pow2,axiom,
! [N2: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N2 @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% binomial_le_pow2
thf(fact_2422_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( A3 = C2 )
& ( B2 = D3 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_2423_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( A3 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_2424_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( B2 = D3 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_2425_subset__eq__atLeast0__lessThan__finite,axiom,
! [N7: set @ nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_2426_choose__row__sum,axiom,
! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ N2 ) @ ( set_ord_atMost @ nat @ N2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% choose_row_sum
thf(fact_2427_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N8: set @ nat] :
? [M3: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N8 )
=> ( ord_less @ nat @ X4 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_2428_bounded__nat__set__is__finite,axiom,
! [N7: set @ nat,N2: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N7 )
=> ( ord_less @ nat @ X3 @ N2 ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% bounded_nat_set_is_finite
thf(fact_2429_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N8: set @ nat] :
? [M3: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N8 )
=> ( ord_less_eq @ nat @ X4 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_2430_choose__square__sum,axiom,
! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( power_power @ nat @ ( binomial @ N2 @ K3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ N2 ) ) ).
% choose_square_sum
thf(fact_2431_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_2432_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_2433_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_2434_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_2435_power4__eq__xxxx,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A] :
( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X @ X ) @ X ) @ X ) ) ) ).
% power4_eq_xxxx
thf(fact_2436_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% power2_eq_square
thf(fact_2437_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_2438_finite__less__ub,axiom,
! [F3: nat > nat,U: nat] :
( ! [N5: nat] : ( ord_less_eq @ nat @ N5 @ ( F3 @ N5 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ ( F3 @ N ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_2439_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ C,G3: B > C > A,R2: B > C > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G3 @ X4 )
@ ( collect @ C
@ ^ [Y4: C] :
( ( member @ C @ Y4 @ B4 )
& ( R2 @ X4 @ Y4 ) ) ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y4: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( G3 @ X4 @ Y4 )
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( R2 @ X4 @ Y4 ) ) ) )
@ B4 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_2440_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ C,G3: B > C > A,R2: B > C > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B4 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G3 @ X4 )
@ ( collect @ C
@ ^ [Y4: C] :
( ( member @ C @ Y4 @ B4 )
& ( R2 @ X4 @ Y4 ) ) ) )
@ A5 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y4: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( G3 @ X4 @ Y4 )
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( R2 @ X4 @ Y4 ) ) ) )
@ B4 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_2441_less__exp,axiom,
! [N2: nat] : ( ord_less @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% less_exp
thf(fact_2442_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_2443_power2__nat__le__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% power2_nat_le_eq_le
thf(fact_2444_power2__nat__le__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% power2_nat_le_imp_le
thf(fact_2445_sum__power2,axiom,
! [K: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ).
% sum_power2
thf(fact_2446_Sum__Ico__nat,axiom,
! [M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_2447_binomial__eq__0,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ N2 @ K )
=> ( ( binomial @ N2 @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2448_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% half_gt_zero_iff
thf(fact_2449_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_2450_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_2451_zero__le__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% zero_le_power2
thf(fact_2452_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( X = Y ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2453_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% power2_le_imp_le
thf(fact_2454_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_2455_binomial__symmetric,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( binomial @ N2 @ K )
= ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_2456_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_2457_ex__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ( P @ M3 ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_2458_all__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( P @ M3 ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less_eq
thf(fact_2459_less__2__cases__iff,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases_iff
thf(fact_2460_less__2__cases,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( N2
= ( zero_zero @ nat ) )
| ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases
thf(fact_2461_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(3)
thf(fact_2462_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( abs_abs @ A @ Y ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_2463_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S: set @ A] :
( ( finite_finite2 @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S )
& ~ ? [Xa2: A] :
( ( member @ A @ Xa2 @ S )
& ( ord_less @ A @ Xa2 @ X3 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_2464_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X6: set @ A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X6 )
& ( ord_less @ A @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ A @ X6 ) ) ) ) ).
% infinite_growing
thf(fact_2465_binomial__le__pow,axiom,
! [R3: nat,N2: nat] :
( ( ord_less_eq @ nat @ R3 @ N2 )
=> ( ord_less_eq @ nat @ ( binomial @ N2 @ R3 ) @ ( power_power @ nat @ N2 @ R3 ) ) ) ).
% binomial_le_pow
thf(fact_2466_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N2 ) ) ) ) ).
% diff_le_diff_pow
thf(fact_2467_binomial__r__part__sum,axiom,
! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% binomial_r_part_sum
thf(fact_2468_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_2469_choose__linear__sum,axiom,
! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ I4 @ ( binomial @ N2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( times_times @ nat @ N2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_2470_sum__mono__inv,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F3: I6 > A,I5: set @ I6,G3: I6 > A,I2: I6] :
( ( ( groups7311177749621191930dd_sum @ I6 @ A @ F3 @ I5 )
= ( groups7311177749621191930dd_sum @ I6 @ A @ G3 @ I5 ) )
=> ( ! [I3: I6] :
( ( member @ I6 @ I3 @ I5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) )
=> ( ( member @ I6 @ I2 @ I5 )
=> ( ( finite_finite2 @ I6 @ I5 )
=> ( ( F3 @ I2 )
= ( G3 @ I2 ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_2471_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_2472_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% mult_numeral_1_right
thf(fact_2473_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A3 )
= A3 ) ) ).
% mult_numeral_1
thf(fact_2474_sum__choose__upper,axiom,
! [M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ K3 @ M )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_2475_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_2476_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_2477_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_2478_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( X @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( Y @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( plus_plus @ A @ ( X @ I4 ) @ ( Y @ I4 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_2479_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_2480_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2481_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( X @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( Y @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I5 )
& ( ( times_times @ A @ ( X @ I4 ) @ ( Y @ I4 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_2482_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( 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 @ X @ Y ) ) ) ) ).
% power2_less_imp_less
thf(fact_2483_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2484_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2485_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2486_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_2487_numeral__code_I2_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit0 @ N2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% numeral_code(2)
thf(fact_2488_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( P @ X4 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( G3 @ X4 ) @ ( zero_zero @ A ) )
@ A5 ) ) ) ) ).
% sum.inter_filter
thf(fact_2489_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_sum
thf(fact_2490_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_2491_filter__preserves__multiset,axiom,
! [A: $tType,M6: A > nat,P: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M6 @ X4 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( P @ X4 ) @ ( M6 @ X4 ) @ ( zero_zero @ nat ) ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_2492_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% zero_le_even_power'
thf(fact_2493_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ Y ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_2494_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,H: B > A,G3: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H @ S )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y4: C] :
( groups7311177749621191930dd_sum @ B @ A @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S )
& ( ( G3 @ X4 )
= Y4 ) ) ) )
@ ( image2 @ B @ C @ G3 @ S ) ) ) ) ) ).
% sum.image_gen
thf(fact_2495_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_2496_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2497_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( P @ X4 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( G3 @ X4 ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_filter
thf(fact_2498_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,H: B > A,G3: B > C] :
( ( finite_finite2 @ B @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H @ S )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y4: C] :
( groups7121269368397514597t_prod @ B @ A @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S )
& ( ( G3 @ X4 )
= Y4 ) ) ) )
@ ( image2 @ B @ C @ G3 @ S ) ) ) ) ) ).
% prod.image_gen
thf(fact_2499_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,N2: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N2 ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% minus_power_mult_self
thf(fact_2500_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N2: nat] :
( ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ ( power_power @ A @ A3 @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_2501_Suc__n__div__2__gt__zero,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_2502_div__2__gt__zero,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_2503_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_2504_zero__less__binomial,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N2 @ K ) ) ) ).
% zero_less_binomial
thf(fact_2505_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A3: B,C2: B,B2: B,D3: B,G3: B > A,H: B > A] :
( ( A3 = C2 )
=> ( ( B2 = D3 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D3 )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_2506_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A3: B,C2: B,B2: B,D3: B,G3: B > A,H: B > A] :
( ( A3 = C2 )
=> ( ( B2 = D3 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D3 )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_2507_choose__mult,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( times_times @ nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
= ( times_times @ nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus @ nat @ N2 @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_2508_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_2509_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_2510_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2511_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(7)
thf(fact_2512_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,P3: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ P3 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ P3 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_2513_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N2: nat,P3: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ P3 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M @ P3 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P3 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_2514_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_diff
thf(fact_2515_binomial__code,axiom,
( binomial
= ( ^ [N: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus @ nat @ N @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N @ K3 ) @ ( one_one @ nat ) ) @ N @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2516_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,P3: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ P3 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ P3 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ P3 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_2517_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2518_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_2519_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 )
= ( zero_zero @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( F3 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_2520_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,T6: set @ C,G3: C > A,I2: C > B,F3: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G3 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T6 )
& ( ( I2 @ Xa2 )
= X3 )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ Xa2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S2 ) @ ( groups7311177749621191930dd_sum @ C @ A @ G3 @ T6 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_2521_sum__strict__mono__ex1,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A5: set @ I6,F3: I6 > A,G3: I6 > A] :
( ( finite_finite2 @ I6 @ A5 )
=> ( ! [X3: I6] :
( ( member @ I6 @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ? [X7: I6] :
( ( member @ I6 @ X7 @ A5 )
& ( ord_less @ A @ ( F3 @ X7 ) @ ( G3 @ X7 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I6 @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ I6 @ A @ G3 @ A5 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_2522_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_2523_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R2: A > A > $o,S: set @ B,H: B > A,G3: B > A] :
( ( R2 @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X12: A,Y12: A,X22: A,Y22: A] :
( ( ( R2 @ X12 @ X22 )
& ( R2 @ Y12 @ Y22 ) )
=> ( R2 @ ( times_times @ A @ X12 @ Y12 ) @ ( times_times @ A @ X22 @ Y22 ) ) )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( R2 @ ( H @ X3 ) @ ( G3 @ X3 ) ) )
=> ( R2 @ ( groups7121269368397514597t_prod @ B @ A @ H @ S ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ S ) ) ) ) ) ) ) ).
% prod.related
thf(fact_2524_ex__power__ivl1,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N5: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N5 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_2525_ex__power__ivl2,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N5: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N5 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_2526_atLeastLessThan__add__Un,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( set_or7035219750837199246ssThan @ nat @ I2 @ ( plus_plus @ nat @ J @ K ) )
= ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I2 @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_2527_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N2: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q7: nat] : ( ord_less @ nat @ Q7 @ N2 ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_2528_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(1)
thf(fact_2529_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(2)
thf(fact_2530_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% sum.in_pairs
thf(fact_2531_mask__eq__sum__exp__nat,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q7: nat] : ( ord_less @ nat @ Q7 @ N2 ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_2532_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% prod.in_pairs
thf(fact_2533_gauss__sum__nat,axiom,
! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_2534_prod__int__eq,axiom,
! [I2: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I2 @ J ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X4: int] : X4
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I2 ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_2535_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% sum.in_pairs_0
thf(fact_2536_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% prod.in_pairs_0
thf(fact_2537_sum__choose__lower,axiom,
! [R3: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R3 @ K3 ) @ K3 )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R3 @ N2 ) ) @ N2 ) ) ).
% sum_choose_lower
thf(fact_2538_choose__rising__sum_I2_J,axiom,
! [N2: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J3 ) @ N2 )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ ( one_one @ nat ) ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_2539_choose__rising__sum_I1_J,axiom,
! [N2: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N2 @ J3 ) @ N2 )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_2540_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,F3: B > A,I2: B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S2 )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I2 @ S2 )
=> ( ( F3 @ I2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_2541_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S2: set @ B,F3: B > A,B4: A,I2: B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S2 )
= B4 )
=> ( ( member @ B @ I2 @ S2 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ B4 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_2542_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X @ M ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2543_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2544_add__mset__in__multiset,axiom,
! [A: $tType,M6: A > nat,A3: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M6 @ X4 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( X4 = A3 ) @ ( suc @ ( M6 @ X4 ) ) @ ( M6 @ X4 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_2545_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,B4: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( member @ B @ X4 @ B4 ) @ ( G3 @ X4 ) @ ( zero_zero @ A ) )
@ A5 ) ) ) ) ).
% sum.inter_restrict
thf(fact_2546_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T2: set @ C,G3: B > C,H: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G3 @ S ) @ T2 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y4: C] :
( groups7311177749621191930dd_sum @ B @ A @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S )
& ( ( G3 @ X4 )
= Y4 ) ) ) )
@ T2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S ) ) ) ) ) ) ).
% sum.group
thf(fact_2547_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3
@ ( minus_minus @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [X4: B] :
( ( G3 @ X4 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_2548_diff__preserves__multiset,axiom,
! [A: $tType,M6: A > nat,N7: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M6 @ X4 ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( M6 @ X4 ) @ ( N7 @ X4 ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_2549_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T2: set @ C,G3: B > C,H: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G3 @ S ) @ T2 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y4: C] :
( groups7121269368397514597t_prod @ B @ A @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S )
& ( ( G3 @ X4 )
= Y4 ) ) ) )
@ T2 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S ) ) ) ) ) ) ).
% prod.group
thf(fact_2550_subset__eq__atLeast0__atMost__finite,axiom,
! [N7: set @ nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( finite_finite2 @ nat @ N7 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_2551_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,B4: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( member @ B @ X4 @ B4 ) @ ( G3 @ X4 ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_restrict
thf(fact_2552_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_2553_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3
@ ( minus_minus @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [X4: B] :
( ( G3 @ X4 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_2554_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K3: A] :
( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A3 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_2555_neg__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_2556_pos__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ B2 @ A3 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_2557_pos__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A3 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_2558_arith__series__nat,axiom,
! [A3: nat,D3: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I4 @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N2 @ D3 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_2559_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_2560_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_2561_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_2562_Sum__Icc__nat,axiom,
! [M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_2563_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( G3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_2564_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat,B2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G3 @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_2565_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) @ ( G3 @ N2 ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_2566_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G3 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N2 ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_2567_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat,B2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G3 @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_2568_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( G3 @ N2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_2569_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G3 @ N2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_2570_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,I2: B,F3: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( member @ B @ I2 @ I5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I5 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_2571_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I5 ) ) ) ) ) ) ).
% sum_pos
thf(fact_2572_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,I2: A,F3: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( member @ A @ I2 @ I5 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I2 ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I5 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_2573_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I5 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_2574_binomial__fact__lemma,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
= ( semiring_char_0_fact @ nat @ N2 ) ) ) ).
% binomial_fact_lemma
thf(fact_2575_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C4: set @ B,A5: set @ B,B4: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C4 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C4 @ A5 ) )
=> ( ( G3 @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C4 @ B4 ) )
=> ( ( H @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B4 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ C4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C4 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_2576_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C4: set @ B,A5: set @ B,B4: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C4 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C4 @ A5 ) )
=> ( ( G3 @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C4 @ B4 ) )
=> ( ( H @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ C4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C4 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B4 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_2577_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T2: set @ B,S: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_2578_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T2: set @ B,S: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T2 )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_2579_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T2: set @ B,S: set @ B,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( H @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T2 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_2580_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T2: set @ B,S: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_2581_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: set @ B,A5: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B4 @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B4 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_2582_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,B4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B4 ) ) ) ) ) ) ).
% sum_diff
thf(fact_2583_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T2: set @ B,S: set @ B,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S @ T2 ) )
=> ( ( G3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S @ T2 ) )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T2 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_2584_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B4: set @ B,A5: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B4 @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B4 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_2585_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ B,A5: set @ B,B4: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C4 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C4 @ A5 ) )
=> ( ( G3 @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C4 @ B4 ) )
=> ( ( H @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B4 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ C4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C4 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_2586_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ B,A5: set @ B,B4: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C4 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C4 @ A5 ) )
=> ( ( G3 @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C4 @ B4 ) )
=> ( ( H @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ C4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C4 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B4 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_2587_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T2: set @ B,S: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_2588_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T2: set @ B,S: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ T2 )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ S ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_2589_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T2: set @ B,S: set @ B,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( H @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_2590_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T2: set @ B,S: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ T2 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_2591_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B4 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_2592_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,B4: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_2593_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B4 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_2594_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,B4: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_2595_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T2: set @ B,S: set @ B,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T2 )
=> ( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( H @ I3 )
= ( one_one @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S @ T2 ) )
=> ( ( G3 @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S @ T2 ) )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T2 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_2596_prod__int__plus__eq,axiom,
! [I2: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I2 @ ( plus_plus @ nat @ I2 @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X4: int] : X4
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I2 ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I2 @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_2597_neg__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_2598_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D3: A,N2: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D3 ) ) ) ) ) ).
% double_arith_series
thf(fact_2599_double__gauss__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum
thf(fact_2600_sum__choose__diagonal,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N2 @ K3 ) @ ( minus_minus @ nat @ M @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( suc @ N2 ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_2601_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N2: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) )
= ( minus_minus @ A @ ( F3 @ N2 ) @ ( F3 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_2602_sum__diff__nat,axiom,
! [A: $tType,B4: set @ A,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B4 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_2603_vandermonde,axiom,
! [M: nat,N2: nat,R3: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus @ nat @ R3 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R3 ) )
= ( binomial @ ( plus_plus @ nat @ M @ N2 ) @ R3 ) ) ).
% vandermonde
thf(fact_2604_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N2: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_2605_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q4: int,R3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q4 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R3 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_2606_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% sum.nested_swap
thf(fact_2607_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_2608_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% prod.nested_swap
thf(fact_2609_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_2610_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_2611_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2612_Sum__Icc__int,axiom,
! [M: int,N2: int] :
( ( ord_less_eq @ int @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X4: int] : X4
@ ( set_or1337092689740270186AtMost @ int @ M @ N2 ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N2 @ ( plus_plus @ int @ N2 @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_2613_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,P: B > $o,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( H @ X4 ) @ ( G3 @ X4 ) )
@ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_2614_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,P: B > $o,H: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( H @ X4 ) @ ( G3 @ X4 ) )
@ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_2615_fact__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N2: nat] :
( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N2 ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% fact_double
thf(fact_2616_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_2617_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_2618_choose__reduce__nat,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( binomial @ N2 @ K )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_2619_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D3: A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_2620_gauss__sum,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum
thf(fact_2621_times__binomial__minus1__eq,axiom,
! [K: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N2 @ K ) )
= ( times_times @ nat @ N2 @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_2622_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N2: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G3 @ N2 ) ) ) ) ) ) ).
% sum.head_if
thf(fact_2623_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q4: int,R3: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q4 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_2624_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N2: nat,M: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) @ ( G3 @ N2 ) ) ) ) ) ) ).
% prod.head_if
thf(fact_2625_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) ) )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_2626_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B4: set @ B,A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B4 @ A5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B4 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_2627_binomial__altdef__nat,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( binomial @ N2 @ K )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N2 ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_2628_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B4 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_2629_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,B4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_2630_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( ( inf_inf @ ( set @ B ) @ A5 @ B4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B4 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_2631_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B4 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_2632_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( ( inf_inf @ ( set @ B ) @ A5 @ B4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B4 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_2633_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A5: set @ A,B4: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_2634_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ B4 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_2635_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ B4 @ A5 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_2636_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N2 ) ) ) ) ).
% pochhammer_double
thf(fact_2637_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_2638_sum__Un__nat,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_2639_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N2: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_2640_binomial,axiom,
! [A3: nat,B2: nat,N2: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power @ nat @ A3 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ).
% binomial
thf(fact_2641_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N2 @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N2 ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_2642_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R3 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R3 @ ( suc @ M ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_2643_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A7: A,N: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% pochhammer_prod
thf(fact_2644_binomial__addition__formula,axiom,
! [N2: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( binomial @ N2 @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_2645_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_2646_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_2647_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( semiring_char_0_fact @ A @ N2 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N2 @ K ) @ N2 ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_2648_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_2649_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B4: set @ A,A5: set @ A,B2: A,F3: A > B] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F3 @ B2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ B4 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_2650_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B4: set @ A,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ B5 ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ A6 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ B4 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_2651_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A5: set @ B,B4: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
=> ( ( F3 @ X3 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_2652_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_2653_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B2: A,N2: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K3 ) ) @ ( power_power @ A @ A3 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% binomial_ring
thf(fact_2654_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,B2: A,N2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N2 )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_2655_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_2656_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_2657_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_2658_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% half_negative_int_iff
thf(fact_2659_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_2660_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R3: A,Q4: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R3 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R3 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R3 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R3 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R3 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R3 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_2661_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2662_nat__bit__induct,axiom,
! [P: nat > $o,N2: 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 @ N2 ) ) ) ) ).
% nat_bit_induct
thf(fact_2663_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ N @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_2664_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_2665_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_2666_finite__interval__int1,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A3 @ I4 )
& ( ord_less_eq @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_2667_finite__interval__int4,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A3 @ I4 )
& ( ord_less @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_2668_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N: nat] : N ) ) ).
% of_nat_id
thf(fact_2669_finite__Int,axiom,
! [A: $tType,F5: set @ A,G5: set @ A] :
( ( ( finite_finite2 @ A @ F5 )
| ( finite_finite2 @ A @ G5 ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G5 ) ) ) ).
% finite_Int
thf(fact_2670_finite__Un,axiom,
! [A: $tType,F5: set @ A,G5: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G5 ) )
= ( ( finite_finite2 @ A @ F5 )
& ( finite_finite2 @ A @ G5 ) ) ) ).
% finite_Un
thf(fact_2671_finite__interval__int3,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A3 @ I4 )
& ( ord_less_eq @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_2672_finite__interval__int2,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A3 @ I4 )
& ( ord_less @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_2673_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less @ nat @ N @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_2674_finite__Collect__subsets,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ B7 @ A5 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_2675_finite__maxlen,axiom,
! [A: $tType,M6: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M6 )
=> ? [N5: nat] :
! [X7: list @ A] :
( ( member @ ( list @ A ) @ X7 @ M6 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X7 ) @ N5 ) ) ) ).
% finite_maxlen
thf(fact_2676_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image2 @ int @ int
@ ^ [X4: int] : ( plus_plus @ int @ X4 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_2677_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_2678_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B,R2: A > B > $o] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B4 )
& ( R2 @ X3 @ Xa2 ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ B4 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A7: A] :
( ( member @ A @ A7 @ A5 )
& ( R2 @ A7 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_2679_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ A3 @ X3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A5 )
=> ( ( ord_less_eq @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_2680_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ X3 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A5 )
=> ( ( ord_less_eq @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_2681_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_2682_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_2683_all__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ A5 )
=> ( P @ ( image2 @ B @ A @ F3 @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_2684_finite__subset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_subset
thf(fact_2685_infinite__super,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% infinite_super
thf(fact_2686_rev__finite__subset,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% rev_finite_subset
thf(fact_2687_finite__UnI,axiom,
! [A: $tType,F5: set @ A,G5: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( finite_finite2 @ A @ G5 )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G5 ) ) ) ) ).
% finite_UnI
thf(fact_2688_Un__infinite,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_2689_infinite__Un,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S @ T2 ) ) )
= ( ~ ( finite_finite2 @ A @ S )
| ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% infinite_Un
thf(fact_2690_finite__psubset__induct,axiom,
! [A: $tType,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A13: set @ A] :
( ( finite_finite2 @ A @ A13 )
=> ( ! [B9: set @ A] :
( ( ord_less @ ( set @ A ) @ B9 @ A13 )
=> ( P @ B9 ) )
=> ( P @ A13 ) ) )
=> ( P @ A5 ) ) ) ).
% finite_psubset_induct
thf(fact_2691_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ ( image2 @ A @ B @ F3 @ A5 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A7: A] :
( ( member @ A @ A7 @ A5 )
& ( ( F3 @ A7 )
= ( F3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_2692_nat__seg__image__imp__finite,axiom,
! [A: $tType,A5: set @ A,F3: nat > A,N2: nat] :
( ( A5
= ( image2 @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) ) )
=> ( finite_finite2 @ A @ A5 ) ) ).
% nat_seg_image_imp_finite
thf(fact_2693_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A8: set @ A] :
? [N: nat,F4: nat > A] :
( A8
= ( image2 @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_2694_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A5 )
=> ( ( ord_less_eq @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_2695_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A5 )
=> ( ( ord_less_eq @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_2696_finite__surj,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ ( image2 @ A @ B @ F3 @ A5 ) )
=> ( finite_finite2 @ B @ B4 ) ) ) ).
% finite_surj
thf(fact_2697_finite__subset__image,axiom,
! [A: $tType,B: $tType,B4: set @ A,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ? [C6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C6 @ A5 )
& ( finite_finite2 @ B @ C6 )
& ( B4
= ( image2 @ B @ A @ F3 @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_2698_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: ( set @ A ) > $o] :
( ( ? [B7: set @ A] :
( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image2 @ B @ A @ F3 @ A5 ) )
& ( P @ B7 ) ) )
= ( ? [B7: set @ B] :
( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A5 )
& ( P @ ( image2 @ B @ A @ F3 @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_2699_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image2 @ B @ A @ F3 @ A5 ) ) )
=> ( P @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A5 ) )
=> ( P @ ( image2 @ B @ A @ F3 @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_2700_not__exp__less__eq__0__int,axiom,
! [N2: nat] :
~ ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ int ) ) ).
% not_exp_less_eq_0_int
thf(fact_2701_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2702_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_2703_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N2 ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N2 @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_2704_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N2 ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N2 @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_2705_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N2 ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N2 @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_2706_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N: nat,A7: A] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2707_unset__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N2 @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_2708_set__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N2 @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_2709_flip__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N2 @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_2710_unset__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% unset_bit_negative_int_iff
thf(fact_2711_set__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% set_bit_negative_int_iff
thf(fact_2712_flip__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% flip_bit_negative_int_iff
thf(fact_2713_unset__bit__less__eq,axiom,
! [N2: nat,K: int] : ( ord_less_eq @ int @ ( bit_se2638667681897837118et_bit @ int @ N2 @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_2714_set__bit__greater__eq,axiom,
! [K: int,N2: nat] : ( ord_less_eq @ int @ K @ ( bit_se5668285175392031749et_bit @ int @ N2 @ K ) ) ).
% set_bit_greater_eq
thf(fact_2715_signed__take__bit__int__less__exp,axiom,
! [N2: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% signed_take_bit_int_less_exp
thf(fact_2716_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_2717_signed__take__bit__int__less__self__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_2718_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N2: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_2719_signed__take__bit__int__less__eq__self__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ K )
= ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_2720_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) )
= ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_2721_signed__take__bit__int__less__eq,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_2722_signed__take__bit__int__eq__self,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_2723_signed__take__bit__int__eq__self__iff,axiom,
! [N2: nat,K: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K )
= K )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_2724_signed__take__bit__int__greater__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_2725_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N2 ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N2 @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_2726_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X )
= Y ) ) ) ) ).
% round_unique
thf(fact_2727_finite__int__iff__bounded__le,axiom,
( ( finite_finite2 @ int )
= ( ^ [S5: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ S5 ) @ ( set_ord_atMost @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_2728_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I4 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% choose_odd_sum
thf(fact_2729_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N2 @ I4 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% choose_even_sum
thf(fact_2730_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X4: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X4 ) ) @ ( archimedean_ceiling @ A @ X4 ) @ ( archim6421214686448440834_floor @ A @ X4 ) ) ) ) ) ).
% round_altdef
thf(fact_2731_finite__nat__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S5: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S5 @ ( set_ord_atMost @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_2732_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) )
= ( M
= ( one_one @ nat ) ) ) ).
% nat_dvd_1_iff_1
thf(fact_2733_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_2734_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_2735_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_2736_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_2737_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_2738_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_2739_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_2740_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_2741_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% dvd_div_mult_self
thf(fact_2742_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( M
= ( suc @ ( zero_zero @ nat ) ) ) ) ).
% dvd_1_iff_1
thf(fact_2743_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_2744_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_2745_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( divide_divide @ A @ B2 @ A3 ) ) ) ) ).
% unit_mult_div_div
thf(fact_2746_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N2: nat,A3: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_2747_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_mult_iff
thf(fact_2748_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A3 @ N2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% even_power
thf(fact_2749_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_2750_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_2751_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N2 ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_2752_even__diff__nat,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) )
= ( ( ord_less @ nat @ M @ N2 )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ).
% even_diff_nat
thf(fact_2753_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A3 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_2754_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( ( numeral_numeral @ nat @ W2 )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_2755_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( 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 @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_2756_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_2757_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2758_dvd__antisym,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd @ nat @ M @ N2 )
=> ( ( dvd_dvd @ nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% dvd_antisym
thf(fact_2759_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
=> ? [B10: A,C7: A] :
( ( A3
= ( times_times @ A @ B10 @ C7 ) )
& ( dvd_dvd @ A @ B10 @ B2 )
& ( dvd_dvd @ A @ C7 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_2760_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P3: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ P3 @ ( times_times @ A @ A3 @ B2 ) )
=> ~ ! [X3: A,Y3: A] :
( ( P3
= ( times_times @ A @ X3 @ Y3 ) )
=> ( ( dvd_dvd @ A @ X3 @ A3 )
=> ~ ( dvd_dvd @ A @ Y3 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_2761_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ~ ! [K2: A] :
( A3
!= ( times_times @ A @ B2 @ K2 ) ) ) ) ).
% dvdE
thf(fact_2762_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A3: A,B2: A,K: A] :
( ( A3
= ( times_times @ A @ B2 @ K ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% dvdI
thf(fact_2763_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B6: A,A7: A] :
? [K3: A] :
( A7
= ( times_times @ A @ B6 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_2764_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult
thf(fact_2765_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult2
thf(fact_2766_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ A3 @ C2 ) ) ) ).
% dvd_mult_left
thf(fact_2767_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) ) ) ).
% dvd_triv_left
thf(fact_2768_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_2769_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% dvd_mult_right
thf(fact_2770_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) ) ) ).
% dvd_triv_right
thf(fact_2771_dvd__diff__nat,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ M )
=> ( ( dvd_dvd @ nat @ K @ N2 )
=> ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% dvd_diff_nat
thf(fact_2772_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ A3 ) )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ B2 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_2773_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ A3 ) )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A3 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_2774_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S2: B] :
? [Z3: B] :
! [X7: B] :
( ( ord_less @ B @ Z3 @ X7 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S2 ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S2 ) ) ) ) ) ).
% pinf(9)
thf(fact_2775_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S2: B] :
? [Z3: B] :
! [X7: B] :
( ( ord_less @ B @ Z3 @ X7 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S2 ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S2 ) ) ) ) ) ) ).
% pinf(10)
thf(fact_2776_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S2: B] :
? [Z3: B] :
! [X7: B] :
( ( ord_less @ B @ X7 @ Z3 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S2 ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S2 ) ) ) ) ) ).
% minf(9)
thf(fact_2777_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S2: B] :
? [Z3: B] :
! [X7: B] :
( ( ord_less @ B @ X7 @ Z3 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S2 ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S2 ) ) ) ) ) ) ).
% minf(10)
thf(fact_2778_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A3 )
= ( times_times @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_2779_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A3 @ B2 )
= ( times_times @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_2780_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_2781_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_2782_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_2783_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_2784_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_2785_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,D3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( dvd_dvd @ A @ D3 @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( divide_divide @ A @ C2 @ D3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_2786_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_2787_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_2788_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_2789_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% div_mult_swap
thf(fact_2790_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% dvd_div_mult
thf(fact_2791_dvd__pos__nat,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( dvd_dvd @ nat @ M @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).
% dvd_pos_nat
thf(fact_2792_nat__dvd__not__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N2 )
=> ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_2793_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N2: nat,M: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N2 ) @ ( power_power @ A @ Y @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_2794_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N2: nat,B2: A,M: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N2 ) @ B2 )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ B2 ) ) ) ) ).
% power_le_dvd
thf(fact_2795_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N2: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% le_imp_power_dvd
thf(fact_2796_dvd__minus__self,axiom,
! [M: nat,N2: nat] :
( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ M ) )
= ( ( ord_less @ nat @ N2 @ M )
| ( dvd_dvd @ nat @ M @ N2 ) ) ) ).
% dvd_minus_self
thf(fact_2797_less__eq__dvd__minus,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( dvd_dvd @ nat @ M @ N2 )
= ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_2798_dvd__diffD1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
=> ( ( dvd_dvd @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( dvd_dvd @ nat @ K @ N2 ) ) ) ) ).
% dvd_diffD1
thf(fact_2799_dvd__diffD,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N2 ) )
=> ( ( dvd_dvd @ nat @ K @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M )
=> ( dvd_dvd @ nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_2800_fact__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N2 ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ).
% fact_dvd
thf(fact_2801_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_2802_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [C3: A] :
( B2
!= ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% unit_dvdE
thf(fact_2803_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T6: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X7: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X7 @ T6 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K4 @ D4 ) ) @ T6 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_2804_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T6: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X7: A,K4: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X7 @ T6 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K4 @ D4 ) ) @ T6 ) ) ) ) ) ).
% inf_period(3)
thf(fact_2805_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D3 )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= ( divide_divide @ A @ D3 @ C2 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A3 @ D3 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_2806_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_2807_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_2808_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= C2 )
= ( B2
= ( times_times @ A @ C2 @ A3 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_2809_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_2810_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_2811_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_2812_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_2813_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A3
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( times_times @ A @ A3 @ B2 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_2814_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= C2 )
= ( A3
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_2815_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% round_mono
thf(fact_2816_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B4: set @ B,A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B4 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_2817_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B4: set @ B,A5: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ B4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ A5 )
=> ( dvd_dvd @ A @ ( F3 @ A6 ) @ ( G3 @ A6 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B4 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_2818_dvd__imp__le,axiom,
! [K: nat,N2: nat] :
( ( dvd_dvd @ nat @ K @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ nat @ K @ N2 ) ) ) ).
% dvd_imp_le
thf(fact_2819_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( dvd_dvd @ nat @ M @ N2 ) ) ) ).
% dvd_mult_cancel
thf(fact_2820_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N2 ) )
= ( dvd_dvd @ nat @ M @ N2 ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_2821_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N2: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N2 ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N2 ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_2822_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N2 ) )
= ( ~ ( dvd_dvd @ nat @ N2 @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_2823_mod__eq__dvd__iff__nat,axiom,
! [N2: nat,M: nat,Q4: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( ( modulo_modulo @ nat @ M @ Q4 )
= ( modulo_modulo @ nat @ N2 @ Q4 ) )
= ( dvd_dvd @ nat @ Q4 @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_2824_floor__le__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% floor_le_round
thf(fact_2825_ceiling__ge__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% ceiling_ge_round
thf(fact_2826_dvd__fact,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N2 ) ) ) ) ).
% dvd_fact
thf(fact_2827_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D6: nat] : ( dvd_dvd @ nat @ D6 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_2828_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top @ nat @ ( dvd_dvd @ nat )
@ ^ [M3: nat,N: nat] :
( ( dvd_dvd @ nat @ M3 @ N )
& ( M3 != N ) )
@ ( zero_zero @ nat ) ) ).
% gcd_nat.ordering_top_axioms
thf(fact_2829_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_2830_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_2831_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B5: A] :
( ( B5
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B5 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B5 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B5 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B5 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A3 )
!= ( times_times @ A @ C2 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_2832_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B5: A] :
( A3
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) ) ) ) ).
% evenE
thf(fact_2833_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M: nat,N2: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N2 ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ) ).
% dvd_power_iff
thf(fact_2834_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N2: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N2 ) ) ) ) ).
% dvd_power
thf(fact_2835_dvd__mult__cancel1,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N2 ) @ M )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_2836_dvd__mult__cancel2,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N2 @ M ) @ M )
= ( N2
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_2837_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N2 @ K ) ) ) @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ).
% choose_dvd
thf(fact_2838_dvd__minus__add,axiom,
! [Q4: nat,N2: nat,R3: nat,M: nat] :
( ( ord_less_eq @ nat @ Q4 @ N2 )
=> ( ( ord_less_eq @ nat @ Q4 @ ( times_times @ nat @ R3 @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N2 @ Q4 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ ( times_times @ nat @ R3 @ M ) @ Q4 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_2839_mod__nat__eqI,axiom,
! [R3: nat,N2: nat,M: nat] :
( ( ord_less @ nat @ R3 @ N2 )
=> ( ( ord_less_eq @ nat @ R3 @ M )
=> ( ( dvd_dvd @ nat @ N2 @ ( minus_minus @ nat @ M @ R3 ) )
=> ( ( modulo_modulo @ nat @ M @ N2 )
= R3 ) ) ) ) ).
% mod_nat_eqI
thf(fact_2840_power__dvd__imp__le,axiom,
! [I2: nat,M: nat,N2: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I2 @ M ) @ ( power_power @ nat @ I2 @ N2 ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I2 )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% power_dvd_imp_le
thf(fact_2841_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A3 ) ) ) ).
% even_two_times_div_two
thf(fact_2842_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A3: A,B2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ).
% power_mono_odd
thf(fact_2843_odd__pos,axiom,
! [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% odd_pos
thf(fact_2844_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N2 ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% dvd_power_iff_le
thf(fact_2845_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A5: set @ B,F3: B > A,B2: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A7: B] : ( divide_divide @ A @ ( F3 @ A7 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A7: B] : ( dvd_dvd @ A @ B2 @ ( F3 @ A7 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F3
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A7: B] :
~ ( dvd_dvd @ A @ B2 @ ( F3 @ A7 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_2846_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: A,M: 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 @ M ) ) ) ) ) ).
% round_diff_minimal
thf(fact_2847_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B5: A] :
( A3
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_2848_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) ) ) ) ).
% zero_le_even_power
thf(fact_2849_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% zero_le_odd_power
thf(fact_2850_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_2851_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat,A3: A,B2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( power_power @ A @ B2 @ N2 ) ) ) ) ) ).
% power_mono_even
thf(fact_2852_finite__transitivity__chain,axiom,
! [A: $tType,A5: set @ A,R2: A > A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X3: A] :
~ ( R2 @ X3 @ X3 )
=> ( ! [X3: A,Y3: A,Z3: A] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A5 )
& ( R2 @ X3 @ Y6 ) ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_2853_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N2 ) )
= ( ( N2
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_2854_unbounded__k__infinite,axiom,
! [K: nat,S: set @ nat] :
( ! [M5: nat] :
( ( ord_less @ nat @ K @ M5 )
=> ? [N9: nat] :
( ( ord_less @ nat @ M5 @ N9 )
& ( member @ nat @ N9 @ S ) ) )
=> ~ ( finite_finite2 @ nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_2855_infinite__nat__iff__unbounded,axiom,
! [S: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S ) )
= ( ! [M3: nat] :
? [N: nat] :
( ( ord_less @ nat @ M3 @ N )
& ( member @ nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_2856_infinite__nat__iff__unbounded__le,axiom,
! [S: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S ) )
= ( ! [M3: nat] :
? [N: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
& ( member @ nat @ N @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_2857_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ) ).
% even_mask_div_iff'
thf(fact_2858_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N2: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N2 ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_2859_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% even_mask_div_iff
thf(fact_2860_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M: nat,N2: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) )
= ( ( ord_less @ nat @ N2 @ M )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M @ N2 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2861_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_2862_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_ge
thf(fact_2863_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_gt
thf(fact_2864_infinite__int__iff__unbounded__le,axiom,
! [S: set @ int] :
( ( ~ ( finite_finite2 @ int @ S ) )
= ( ! [M3: int] :
? [N: int] :
( ( ord_less_eq @ int @ M3 @ ( abs_abs @ int @ N ) )
& ( member @ int @ N @ S ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_2865_infinite__int__iff__unbounded,axiom,
! [S: set @ int] :
( ( ~ ( finite_finite2 @ int @ S ) )
= ( ! [M3: int] :
? [N: int] :
( ( ord_less @ int @ M3 @ ( abs_abs @ int @ N ) )
& ( member @ int @ N @ S ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_2866_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ X ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_2867_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N2: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N2 ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N2 ) ) ) ).
% round_unique'
thf(fact_2868_in__finite__psubset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A5 @ B4 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
& ( finite_finite2 @ A @ B4 ) ) ) ).
% in_finite_psubset
thf(fact_2869_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_2870_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X4: nat] :
( collect @ nat
@ ^ [N: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% set_decode_def
thf(fact_2871_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N: nat,A7: A] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_2872_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q7: A,R4: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R4 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q7 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R4 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q7 ) @ R4 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_2873_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N2: nat] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% one_mod_2_pow_eq
thf(fact_2874_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_2875_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_2876_of__nat__of__bool,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o] :
( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% of_nat_of_bool
thf(fact_2877_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A5: set @ ( product_prod @ A @ B ),F3: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ A5 )
=> ( member @ C @ ( F3 @ A3 @ B2 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ A5 ) ) ) ).
% pair_imageI
thf(fact_2878_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A3: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( F3 @ A3 @ B2 ) ) ).
% case_prod_conv
thf(fact_2879_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F3 ) )
= F3 ) ).
% curry_case_prod
thf(fact_2880_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F3 ) )
= F3 ) ).
% case_prod_curry
thf(fact_2881_take__bit__of__Suc__0,axiom,
! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% take_bit_of_Suc_0
thf(fact_2882_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_2883_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_2884_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_2885_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_2886_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% take_bit_of_1
thf(fact_2887_sgn__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N2: nat] :
( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N2 ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% sgn_of_nat
thf(fact_2888_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,F3: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X4 ) ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_2889_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,P: B > $o,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X4 ) ) @ ( F3 @ X4 ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_2890_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N2 @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% take_bit_of_exp
thf(fact_2891_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_2892_nested__case__prod__simp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F4: B > C > D > A,X4: product_prod @ B @ C,Y4: D] :
( product_case_prod @ B @ C @ A
@ ^ [A7: B,B6: C] : ( F4 @ A7 @ B6 @ Y4 )
@ X4 ) ) ) ).
% nested_case_prod_simp
thf(fact_2893_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F4: B > C > D > A,X4: product_prod @ B @ C,Y4: D] :
( product_case_prod @ B @ C @ A
@ ^ [L2: B,R4: C] : ( F4 @ L2 @ R4 @ Y4 )
@ X4 ) ) ) ).
% case_prod_app
thf(fact_2894_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X13: A,X23: B] : ( H @ ( F3 @ X13 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_2895_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A,B2: A,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ B2 ) )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_2896_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q4 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ Q4 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_2897_take__bit__nat__less__eq__self,axiom,
! [N2: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ M ) ).
% take_bit_nat_less_eq_self
thf(fact_2898_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_2899_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q4: product_prod @ A @ B,F3: A > B > C,G3: A > B > C,P3: product_prod @ A @ B] :
( ! [X3: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y3 )
= Q4 )
=> ( ( F3 @ X3 @ Y3 )
= ( G3 @ X3 @ Y3 ) ) )
=> ( ( P3 = Q4 )
=> ( ( product_case_prod @ A @ B @ C @ F3 @ P3 )
= ( product_case_prod @ A @ B @ C @ G3 @ Q4 ) ) ) ) ).
% split_cong
thf(fact_2900_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X1: A,X2: B] :
( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= ( F3 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_2901_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P3 )
= P3 ) ).
% case_prod_Pair_iden
thf(fact_2902_take__bit__nat__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ) ).
% take_bit_nat_eq
thf(fact_2903_nat__take__bit__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) )
= ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( nat2 @ K ) ) ) ) ).
% nat_take_bit_eq
thf(fact_2904_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G3: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F3 @ X3 @ Y3 )
= ( G3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F3 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_2905_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y4: B] : ( F3 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_2906_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z2
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_2907_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_2908_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_2909_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N2: nat,K: int] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_2910_take__bit__nonnegative,axiom,
! [N2: nat,K: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ).
% take_bit_nonnegative
thf(fact_2911_take__bit__int__less__eq__self__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_2912_not__take__bit__negative,axiom,
! [N2: nat,K: int] :
~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) ) ).
% not_take_bit_negative
thf(fact_2913_take__bit__int__greater__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% take_bit_int_greater_self_iff
thf(fact_2914_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N2: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N2 @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N2 ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A3 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_2915_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_2916_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_2917_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A3 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_2918_zdvd__antisym__nonneg,axiom,
! [M: int,N2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( dvd_dvd @ int @ M @ N2 )
=> ( ( dvd_dvd @ int @ N2 @ M )
=> ( M = N2 ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_2919_zdvd__not__zless,axiom,
! [M: int,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less @ int @ M @ N2 )
=> ~ ( dvd_dvd @ int @ N2 @ M ) ) ) ).
% zdvd_not_zless
thf(fact_2920_uncurry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( uncurry @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% uncurry_def
thf(fact_2921_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc5280177257484947105e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_2922_finite__divisors__int,axiom,
! [I2: int] :
( ( I2
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D6: int] : ( dvd_dvd @ int @ D6 @ I2 ) ) ) ) ).
% finite_divisors_int
thf(fact_2923_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N2: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N2 @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_2924_zdvd__imp__le,axiom,
! [Z2: int,N2: int] :
( ( dvd_dvd @ int @ Z2 @ N2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ int @ Z2 @ N2 ) ) ) ).
% zdvd_imp_le
thf(fact_2925_dvd__imp__le__int,axiom,
! [I2: int,D3: int] :
( ( I2
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D3 @ I2 )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D3 ) @ ( abs_abs @ int @ I2 ) ) ) ) ).
% dvd_imp_le_int
thf(fact_2926_subset__decode__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
=> ( ord_less_eq @ nat @ M @ N2 ) ) ).
% subset_decode_imp_le
thf(fact_2927_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( ( L
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_2928_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_2929_take__bit__nat__eq__self,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M )
= M ) ) ).
% take_bit_nat_eq_self
thf(fact_2930_take__bit__nat__less__exp,axiom,
! [N2: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% take_bit_nat_less_exp
thf(fact_2931_take__bit__nat__eq__self__iff,axiom,
! [N2: nat,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M )
= M )
= ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_2932_take__bit__int__less__exp,axiom,
! [N2: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% take_bit_int_less_exp
thf(fact_2933_take__bit__nat__less__self__iff,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ M ) @ M )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_2934_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A3: A] :
( ! [A6: A] :
( ( ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: A,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 @ A3 ) ) ) ) ).
% bits_induct
thf(fact_2935_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_2936_take__bit__int__less__self__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_2937_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M3: nat,N: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M3 @ N ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M3 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q7: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q7 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M3 @ N ) @ N ) ) ) ) ) ).
% divmod_nat_if
thf(fact_2938_nat__dvd__iff,axiom,
! [Z2: int,M: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ Z2 ) @ M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( dvd_dvd @ int @ Z2 @ ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_dvd_iff
thf(fact_2939_take__bit__int__eq__self__iff,axiom,
! [N2: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K )
= K )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_2940_take__bit__int__eq__self,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_2941_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N2: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M @ N2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% exp_mod_exp
thf(fact_2942_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_2943_take__bit__int__less__eq,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_2944_take__bit__int__greater__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_2945_even__nat__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat2 @ K ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).
% even_nat_iff
thf(fact_2946_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q7: nat,R4: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R4 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q7 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R4 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q7 ) @ R4 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_2947_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N2: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N2 @ M ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N2 ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_2948_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q7: int,R4: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R4 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q7 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R4 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q7 ) @ R4 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_2949_take__bit__minus__small__eq,axiom,
! [K: int,N2: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N2 @ ( uminus_uminus @ int @ K ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_2950_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N2: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int )
@ ^ [Q7: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ int @ Q7 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N2 ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_2951_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q7: A,R4: A] : ( product_Pair @ A @ A @ Q7 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R4 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_2952_Divides_Oadjust__div__eq,axiom,
! [Q4: int,R3: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q4 @ R3 ) )
= ( plus_plus @ int @ Q4
@ ( zero_neq_one_of_bool @ int
@ ( R3
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_2953_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M3: num,N: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M3 @ N ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M3 ) ) @ ( unique1321980374590559556d_step @ A @ N @ ( unique8689654367752047608divmod @ A @ M3 @ ( bit0 @ N ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2954_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_2955_case__prodI,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B2: B] :
( ( F3 @ A3 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% case_prodI
thf(fact_2956_case__prodI2,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,C2: A > B > $o] :
( ! [A6: A,B5: B] :
( ( P3
= ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( C2 @ A6 @ B5 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P3 ) ) ).
% case_prodI2
thf(fact_2957_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A3: B,B2: C] :
( ( member @ A @ Z2 @ ( C2 @ A3 @ B2 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_2958_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P3: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A6: A,B5: B] :
( ( P3
= ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( member @ C @ Z2 @ ( C2 @ A6 @ B5 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P3 ) ) ) ).
% mem_case_prodI2
thf(fact_2959_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P3: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A6: A,B5: B] :
( ( ( product_Pair @ A @ B @ A6 @ B5 )
= P3 )
=> ( C2 @ A6 @ B5 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P3 @ X ) ) ).
% case_prodI2'
thf(fact_2960_take__bit__num__simps_I2_J,axiom,
! [N2: nat] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(2)
thf(fact_2961_take__bit__num__simps_I5_J,axiom,
! [R3: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R3 ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(5)
thf(fact_2962_take__bit__num__simps_I3_J,axiom,
! [N2: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q7: num] : ( some @ num @ ( bit0 @ Q7 ) )
@ ( bit_take_bit_num @ N2 @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_2963_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num] :
( ( unique8689654367752047608divmod @ A @ M @ one2 )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_2964_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N2 ) ) ) ) ).
% take_bit_numeral_numeral
thf(fact_2965_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P3: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P3 ) )
=> ~ ! [X3: B,Y3: C] :
( ( P3
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_2966_case__prodD,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( F3 @ A3 @ B2 ) ) ).
% case_prodD
thf(fact_2967_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P3: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P3 )
=> ~ ! [X3: A,Y3: B] :
( ( P3
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_2968_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R2: A > B > C > $o,A3: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R2 @ ( product_Pair @ A @ B @ A3 @ B2 ) @ C2 )
=> ( R2 @ A3 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_2969_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P3: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P3 @ Z2 )
=> ~ ! [X3: A,Y3: B] :
( ( P3
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_2970_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q7: int,R4: int] :
( plus_plus @ int @ Q7
@ ( zero_neq_one_of_bool @ int
@ ( R4
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_2971_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A5: A > B > $o,B4: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B4 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_2972_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_2973_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_2974_execute__bind__case,axiom,
! [A: $tType,B: $tType,F3: heap_Time_Heap @ B,G3: B > ( heap_Time_Heap @ A ),H: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( heap_Time_bind @ B @ A @ F3 @ G3 ) @ H )
= ( case_option @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ( product_case_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [X4: B] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H7: heap_ext @ product_unit,N: nat] : ( heap_Time_timeFrame @ A @ N @ ( heap_Time_execute @ A @ ( G3 @ X4 ) @ H7 ) ) ) )
@ ( heap_Time_execute @ B @ F3 @ H ) ) ) ).
% execute_bind_case
thf(fact_2975_map__to__set__def,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K3: A,V3: B] :
( ( M3 @ K3 )
= ( some @ B @ V3 ) ) ) ) ) ) ).
% map_to_set_def
thf(fact_2976_take__bit__num__eq__Some__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N2: num,Q4: num] :
( ( ( bit_take_bit_num @ M @ N2 )
= ( some @ num @ Q4 ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N2 ) )
= ( numeral_numeral @ A @ Q4 ) ) ) ) ).
% take_bit_num_eq_Some_imp
thf(fact_2977_Heap__Time__Monad_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( heap_Time_bind @ A @ B )
= ( ^ [F4: heap_Time_Heap @ A,G4: A > ( heap_Time_Heap @ B )] :
( heap_Time_Heap2 @ B
@ ^ [H6: heap_ext @ product_unit] :
( case_option @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [R4: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H7: heap_ext @ product_unit,N: nat] : ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G4 @ R4 ) @ H7 ) ) ) )
@ ( heap_Time_execute @ A @ F4 @ H6 ) ) ) ) ) ).
% Heap_Time_Monad.bind_def
thf(fact_2978_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A8: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A8 @ B7 )
& ( finite_finite2 @ A @ B7 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_2979_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M3: A > ( option @ B ),P4: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K3: A,V3: B] :
( ( ( M3 @ K3 )
= ( some @ B @ V3 ) )
& ( P4 @ ( product_Pair @ A @ B @ K3 @ V3 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_2980_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M3: num,N: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M3 ) @ ( numeral_numeral @ int @ N ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M3 ) @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% divmod_int_def
thf(fact_2981_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M3: num,N: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M3 ) @ ( numeral_numeral @ A @ N ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M3 ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ) ).
% divmod_def
thf(fact_2982_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M3: num,N: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M3 ) @ ( numeral_numeral @ nat @ N ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M3 ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_2983_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F4: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] :
( ( ord_less @ nat @ ( F4 @ X4 ) @ ( F4 @ Y4 ) )
| ( ( ord_less_eq @ nat @ ( F4 @ X4 ) @ ( F4 @ Y4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_2984_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q7: A,R4: A] : ( product_Pair @ A @ A @ Q7 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R4 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N2 ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_2985_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D6: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z9: int,Z7: int] :
( ( ord_less_eq @ int @ D6 @ Z7 )
& ( ord_less @ int @ Z9 @ Z7 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_2986_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D6: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z9: int,Z7: int] :
( ( ord_less_eq @ int @ D6 @ Z9 )
& ( ord_less @ int @ Z9 @ Z7 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_2987_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( ( ord_less @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_2988_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B6: B] :
( P
& ( Q @ A7 @ B6 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_2989_semiring__norm_I80_J,axiom,
! [M: num,N2: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less @ num @ M @ N2 ) ) ).
% semiring_norm(80)
thf(fact_2990_semiring__norm_I73_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_eq @ num @ M @ N2 ) ) ).
% semiring_norm(73)
thf(fact_2991_semiring__norm_I72_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_eq @ num @ M @ N2 ) ) ).
% semiring_norm(72)
thf(fact_2992_semiring__norm_I81_J,axiom,
! [M: num,N2: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less @ num @ M @ N2 ) ) ).
% semiring_norm(81)
thf(fact_2993_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_2994_semiring__norm_I77_J,axiom,
! [N2: num] : ( ord_less @ num @ one2 @ ( bit1 @ N2 ) ) ).
% semiring_norm(77)
thf(fact_2995_semiring__norm_I74_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less @ num @ M @ N2 ) ) ).
% semiring_norm(74)
thf(fact_2996_semiring__norm_I79_J,axiom,
! [M: num,N2: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( ord_less_eq @ num @ M @ N2 ) ) ).
% semiring_norm(79)
thf(fact_2997_take__bit__num__simps_I4_J,axiom,
! [N2: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_2998_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N2: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_2999_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N2: num] :
( ( ( ord_less_eq @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M @ N2 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N2 ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_3000_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu2: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_3001_xor__num_Ocases,axiom,
! [X: product_prod @ num @ num] :
( ( X
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N5: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N5 ) ) )
=> ( ! [N5: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N5 ) ) )
=> ( ! [M5: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) )
=> ( ! [M5: num,N5: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N5 ) ) )
=> ( ! [M5: num,N5: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N5 ) ) )
=> ( ! [M5: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) )
=> ( ! [M5: num,N5: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N5 ) ) )
=> ~ ! [M5: num,N5: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_3002_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N2: num] :
( ( numeral_numeral @ A @ ( bit1 @ N2 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_3003_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_3004_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A3 @ A3 ) @ A3 ) ) ) ).
% power3_eq_cube
thf(fact_3005_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N2: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_3006_divmod__BitM__2__eq,axiom,
! [M: num] :
( ( unique8689654367752047608divmod @ int @ ( bitM @ M ) @ ( bit0 @ one2 ) )
= ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% divmod_BitM_2_eq
thf(fact_3007_take__bit__num__simps_I6_J,axiom,
! [R3: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R3 ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q7: num] : ( some @ num @ ( bit0 @ Q7 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_3008_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_3009_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N2 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.triangle_reindex
thf(fact_3010_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N: nat,M3: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M3 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ ( numeral_numeral @ nat @ M3 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_3011_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K: A] :
( ( member @ A @ I2 @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I2 @ K ) ) ) ).
% lessThan_iff
thf(fact_3012_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% lessThan_subset_iff
thf(fact_3013_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_3014_less__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_less @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less @ nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_3015_less__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( ord_less @ nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% less_numeral_Suc
thf(fact_3016_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( G3 @ N2 ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_3017_le__Suc__numeral,axiom,
! [N2: nat,K: num] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less_eq @ nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_3018_le__numeral__Suc,axiom,
! [K: num,N2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N2 ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% le_numeral_Suc
thf(fact_3019_take__bit__num__simps_I7_J,axiom,
! [R3: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R3 ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_3020_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
( ( set_ord_lessThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_3021_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_3022_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan @ nat @ N2 )
= ( bot_bot @ ( set @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_3023_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N2: A] :
( ( ( set_ord_lessThan @ A @ N2 )
= ( bot_bot @ ( set @ A ) ) )
= ( N2
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_3024_finite__nat__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S5: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S5 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_3025_finite__nat__bounded,axiom,
! [S: set @ nat] :
( ( finite_finite2 @ nat @ S )
=> ? [K2: nat] : ( ord_less_eq @ ( set @ nat ) @ S @ ( set_ord_lessThan @ nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_3026_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N2: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N2 ) )
= ( ord_less @ A @ M @ N2 ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_3027_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ N2 @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_3028_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ N2 @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_3029_ivl__disj__un__one_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_3030_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_3031_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_3032_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_3033_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N2: A] :
( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N2 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( minus_minus @ nat @ ( P @ X4 ) @ ( Q @ X4 ) )
@ ( set_ord_lessThan @ A @ N2 ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_3034_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_3035_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ ( suc @ N ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_3036_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ N ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F3 @ M ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_3037_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3038_numeral__num__of__nat,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( numeral_numeral @ nat @ ( num_of_nat @ N2 ) )
= N2 ) ) ).
% numeral_num_of_nat
thf(fact_3039_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_3040_num__of__nat__One,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( one_one @ nat ) )
=> ( ( num_of_nat @ N2 )
= one2 ) ) ).
% num_of_nat_One
thf(fact_3041_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_3042_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( F3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_3043_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,K: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M3 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M3 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_3044_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,K: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M3 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M3 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N2 @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_3045_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I4 ) @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.nested_swap'
thf(fact_3046_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I4 ) @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A3 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N2 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% prod.nested_swap'
thf(fact_3047_fact__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: num] :
( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ K ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( semiring_char_0_fact @ A @ ( pred_numeral @ K ) ) ) ) ) ).
% fact_numeral
thf(fact_3048_ivl__disj__un__one_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_3049_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_3050_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N2 ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_3051_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N2: nat] :
( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N2 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_3052_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
=> ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
= ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_3053_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N2 ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_3054_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N2 ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_3055_num__of__nat__double,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( plus_plus @ nat @ N2 @ N2 ) )
= ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).
% num_of_nat_double
thf(fact_3056_num__of__nat__plus__distrib,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M @ N2 ) )
= ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_3057_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) )
= ( semiring_1_of_nat @ A @ N2 ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N2 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_3058_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N2 ) ) @ ( power_power @ A @ Y @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_3059_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N2 ) @ ( power_power @ A @ Y @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N2 @ ( suc @ I4 ) ) ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_3060_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_3061_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N2: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N2 ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_3062_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > nat > A,N2: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ).
% sum.triangle_reindex
thf(fact_3063_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N2 ) ) ) ) ) ) ).
% mask_numeral
thf(fact_3064_horner__sum__of__bool__2__less,axiom,
! [Bs: list @ $o] : ( ord_less @ int @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Bs ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ $o ) @ Bs ) ) ) ).
% horner_sum_of_bool_2_less
thf(fact_3065_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_3066_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_3067_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q7: code_integer,R4: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R4 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q7 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R4 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q7 ) @ R4 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_3068_mask__nat__positive__iff,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% mask_nat_positive_iff
thf(fact_3069_xor__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_3070_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
!= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% xor_negative_int_iff
thf(fact_3071_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_3072_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_3073_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M3: num,N: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M3 ) @ ( numeral_numeral @ code_integer @ N ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M3 ) @ ( numeral_numeral @ code_integer @ N ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_3074_exhaustive__integer_H_Ocases,axiom,
! [X: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F2: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I3: code_integer] :
( X
!= ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F2 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I3 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_3075_full__exhaustive__integer_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F2: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I3: code_integer] :
( X
!= ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F2 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I3 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_3076_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_3077_less__eq__integer__code_I1_J,axiom,
ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ).
% less_eq_integer_code(1)
thf(fact_3078_sgn__integer__code,axiom,
( ( sgn_sgn @ code_integer )
= ( ^ [K3: code_integer] :
( if @ code_integer
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% sgn_integer_code
thf(fact_3079_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_3080_less__eq__mask,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ).
% less_eq_mask
thf(fact_3081_XOR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) ) ) ) ).
% XOR_lower
thf(fact_3082_mask__nonnegative__int,axiom,
! [N2: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N2 ) ) ).
% mask_nonnegative_int
thf(fact_3083_not__mask__negative__int,axiom,
! [N2: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N2 ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_3084_less__mask,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
=> ( ord_less @ nat @ N2 @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) ) ) ).
% less_mask
thf(fact_3085_mask__nat__less__exp,axiom,
! [N2: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ).
% mask_nat_less_exp
thf(fact_3086_finite__int__iff__bounded,axiom,
( ( finite_finite2 @ int )
= ( ^ [S5: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ S5 ) @ ( set_ord_lessThan @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_3087_XOR__upper,axiom,
! [X: int,N2: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% XOR_upper
thf(fact_3088_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A7: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( times_times @ A @ ( F4 @ ( nth @ B @ Xs3 @ N ) ) @ ( power_power @ A @ A7 @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_3089_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K3: int] :
( if @ code_integer @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K3 ) ) )
@ ( if @ code_integer
@ ( K3
= ( zero_zero @ int ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_3090_integer__of__num_I3_J,axiom,
! [N2: num] :
( ( code_integer_of_num @ ( bit1 @ N2 ) )
= ( plus_plus @ code_integer @ ( plus_plus @ code_integer @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) @ ( one_one @ code_integer ) ) ) ).
% integer_of_num(3)
thf(fact_3091_take__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N: nat,M3: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A7: nat,X4: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P6: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q7: num] : ( some @ num @ ( bit0 @ Q7 ) )
@ ( bit_take_bit_num @ O @ P6 ) )
@ ^ [P6: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
@ X4 )
@ A7 )
@ ( product_Pair @ nat @ num @ N @ M3 ) ) ) ) ).
% take_bit_num_code
thf(fact_3092_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A7: B] :
( ( member @ B @ A7 @ A5 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ A7 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_3093_card__Collect__less__nat,axiom,
! [N2: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) )
= N2 ) ).
% card_Collect_less_nat
thf(fact_3094_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_3095_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_3096_card__Collect__le__nat,axiom,
! [N2: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N2 ) ) )
= ( suc @ N2 ) ) ).
% card_Collect_le_nat
thf(fact_3097_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_3098_prod__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: A,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : Y
@ A5 )
= ( power_power @ A @ Y @ ( finite_card @ B @ A5 ) ) ) ) ).
% prod_constant
thf(fact_3099_case__nat__numeral,axiom,
! [A: $tType,A3: A,F3: nat > A,V2: num] :
( ( case_nat @ A @ A3 @ F3 @ ( numeral_numeral @ nat @ V2 ) )
= ( F3 @ ( pred_numeral @ V2 ) ) ) ).
% case_nat_numeral
thf(fact_3100_card__0__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_3101_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y: A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : Y
@ A5 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ Y ) ) ) ).
% sum_constant
thf(fact_3102_case__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F3: nat > A,V2: num,N2: nat] :
( ( case_nat @ A @ A3 @ F3 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) )
= ( F3 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) ) ) ).
% case_nat_add_eq_if
thf(fact_3103_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,P: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( zero_neq_one_of_bool @ A @ ( P @ X4 ) )
@ A5 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_3104_abs__integer__code,axiom,
( ( abs_abs @ code_integer )
= ( ^ [K3: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K3 ) @ K3 ) ) ) ).
% abs_integer_code
thf(fact_3105_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_3106_less__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less @ int @ Xa @ X ) ) ).
% less_integer.abs_eq
thf(fact_3107_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_3108_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F22: nat > A,Nat: nat] :
( ( H @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H @ F1 )
@ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_3109_num_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F22: num > A,F32: num > A,Num: num] :
( ( H @ ( case_num @ A @ F1 @ F22 @ F32 @ Num ) )
= ( case_num @ B @ ( H @ F1 )
@ ^ [X4: num] : ( H @ ( F22 @ X4 ) )
@ ^ [X4: num] : ( H @ ( F32 @ X4 ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_3110_n__subsets,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ A5 )
& ( ( finite_card @ A @ B7 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A5 ) @ K ) ) ) ).
% n_subsets
thf(fact_3111_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F22: nat > A] :
( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(4)
thf(fact_3112_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X2: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X2 ) )
= ( F22 @ X2 ) ) ).
% old.nat.simps(5)
thf(fact_3113_infinite__arbitrarily__large,axiom,
! [A: $tType,A5: set @ A,N2: nat] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [B11: set @ A] :
( ( finite_finite2 @ A @ B11 )
& ( ( finite_card @ A @ B11 )
= N2 )
& ( ord_less_eq @ ( set @ A ) @ B11 @ A5 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_3114_card__subset__eq,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ( finite_card @ A @ A5 )
= ( finite_card @ A @ B4 ) )
=> ( A5 = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_3115_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B4: set @ A,A5: set @ B,R3: B > A > $o] :
( ( finite_finite2 @ A @ B4 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ A5 )
=> ? [B12: A] :
( ( member @ A @ B12 @ B4 )
& ( R3 @ A6 @ B12 ) ) )
=> ( ! [A14: B,A24: B,B5: A] :
( ( member @ B @ A14 @ A5 )
=> ( ( member @ B @ A24 @ A5 )
=> ( ( member @ A @ B5 @ B4 )
=> ( ( R3 @ A14 @ B5 )
=> ( ( R3 @ A24 @ B5 )
=> ( A14 = A24 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ ( finite_card @ A @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_3116_less__eq__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less_eq @ int @ Xa @ X ) ) ).
% less_eq_integer.abs_eq
thf(fact_3117_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S2: set @ A,T6: set @ B,R2: A > B > $o,K: B > nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T6 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S2 )
& ( R2 @ I4 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T6 )
& ( R2 @ I4 @ J3 ) ) ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ nat @ K @ T6 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_3118_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_3119_card__eq__0__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% card_eq_0_iff
thf(fact_3120_card__ge__0__finite,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A5 ) )
=> ( finite_finite2 @ A @ A5 ) ) ).
% card_ge_0_finite
thf(fact_3121_card__image__le,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) @ ( finite_card @ A @ A5 ) ) ) ).
% card_image_le
thf(fact_3122_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F5: set @ A,C4: nat] :
( ! [G6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G6 @ F5 )
=> ( ( finite_finite2 @ A @ G6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G6 ) @ C4 ) ) )
=> ( ( finite_finite2 @ A @ F5 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F5 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_3123_card__seteq,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B4 ) @ ( finite_card @ A @ A5 ) )
=> ( A5 = B4 ) ) ) ) ).
% card_seteq
thf(fact_3124_card__mono,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) ) ) ) ).
% card_mono
thf(fact_3125_obtain__subset__with__card__n,axiom,
! [A: $tType,N2: nat,S: set @ A] :
( ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ S ) )
=> ~ ! [T7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T7 @ S )
=> ( ( ( finite_card @ A @ T7 )
= N2 )
=> ~ ( finite_finite2 @ A @ T7 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_3126_card__less__sym__Diff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_3127_card__le__sym__Diff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_3128_psubset__card__mono,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_3129_card__Un__le,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) ) ) ).
% card_Un_le
thf(fact_3130_card__less__Suc2,axiom,
! [M6: set @ nat,I2: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M6 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M6 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M6 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_3131_card__less__Suc,axiom,
! [M6: set @ nat,I2: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M6 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M6 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M6 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_3132_card__less,axiom,
! [M6: set @ nat,I2: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M6 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M6 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_3133_subset__card__intvl__is__intvl,axiom,
! [A5: set @ nat,K: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A5 ) ) ) )
=> ( A5
= ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A5 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_3134_sum__Suc,axiom,
! [A: $tType,F3: A > nat,A5: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( suc @ ( F3 @ X4 ) )
@ A5 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( finite_card @ A @ A5 ) ) ) ).
% sum_Suc
thf(fact_3135_sum__multicount,axiom,
! [A: $tType,B: $tType,S: set @ A,T2: set @ B,R2: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S )
& ( R2 @ I4 @ X3 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T2 )
& ( R2 @ I4 @ J3 ) ) ) )
@ S )
= ( times_times @ nat @ K @ ( finite_card @ B @ T2 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_3136_card__gt__0__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A5 ) )
= ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A5 ) ) ) ).
% card_gt_0_iff
thf(fact_3137_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,K5: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ K5 @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) ) ) ) ).
% sum_bounded_below
thf(fact_3138_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ K5 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above
thf(fact_3139_surj__card__le,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ ( image2 @ A @ B @ F3 @ A5 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B4 ) @ ( finite_card @ A @ A5 ) ) ) ) ).
% surj_card_le
thf(fact_3140_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( X4 = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_3141_card__Diff__subset,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_3142_card__psubset,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) )
=> ( ord_less @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% card_psubset
thf(fact_3143_card__Un__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_3144_diff__card__le__card__Diff,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_3145_card__Diff__subset__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_3146_subset__eq__atLeast0__lessThan__card,axiom,
! [N7: set @ nat,N2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N7 ) @ N2 ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_3147_card__sum__le__nat__sum,axiom,
! [S: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ S ) ) ).
% card_sum_le_nat_sum
thf(fact_3148_odd__card__imp__not__empty,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A5 ) )
=> ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_3149_card__Un__disjoint,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_3150_integer__of__num_I2_J,axiom,
! [N2: num] :
( ( code_integer_of_num @ ( bit0 @ N2 ) )
= ( plus_plus @ code_integer @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) ) ).
% integer_of_num(2)
thf(fact_3151_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A,N2: A,K: nat] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N2 )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( power_power @ A @ N2 @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_3152_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A5: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_3153_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less @ A @ ( F3 @ I3 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A5 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_3154_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S: set @ A,R2: set @ B,G3: A > B,F3: B > C] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ R2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G3 @ S ) @ R2 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X4: A] : ( F3 @ ( G3 @ X4 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y4: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S )
& ( ( G3 @ X4 )
= Y4 ) ) ) ) )
@ ( F3 @ Y4 ) )
@ R2 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_3155_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A,C2: A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C2 )
@ S )
= ( times_times @ A @ ( B2 @ A3 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C2 )
@ S )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_3156_card__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N2 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A5 ) ) @ ( set_ord_atMost @ nat @ N2 ) ) ) ) ).
% card_lists_length_le
thf(fact_3157_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if @ num @ ( ord_less_eq @ code_integer @ K3 @ ( one_one @ code_integer ) ) @ one2
@ ( product_case_prod @ code_integer @ code_integer @ num
@ ^ [L2: code_integer,J3: code_integer] :
( if @ num
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) )
@ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one2 ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_3158_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
@ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K3 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_3159_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K3: code_integer] :
( if @ int @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K3 ) ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ int )
@ ( product_case_prod @ code_integer @ code_integer @ int
@ ^ [L2: code_integer,J3: code_integer] :
( if @ int
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_3160_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K3 @ L2 ) @ ( modulo_modulo @ code_integer @ K3 @ L2 ) ) ) ) ).
% divmod_integer_def
thf(fact_3161_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_3162_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B4 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_3163_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys3 )
=> ( ( ( set2 @ A @ Ys3 )
= ( set2 @ A @ Xs ) )
=> ( Ys3 = Xs ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_3164_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat
!= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $false
@ ^ [Uu2: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_3165_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat
= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $true
@ ^ [Uu2: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_3166_int__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
= ( ord_less @ code_integer @ X @ Y ) ) ).
% int_of_integer_less_iff
thf(fact_3167_integer__less__iff,axiom,
( ( ord_less @ code_integer )
= ( ^ [K3: code_integer,L2: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_iff
thf(fact_3168_less__integer_Orep__eq,axiom,
( ( ord_less @ code_integer )
= ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_3169_integer__less__eq__iff,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_eq_iff
thf(fact_3170_less__eq__integer_Orep__eq,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_3171_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_3172_nth__mem,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N2 ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_3173_list__ball__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth @ A @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_3174_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_3175_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_3176_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_3177_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_3178_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_3179_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
= ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N2 ) ) ).
% less_eq_nat.simps(2)
thf(fact_3180_in__set__image__conv__nth,axiom,
! [B: $tType,A: $tType,F3: B > A,X: B,L: list @ B] :
( ( member @ A @ ( F3 @ X ) @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ L ) ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ B ) @ L ) )
& ( ( F3 @ ( nth @ B @ L @ I4 ) )
= ( F3 @ X ) ) ) ) ) ).
% in_set_image_conv_nth
thf(fact_3181_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list @ A,L3: list @ A,F3: A > B] :
( ( ( size_size @ ( list @ A ) @ L )
= ( size_size @ ( list @ A ) @ L3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F3 @ ( nth @ A @ L @ I3 ) )
= ( F3 @ ( nth @ A @ L3 @ I3 ) ) ) )
=> ( ( image2 @ A @ B @ F3 @ ( set2 @ A @ L ) )
= ( image2 @ A @ B @ F3 @ ( set2 @ A @ L3 ) ) ) ) ) ).
% set_image_eq_pointwiseI
thf(fact_3182_finite__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N2 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3183_diff__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N2 ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K3: nat] : K3
@ ( minus_minus @ nat @ M @ N2 ) ) ) ).
% diff_Suc
thf(fact_3184_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ~ ! [L4: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L4 )
=> ( ( ( set2 @ A @ L4 )
= A5 )
=> ( ( size_size @ ( list @ A ) @ L4 )
!= ( finite_card @ A @ A5 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_3185_card__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N2 ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A5 ) @ N2 ) ) ) ).
% card_lists_length_eq
thf(fact_3186_finite__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N2 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3187_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( if @ ( product_prod @ code_integer @ $o )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
@ ^ [R4: code_integer,S6: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R4 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ S6 ) )
@ ( S6
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_3188_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if @ nat @ ( ord_less_eq @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
@ ( product_case_prod @ code_integer @ code_integer @ nat
@ ^ [L2: code_integer,J3: code_integer] :
( if @ nat
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( one_one @ nat ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_3189_set__union,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs @ Ys3 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) ) ) ).
% set_union
thf(fact_3190_card__lists__distinct__length__eq,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A5 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_3191_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A5: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A5 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_3192_nat__of__integer__non__positive,axiom,
! [K: code_integer] :
( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) )
=> ( ( code_nat_of_integer @ K )
= ( zero_zero @ nat ) ) ) ).
% nat_of_integer_non_positive
thf(fact_3193_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A5: set @ A,N2: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N2 )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_3194_distinct__finite__set,axiom,
! [A: $tType,X: set @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys2: list @ A] :
( ( ( set2 @ A @ Ys2 )
= X )
& ( distinct @ A @ Ys2 ) ) ) ) ).
% distinct_finite_set
thf(fact_3195_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_3196_distinct__length__le,axiom,
! [A: $tType,Ys3: list @ A,Xs: list @ A] :
( ( distinct @ A @ Ys3 )
=> ( ( ( set2 @ A @ Ys3 )
= ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys3 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% distinct_length_le
thf(fact_3197_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys3 )
=> ( ( distinct @ A @ Ys3 )
=> ( ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys3 ) )
=> ( Xs = Ys3 ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_3198_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list @ A,I2: nat,J: nat] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Xs @ J ) )
= ( I2 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_3199_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs3: list @ A] :
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( I4 != J3 )
=> ( ( nth @ A @ Xs3 @ I4 )
!= ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_3200_finite__set__image,axiom,
! [A: $tType,A5: set @ ( list @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A5 ) )
=> ( ! [Xs2: list @ A] :
( ( member @ ( list @ A ) @ Xs2 @ A5 )
=> ( distinct @ A @ Xs2 ) )
=> ( finite_finite2 @ ( list @ A ) @ A5 ) ) ) ).
% finite_set_image
thf(fact_3201_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_3202_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [X3: list @ A] :
( ( ( set2 @ A @ X3 )
= A5 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X3 )
& ( distinct @ A @ X3 )
& ! [Y6: list @ A] :
( ( ( ( set2 @ A @ Y6 )
= A5 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y6 )
& ( distinct @ A @ Y6 ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_3203_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [X3: nat] :
( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X3 )
= X )
& ! [Y6: nat] :
( ( ( ord_less @ nat @ Y6 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y6 )
= X ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_3204_distinct__finite__subset,axiom,
! [A: $tType,X: set @ A] :
( ( finite_finite2 @ A @ X )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys2 ) @ X )
& ( distinct @ A @ Ys2 ) ) ) ) ) ).
% distinct_finite_subset
thf(fact_3205_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_3206_distinct__sorted__strict__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I2: nat,J: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ A @ ( nth @ A @ L @ I2 ) @ ( nth @ A @ L @ J ) )
= ( ord_less @ nat @ I2 @ J ) ) ) ) ) ) ) ).
% distinct_sorted_strict_mono_iff
thf(fact_3207_distinct__sorted__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I2: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( distinct @ A @ L )
=> ( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ord_less @ A @ ( nth @ A @ L @ I2 ) @ ( nth @ A @ L @ J ) ) ) ) ) ) ) ).
% distinct_sorted_mono
thf(fact_3208_distinct__sorted__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I2: nat,J: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I2 ) @ ( nth @ A @ L @ J ) )
= ( ord_less_eq @ nat @ I2 @ J ) ) ) ) ) ) ) ).
% distinct_sorted_mono_iff
thf(fact_3209_nat__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ Y )
=> ( ( ord_less @ nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
= ( ord_less @ code_integer @ X @ Y ) ) ) ) ).
% nat_of_integer_less_iff
thf(fact_3210_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_3211_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_3212_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L2 )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R4: code_integer,S6: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S6
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L2 @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R4: code_integer,S6: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S6
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L2 ) @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_3213_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_3214_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X6 )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X6 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_3215_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_3216_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys3 ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys3 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_3217_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B,X: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_Pair @ A @ C @ X @ Y ) )
= ( product_Pair @ A @ B @ X @ ( F3 @ Y ) ) ) ).
% apsnd_conv
thf(fact_3218_set__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys3: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys3 ) )
=> ( ( set2 @ A @ Zs )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) ) ) ) ).
% set_shuffles
thf(fact_3219_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_3220_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,Zs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ A @ Ys3 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys3 ) )
=> ( distinct @ A @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_3221_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X4: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_3222_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_3223_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_3224_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Bs: list @ $o,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N2 )
= ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ $o ) @ Bs ) )
& ( nth @ $o @ Bs @ N2 ) ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_3225_rec__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F3: nat > A > A,V2: num,N2: nat] :
( ( rec_nat @ A @ A3 @ F3 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N2 ) )
= ( F3 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) @ ( rec_nat @ A @ A3 @ F3 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N2 ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_3226_bit__0__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( zero_zero @ A ) )
= ( bot_bot @ ( nat > $o ) ) ) ) ).
% bit_0_eq
thf(fact_3227_old_Onat_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: nat > T > T] :
( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(6)
thf(fact_3228_old_Onat_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: nat > T > T,Nat: nat] :
( ( rec_nat @ T @ F1 @ F22 @ ( suc @ Nat ) )
= ( F22 @ Nat @ ( rec_nat @ T @ F1 @ F22 @ Nat ) ) ) ).
% old.nat.simps(7)
thf(fact_3229_signed__take__bit__nonnegative__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_3230_signed__take__bit__negative__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ).
% signed_take_bit_negative_iff
thf(fact_3231_rec__nat__numeral,axiom,
! [A: $tType,A3: A,F3: nat > A > A,V2: num] :
( ( rec_nat @ A @ A3 @ F3 @ ( numeral_numeral @ nat @ V2 ) )
= ( F3 @ ( pred_numeral @ V2 ) @ ( rec_nat @ A @ A3 @ F3 @ ( pred_numeral @ V2 ) ) ) ) ).
% rec_nat_numeral
thf(fact_3232_bit__nat__iff,axiom,
! [K: int,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K ) @ N2 )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) ).
% bit_nat_iff
thf(fact_3233_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) @ N2 )
= ( ( ord_less @ nat @ N2 @ M )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 ) ) ) ) ).
% bit_take_bit_iff
thf(fact_3234_rec__nat__0__imp,axiom,
! [A: $tType,F3: nat > A,F1: A,F22: nat > A > A] :
( ( F3
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F3 @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_3235_rec__nat__Suc__imp,axiom,
! [A: $tType,F3: nat > A,F1: A,F22: nat > A > A,N2: nat] :
( ( F3
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F3 @ ( suc @ N2 ) )
= ( F22 @ N2 @ ( F3 @ N2 ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_3236_bit__numeral__rec_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W2: num,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W2 ) ) @ N2 )
= ( case_nat @ $o @ $false @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W2 ) ) @ N2 ) ) ) ).
% bit_numeral_rec(1)
thf(fact_3237_bit__numeral__rec_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W2: num,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W2 ) ) @ N2 )
= ( case_nat @ $o @ $true @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W2 ) ) @ N2 ) ) ) ).
% bit_numeral_rec(2)
thf(fact_3238_bit__imp__take__bit__positive,axiom,
! [N2: nat,M: nat,K: int] :
( ( ord_less @ nat @ N2 @ M )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ N2 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_3239_int__bit__bound,axiom,
! [K: int] :
~ ! [N5: nat] :
( ! [M4: nat] :
( ( ord_less_eq @ nat @ N5 @ M4 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M4 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N5 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N5 ) ) ) ) ) ).
% int_bit_bound
thf(fact_3240_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A,N2: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N2 )
= ( ( ( N2
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N2 ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_3241_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X23: nat] : X23 ) ) ).
% pred_def
thf(fact_3242_ranI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A3: B,B2: A] :
( ( ( M @ A3 )
= ( some @ A @ B2 ) )
=> ( member @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ).
% ranI
thf(fact_3243_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N: nat,A7: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A7 ) @ N ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A7 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A7 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_3244_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A5 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A5 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A5 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_3245_finite__enumerate,axiom,
! [S: set @ nat] :
( ( finite_finite2 @ nat @ S )
=> ? [R: nat > nat] :
( ( strict_mono_on @ nat @ nat @ R @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S ) ) )
& ! [N9: nat] :
( ( ord_less @ nat @ N9 @ ( finite_card @ nat @ S ) )
=> ( member @ nat @ ( R @ N9 ) @ S ) ) ) ) ).
% finite_enumerate
thf(fact_3246_card__partition,axiom,
! [A: $tType,C4: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ ( set @ A ) @ C4 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
=> ( ! [C3: set @ A] :
( ( member @ ( set @ A ) @ C3 @ C4 )
=> ( ( finite_card @ A @ C3 )
= K ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C4 )
=> ( ( member @ ( set @ A ) @ C22 @ C4 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C4 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_3247_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_3248_or__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% or_nonnegative_int_iff
thf(fact_3249_push__bit__nonnegative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_3250_or__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
| ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% or_negative_int_iff
thf(fact_3251_push__bit__negative__int__iff,axiom,
! [N2: nat,K: int] :
( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% push_bit_negative_int_iff
thf(fact_3252_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastAtMost
thf(fact_3253_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastAtMost
thf(fact_3254_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastLessThan
thf(fact_3255_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastLessThan
thf(fact_3256_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,C2: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A5 ) )
= C2 ) ) ) ).
% cSUP_const
thf(fact_3257_finite__UN__I,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A5 )
=> ( finite_finite2 @ B @ ( B4 @ A6 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3258_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% or_numerals(3)
thf(fact_3259_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ A3 )
= ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_3260_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_3261_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_3262_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z2: A,X6: set @ A] :
( ( member @ A @ Z2 @ X6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= Z2 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_3263_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X6: set @ A,A3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ A3 ) )
=> ( ! [Y3: A] :
( ! [X7: A] :
( ( member @ A @ X7 @ X6 )
=> ( ord_less_eq @ A @ X7 @ Y3 ) )
=> ( ord_less_eq @ A @ A3 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= A3 ) ) ) ) ).
% cSup_eq
thf(fact_3264_Some__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ A @ ( complete_Sup_Sup @ A @ A5 ) )
= ( complete_Sup_Sup @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) ) ) ) ) ).
% Some_Sup
thf(fact_3265_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X6 ) @ Z2 ) ) ) ) ).
% cSup_least
thf(fact_3266_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A3: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ X3 @ A3 ) )
=> ( ! [Y3: A] :
( ! [X7: A] :
( ( member @ A @ X7 @ X6 )
=> ( ord_less_eq @ A @ X7 @ Y3 ) )
=> ( ord_less_eq @ A @ A3 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= A3 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_3267_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,X: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( member @ A @ X @ X6 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X6 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_3268_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X6: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X6 ) )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ) ) ).
% less_cSupE
thf(fact_3269_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X6 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ X6 )
& ( ord_less @ A @ Z2 @ X3 ) ) ) ) ) ).
% less_cSupD
thf(fact_3270_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,X: A,A3: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( member @ A @ X @ X6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less @ A @ X3 @ A3 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X6 ) @ A3 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_3271_OR__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) ) ) ) ).
% OR_lower
thf(fact_3272_or__greater__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) ) ) ).
% or_greater_eq
thf(fact_3273_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N2 )
= ( ( ord_less_eq @ nat @ M @ N2 )
& ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_3274_bit__push__bit__iff__nat,axiom,
! [M: nat,Q4: nat,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q4 ) @ N2 )
= ( ( ord_less_eq @ nat @ M @ N2 )
& ( bit_se5641148757651400278ts_bit @ nat @ Q4 @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_3275_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B,A3: B] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
=> ( ( member @ B @ A3 @ A5 )
=> ( finite_finite2 @ A @ ( B4 @ A3 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_3276_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ B )
=> ! [A5: set @ A,F3: A > B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) )
= ( complete_Sup_Sup @ ( option @ B )
@ ( image2 @ A @ ( option @ B )
@ ^ [X4: A] : ( some @ B @ ( F3 @ X4 ) )
@ A5 ) ) ) ) ) ).
% Some_SUP
thf(fact_3277_card__Union__le__sum__card,axiom,
! [A: $tType,U3: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_3278_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F3: A > B,A5: set @ A,X: A,Y: A] :
( ( strict_mono_on @ A @ B @ F3 @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y @ A5 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_3279_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F3: A > B,A5: set @ A,R3: A,S2: A] :
( ( strict_mono_on @ A @ B @ F3 @ A5 )
=> ( ( member @ A @ R3 @ A5 )
=> ( ( member @ A @ S2 @ A5 )
=> ( ( ord_less @ A @ R3 @ S2 )
=> ( ord_less @ B @ ( F3 @ R3 ) @ ( F3 @ S2 ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_3280_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A5: set @ A,F3: A > B] :
( ! [R: A,S7: A] :
( ( member @ A @ R @ A5 )
=> ( ( member @ A @ S7 @ A5 )
=> ( ( ord_less @ A @ R @ S7 )
=> ( ord_less @ B @ ( F3 @ R ) @ ( F3 @ S7 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F3 @ A5 ) ) ) ).
% strict_mono_onI
thf(fact_3281_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F4: A > B,A8: set @ A] :
! [R4: A,S6: A] :
( ( ( member @ A @ R4 @ A8 )
& ( member @ A @ S6 @ A8 )
& ( ord_less @ A @ R4 @ S6 ) )
=> ( ord_less @ B @ ( F4 @ R4 ) @ ( F4 @ S6 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_3282_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F3: B > ( set @ A ),G3: C > ( set @ B ),X: C,X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( G3 @ X ) ) ) @ X6 )
= ( ord_less_eq @ ( set @ B ) @ ( G3 @ X )
@ ( collect @ B
@ ^ [X4: B] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) ) ) ) ).
% UN_image_subset
thf(fact_3283_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_3284_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_3285_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F3: B > A,M6: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ M6 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ M6 ) ) ) ) ).
% cSUP_least
thf(fact_3286_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,A3: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X6 ) @ A3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X6 )
=> ( ord_less @ A @ X4 @ A3 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_3287_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( member @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ A5 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A5 ) @ A5 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_3288_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S ) ) @ A3 ) ) ) ) ).
% cSup_abs_le
thf(fact_3289_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U3: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ U3 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_3290_Union__image__empty,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= A5 ) ).
% Union_image_empty
thf(fact_3291_UN__le__add__shift,axiom,
! [A: $tType,M6: nat > ( set @ A ),K: nat,N2: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I4: nat] : ( M6 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_ord_atMost @ nat @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M6 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_3292_UN__le__add__shift__strict,axiom,
! [A: $tType,M6: nat > ( set @ A ),K: nat,N2: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I4: nat] : ( M6 @ ( plus_plus @ nat @ I4 @ K ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M6 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N2 @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_3293_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S ) @ L ) ) @ E3 ) ) ) ) ).
% cSup_asclose
thf(fact_3294_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_3295_dvd__partition,axiom,
! [A: $tType,C4: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C4 )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ X3 ) ) )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C4 )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C4 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_3296_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N: nat,A7: A] : ( times_times @ A @ A7 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_3297_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_3298_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,A5: B > ( set @ C ),G3: C > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( finite_finite2 @ C @ ( A5 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X3 ) @ ( A5 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G3 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ G3 @ ( A5 @ X4 ) )
@ I5 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_3299_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,A5: B > ( set @ C ),G3: C > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( finite_finite2 @ C @ ( A5 @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X3 ) @ ( A5 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G3 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ G3 @ ( A5 @ X4 ) )
@ I5 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_3300_card__UN__le,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B )] :
( ( finite_finite2 @ A @ I5 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A5 @ I4 ) )
@ I5 ) ) ) ).
% card_UN_le
thf(fact_3301_UN__le__eq__Un0,axiom,
! [A: $tType,M6: nat > ( set @ A ),N2: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M6 @ ( set_ord_atMost @ nat @ N2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M6 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ) @ ( M6 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_3302_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B )] :
( ( finite_finite2 @ A @ I5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I5 )
=> ( finite_finite2 @ B @ ( A5 @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I5 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ I5 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ ( A5 @ X3 ) @ ( A5 @ Xa3 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A5 @ I4 ) )
@ I5 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_3303_mask__Suc__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N2: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ).
% mask_Suc_double
thf(fact_3304_OR__upper,axiom,
! [X: int,N2: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% OR_upper
thf(fact_3305_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N: nat,A7: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A7 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% take_bit_sum
thf(fact_3306_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ C,A5: C > ( set @ D ),B4: set @ D] :
( ( ( C4
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C4 ) ) @ B4 ) ) ) ) ).
% UN_simps(2)
thf(fact_3307_UN__simps_I3_J,axiom,
! [E: $tType,F: $tType,C4: set @ F,A5: set @ E,B4: F > ( set @ E )] :
( ( ( C4
= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F @ ( set @ E )
@ ^ [X4: F] : ( sup_sup @ ( set @ E ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) )
= ( bot_bot @ ( set @ E ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F @ ( set @ E )
@ ^ [X4: F] : ( sup_sup @ ( set @ E ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) )
= ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F @ ( set @ E ) @ B4 @ C4 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_3308_UN__Un,axiom,
! [A: $tType,B: $tType,M6: B > ( set @ A ),A5: set @ B,B4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M6 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M6 @ A5 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M6 @ B4 ) ) ) ) ).
% UN_Un
thf(fact_3309_UN__constant,axiom,
! [B: $tType,A: $tType,A5: set @ B,C2: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] : C2
@ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] : C2
@ A5 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_3310_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A5 ) )
= F3 ) ) ) ).
% SUP_const
thf(fact_3311_subset__mset_OcSUP__const,axiom,
! [B: $tType,A: $tType,A5: set @ B,C2: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X4: B] : C2
@ A5 ) )
= C2 ) ) ).
% subset_mset.cSUP_const
thf(fact_3312_Sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X4: A] :
( complete_Sup_Sup @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F4: A > B] : ( F4 @ X4 )
@ A8 ) ) ) ) ) ).
% Sup_apply
thf(fact_3313_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( X4
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_3314_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( ( complete_Sup_Sup @ A @ A5 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( X4
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_3315_Union__Un__distrib,axiom,
! [A: $tType,A5: set @ ( set @ A ),B4: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) ) ) ).
% Union_Un_distrib
thf(fact_3316_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_3317_SUP__identity__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A )
=> ! [A5: set @ A] :
( ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [X4: A] : X4
@ A5 ) )
= ( complete_Sup_Sup @ A @ A5 ) ) ) ).
% SUP_identity_eq
thf(fact_3318_SUP__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [F3: C > B > A,A5: set @ C,X: B] :
( ( complete_Sup_Sup @ ( B > A ) @ ( image2 @ C @ ( B > A ) @ F3 @ A5 ) @ X )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [Y4: C] : ( F3 @ Y4 @ X )
@ A5 ) ) ) ) ).
% SUP_apply
thf(fact_3319_UN__iff,axiom,
! [A: $tType,B: $tType,B2: A,B4: B > ( set @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( member @ A @ B2 @ ( B4 @ X4 ) ) ) ) ) ).
% UN_iff
thf(fact_3320_UN__I,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B2: B,B4: A > ( set @ B )] :
( ( member @ A @ A3 @ A5 )
=> ( ( member @ B @ B2 @ ( B4 @ A3 ) )
=> ( member @ B @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ) ).
% UN_I
thf(fact_3321_empty__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( none @ A ) ) ) ).
% empty_Sup
thf(fact_3322_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_3323_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( bot_bot @ A )
@ A5 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_3324_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B4 @ A5 ) )
= ( bot_bot @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B4 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_3325_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B4 @ A5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B4 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_3326_Sup__multiset__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( multiset @ A ) @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% Sup_multiset_empty
thf(fact_3327_Sup__SUP__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( A > $o ) )
= ( ^ [S5: set @ ( A > $o ),X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S5 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_3328_Sup__set__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X4: A] : ( complete_Sup_Sup @ $o @ ( image2 @ ( set @ A ) @ $o @ ( member @ A @ X4 ) @ A8 ) ) ) ) ) ).
% Sup_set_def
thf(fact_3329_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R3: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( R3 @ I4 ) )
@ S ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_3330_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,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ I4 )
@ S ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_3331_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_3332_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S5: set @ ( A > B > $o ),X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( 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 ) @ S5 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_3333_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R3: B > ( set @ A ),S: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I4: B,X4: A] : ( member @ A @ X4 @ ( R3 @ I4 ) )
@ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R3 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_3334_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A5: set @ A] :
( ( Inf
@ ( image2 @ A @ A
@ ^ [X4: A] : X4
@ A5 ) )
= ( Inf @ A5 ) ) ).
% Inf.INF_identity_eq
thf(fact_3335_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A5: set @ A] :
( ( Sup
@ ( image2 @ A @ A
@ ^ [X4: A] : X4
@ A5 ) )
= ( Sup @ A5 ) ) ).
% Sup.SUP_identity_eq
thf(fact_3336_Sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X4: A] :
( complete_Sup_Sup @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F4: A > B] : ( F4 @ X4 )
@ A8 ) ) ) ) ) ).
% Sup_fun_def
thf(fact_3337_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,X: A] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [Y3: A] :
( ! [Z6: A] :
( ( member @ A @ Z6 @ A5 )
=> ( ord_less_eq @ A @ Z6 @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ A5 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_3338_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ! [A6: A] :
( ( member @ A @ A6 @ A5 )
=> ? [X7: A] :
( ( member @ A @ X7 @ B4 )
& ( ord_less_eq @ A @ A6 @ X7 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B4 ) ) ) ) ).
% Sup_mono
thf(fact_3339_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,Z2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ Z2 ) ) ) ).
% Sup_least
thf(fact_3340_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ).
% Sup_upper
thf(fact_3341_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ B2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ B2 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_3342_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A5: set @ A,V2: A] :
( ( member @ A @ U @ A5 )
=> ( ( ord_less_eq @ A @ V2 @ U )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% Sup_upper2
thf(fact_3343_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,S: set @ A] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ S ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S )
& ( ord_less @ A @ A3 @ X4 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_3344_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_3345_Union__empty__conv,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_3346_empty__Union__conv,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A5 ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( X4
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_3347_Union__mono,axiom,
! [A: $tType,A5: set @ ( set @ A ),B4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) ) ) ).
% Union_mono
thf(fact_3348_Union__least,axiom,
! [A: $tType,A5: set @ ( set @ A ),C4: set @ A] :
( ! [X14: set @ A] :
( ( member @ ( set @ A ) @ X14 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ X14 @ C4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) @ C4 ) ) ).
% Union_least
thf(fact_3349_Union__upper,axiom,
! [A: $tType,B4: set @ A,A5: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B4 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) ) ) ).
% Union_upper
thf(fact_3350_Union__subsetI,axiom,
! [A: $tType,A5: set @ ( set @ A ),B4: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A5 )
=> ? [Y6: set @ A] :
( ( member @ ( set @ A ) @ Y6 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ X3 @ Y6 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) ) ) ).
% Union_subsetI
thf(fact_3351_SUP__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > C > A,B4: set @ C,A5: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ ( F3 @ I4 ) @ B4 ) )
@ A5 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [J3: C] :
( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( F3 @ I4 @ J3 )
@ A5 ) )
@ B4 ) ) ) ) ).
% SUP_commute
thf(fact_3352_image__Union,axiom,
! [A: $tType,B: $tType,F3: B > A,S: set @ ( set @ B )] :
( ( image2 @ B @ A @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ S ) ) ) ).
% image_Union
thf(fact_3353_Int__Union2,axiom,
! [A: $tType,B4: set @ ( set @ A ),A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) @ A5 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [C8: set @ A] : ( inf_inf @ ( set @ A ) @ C8 @ A5 )
@ B4 ) ) ) ).
% Int_Union2
thf(fact_3354_Int__Union,axiom,
! [A: $tType,A5: set @ A,B4: set @ ( set @ A )] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 ) @ B4 ) ) ) ).
% Int_Union
thf(fact_3355_UN__UN__flatten,axiom,
! [A: $tType,B: $tType,C: $tType,C4: B > ( set @ A ),B4: C > ( set @ B ),A5: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ C4 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [Y4: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ C4 @ ( B4 @ Y4 ) ) )
@ A5 ) ) ) ).
% UN_UN_flatten
thf(fact_3356_UN__E,axiom,
! [A: $tType,B: $tType,B2: A,B4: B > ( set @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
=> ~ ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ~ ( member @ A @ B2 @ ( B4 @ X3 ) ) ) ) ).
% UN_E
thf(fact_3357_UN__extend__simps_I8_J,axiom,
! [P9: $tType,O2: $tType,B4: O2 > ( set @ P9 ),A5: set @ ( set @ O2 )] :
( ( complete_Sup_Sup @ ( set @ P9 )
@ ( image2 @ ( set @ O2 ) @ ( set @ P9 )
@ ^ [Y4: set @ O2] : ( complete_Sup_Sup @ ( set @ P9 ) @ ( image2 @ O2 @ ( set @ P9 ) @ B4 @ Y4 ) )
@ A5 ) )
= ( complete_Sup_Sup @ ( set @ P9 ) @ ( image2 @ O2 @ ( set @ P9 ) @ B4 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A5 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_3358_UN__extend__simps_I9_J,axiom,
! [S8: $tType,R8: $tType,Q9: $tType,C4: R8 > ( set @ S8 ),B4: Q9 > ( set @ R8 ),A5: set @ Q9] :
( ( complete_Sup_Sup @ ( set @ S8 )
@ ( image2 @ Q9 @ ( set @ S8 )
@ ^ [X4: Q9] : ( complete_Sup_Sup @ ( set @ S8 ) @ ( image2 @ R8 @ ( set @ S8 ) @ C4 @ ( B4 @ X4 ) ) )
@ A5 ) )
= ( complete_Sup_Sup @ ( set @ S8 ) @ ( image2 @ R8 @ ( set @ S8 ) @ C4 @ ( complete_Sup_Sup @ ( set @ R8 ) @ ( image2 @ Q9 @ ( set @ R8 ) @ B4 @ A5 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_3359_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,A5: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A5 ) )
= ( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less @ A @ Y4 @ X4 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_3360_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B4: set @ C,F3: B > A,G3: C > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ? [X7: C] :
( ( member @ C @ X7 @ B4 )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ X7 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B4 )
=> ? [X7: B] :
( ( member @ B @ X7 @ A5 )
& ( ord_less_eq @ A @ ( G3 @ J2 ) @ ( F3 @ X7 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B4 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_3361_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,U: A] :
( ! [V4: A] :
( ( member @ A @ V4 @ A5 )
=> ( ord_less_eq @ A @ U @ V4 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_3362_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B4 ) ) ) ) ).
% Sup_subset_mono
thf(fact_3363_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,X: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ( F3 @ I3 )
= X ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= X ) ) ) ) ).
% SUP_eq_const
thf(fact_3364_Sup__union__distrib,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B4 ) ) ) ) ).
% Sup_union_distrib
thf(fact_3365_Union__disjoint,axiom,
! [A: $tType,C4: set @ ( set @ A ),A5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C4 ) @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ X4 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_3366_Union__Int__subset,axiom,
! [A: $tType,A5: set @ ( set @ A ),B4: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A5 @ B4 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) ) ) ).
% Union_Int_subset
thf(fact_3367_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A,X: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ X ) )
=> ( ! [Y3: A] :
( ! [I: B] :
( ( member @ B @ I @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_3368_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B4: set @ C,F3: B > A,G3: C > A] :
( ! [N5: B] :
( ( member @ B @ N5 @ A5 )
=> ? [X7: C] :
( ( member @ C @ X7 @ B4 )
& ( ord_less_eq @ A @ ( F3 @ N5 ) @ ( G3 @ X7 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B4 ) ) ) ) ) ).
% SUP_mono
thf(fact_3369_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A,U: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_3370_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G3: B > A,A5: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ A5 ) ) ) ) ) ).
% SUP_mono'
thf(fact_3371_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A5: set @ B,F3: B > A] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% SUP_upper
thf(fact_3372_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A5: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_3373_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A5: set @ B,U: A,F3: B > A] :
( ( member @ B @ I2 @ A5 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ I2 ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_3374_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A5: set @ B,Y: A,I2: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ Y )
=> ( ( member @ B @ I2 @ A5 )
=> ( ord_less @ A @ ( F3 @ I2 ) @ Y ) ) ) ) ).
% SUP_lessD
thf(fact_3375_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,F3: B > A,A5: set @ B] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ord_less @ A @ A3 @ ( F3 @ X4 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_3376_SUP__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K: B,I5: set @ B,A5: B > A] :
( ( member @ B @ K @ I5 )
=> ( ( sup_sup @ A @ ( A5 @ K ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ A5 @ I5 ) ) )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ A5 @ I5 ) ) ) ) ) ).
% SUP_absorb
thf(fact_3377_complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A5: set @ B,G3: B > A] :
( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ A5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A7: B] : ( sup_sup @ A @ ( F3 @ A7 ) @ ( G3 @ A7 ) )
@ A5 ) ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_3378_UN__extend__simps_I10_J,axiom,
! [V5: $tType,U4: $tType,T: $tType,B4: U4 > ( set @ V5 ),F3: T > U4,A5: set @ T] :
( ( complete_Sup_Sup @ ( set @ V5 )
@ ( image2 @ T @ ( set @ V5 )
@ ^ [A7: T] : ( B4 @ ( F3 @ A7 ) )
@ A5 ) )
= ( complete_Sup_Sup @ ( set @ V5 ) @ ( image2 @ U4 @ ( set @ V5 ) @ B4 @ ( image2 @ T @ U4 @ F3 @ A5 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_3379_image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,B4: C > ( set @ B ),A5: set @ C] :
( ( image2 @ B @ A @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image2 @ B @ A @ F3 @ ( B4 @ X4 ) )
@ A5 ) ) ) ).
% image_UN
thf(fact_3380_UN__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( bot_bot @ ( set @ A ) )
@ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_3381_UN__empty,axiom,
! [B: $tType,A: $tType,B4: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_3382_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B4 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_3383_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B4 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_3384_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ B4 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ ( A5 @ X4 ) @ B4 ) ) ) ) ).
% UN_subset_iff
thf(fact_3385_UN__upper,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ A @ A3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B4 @ A3 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ).
% UN_upper
thf(fact_3386_UN__least,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: A > ( set @ B ),C4: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B4 @ X3 ) @ C4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) @ C4 ) ) ).
% UN_least
thf(fact_3387_UN__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,F3: A > ( set @ B ),G3: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G3 @ B4 ) ) ) ) ) ).
% UN_mono
thf(fact_3388_UN__extend__simps_I5_J,axiom,
! [I6: $tType,J4: $tType,A5: set @ I6,B4: J4 > ( set @ I6 ),C4: set @ J4] :
( ( inf_inf @ ( set @ I6 ) @ A5 @ ( complete_Sup_Sup @ ( set @ I6 ) @ ( image2 @ J4 @ ( set @ I6 ) @ B4 @ C4 ) ) )
= ( complete_Sup_Sup @ ( set @ I6 )
@ ( image2 @ J4 @ ( set @ I6 )
@ ^ [X4: J4] : ( inf_inf @ ( set @ I6 ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) ) ) ).
% UN_extend_simps(5)
thf(fact_3389_UN__extend__simps_I4_J,axiom,
! [H10: $tType,G: $tType,A5: G > ( set @ H10 ),C4: set @ G,B4: set @ H10] :
( ( inf_inf @ ( set @ H10 ) @ ( complete_Sup_Sup @ ( set @ H10 ) @ ( image2 @ G @ ( set @ H10 ) @ A5 @ C4 ) ) @ B4 )
= ( complete_Sup_Sup @ ( set @ H10 )
@ ( image2 @ G @ ( set @ H10 )
@ ^ [X4: G] : ( inf_inf @ ( set @ H10 ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) ) ) ).
% UN_extend_simps(4)
thf(fact_3390_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: B > ( set @ A ),I5: set @ B] :
( ( inf_inf @ ( set @ A ) @ B4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I4: B] : ( inf_inf @ ( set @ A ) @ B4 @ ( A5 @ I4 ) )
@ I5 ) ) ) ).
% Int_UN_distrib
thf(fact_3391_Int__UN__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B,B4: C > ( set @ A ),J5: set @ C] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B4 @ J5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I4: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J3: C] : ( inf_inf @ ( set @ A ) @ ( A5 @ I4 ) @ ( B4 @ J3 ) )
@ J5 ) )
@ I5 ) ) ) ).
% Int_UN_distrib2
thf(fact_3392_UN__absorb,axiom,
! [B: $tType,A: $tType,K: A,I5: set @ A,A5: A > ( set @ B )] :
( ( member @ A @ K @ I5 )
=> ( ( sup_sup @ ( set @ B ) @ ( A5 @ K ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) ) ) ).
% UN_absorb
thf(fact_3393_UN__Un__distrib,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B4: B > ( set @ A ),I5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I4: B] : ( sup_sup @ ( set @ A ) @ ( A5 @ I4 ) @ ( B4 @ I4 ) )
@ I5 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ I5 ) ) ) ) ).
% UN_Un_distrib
thf(fact_3394_Un__Union__image,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B4: B > ( set @ A ),C4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( sup_sup @ ( set @ A ) @ ( A5 @ X4 ) @ ( B4 @ X4 ) )
@ C4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ C4 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C4 ) ) ) ) ).
% Un_Union_image
thf(fact_3395_UN__extend__simps_I6_J,axiom,
! [L5: $tType,K6: $tType,A5: K6 > ( set @ L5 ),C4: set @ K6,B4: set @ L5] :
( ( minus_minus @ ( set @ L5 ) @ ( complete_Sup_Sup @ ( set @ L5 ) @ ( image2 @ K6 @ ( set @ L5 ) @ A5 @ C4 ) ) @ B4 )
= ( complete_Sup_Sup @ ( set @ L5 )
@ ( image2 @ K6 @ ( set @ L5 )
@ ^ [X4: K6] : ( minus_minus @ ( set @ L5 ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) ) ) ).
% UN_extend_simps(6)
thf(fact_3396_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X )
=> ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ord_less @ A @ Y4 @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_3397_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,C2: A,F3: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ C2 @ ( F3 @ I3 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= C2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( ( F3 @ X4 )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_3398_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B4 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_3399_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B4: set @ B,F3: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B4 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_3400_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,C2: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y4: B] : C2
@ A5 ) )
= ( bot_bot @ A ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y4: B] : C2
@ A5 ) )
= C2 ) ) ) ) ).
% SUP_constant
thf(fact_3401_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_3402_SUP__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M6: B > A,A5: set @ B,B4: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M6 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M6 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M6 @ B4 ) ) ) ) ) ).
% SUP_union
thf(fact_3403_SUP__UNION,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G3: C > ( set @ B ),A5: set @ C] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ G3 @ A5 ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [Y4: C] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( G3 @ Y4 ) ) )
@ A5 ) ) ) ) ).
% SUP_UNION
thf(fact_3404_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C4: set @ C,A5: C > ( set @ D ),B4: set @ D] :
( ( ( C4
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C4 ) ) @ B4 )
= B4 ) )
& ( ( C4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C4 ) ) @ B4 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3405_UN__extend__simps_I3_J,axiom,
! [E: $tType,F: $tType,C4: set @ F,A5: set @ E,B4: F > ( set @ E )] :
( ( ( C4
= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F @ ( set @ E ) @ B4 @ C4 ) ) )
= A5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F @ ( set @ E ) @ B4 @ C4 ) ) )
= ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F @ ( set @ E )
@ ^ [X4: F] : ( sup_sup @ ( set @ E ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3406_finite__subset__Union,axiom,
! [A: $tType,A5: set @ A,B13: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ B13 ) )
=> ~ ! [F11: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F11 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F11 @ B13 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ F11 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_3407_card__UNION,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A5 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A5 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I7: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I7 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I7 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I7: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I7 @ A5 )
& ( I7
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_3408_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,B4: set @ B,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ B4 ) ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B6: B] : ( inf_inf @ A @ ( F3 @ B6 ) @ A3 )
@ B4 ) ) ) ) ).
% SUP_inf
thf(fact_3409_Sup__inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B4: set @ A,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B4 ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [B6: A] : ( inf_inf @ A @ B6 @ A3 )
@ B4 ) ) ) ) ).
% Sup_inf
thf(fact_3410_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F3: B > A,B4: set @ B] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ B4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B6: B] : ( inf_inf @ A @ A3 @ ( F3 @ B6 ) )
@ B4 ) ) ) ) ).
% inf_SUP
thf(fact_3411_INF__identity__eq,axiom,
! [A: $tType] :
( ( complete_Inf @ A )
=> ! [A5: set @ A] :
( ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [X4: A] : X4
@ A5 ) )
= ( complete_Inf_Inf @ A @ A5 ) ) ) ).
% INF_identity_eq
thf(fact_3412_INT__I,axiom,
! [B: $tType,A: $tType,A5: set @ A,B2: B,B4: A > ( set @ B )] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ B @ B2 @ ( B4 @ X3 ) ) )
=> ( member @ B @ B2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ).
% INT_I
thf(fact_3413_INT__iff,axiom,
! [A: $tType,B: $tType,B2: A,B4: B > ( set @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( member @ A @ B2 @ ( B4 @ X4 ) ) ) ) ) ).
% INT_iff
thf(fact_3414_Inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X4: A] :
( complete_Inf_Inf @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F4: A > B] : ( F4 @ X4 )
@ A8 ) ) ) ) ) ).
% Inf_apply
thf(fact_3415_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( ( complete_Inf_Inf @ A @ A5 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A5 )
& ( ord_less @ A @ Y4 @ X4 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_3416_SUP2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A3: A,A5: set @ A,B4: A > B > C > $o,B2: B,C2: C] :
( ( member @ A @ A3 @ A5 )
=> ( ( B4 @ A3 @ B2 @ C2 )
=> ( complete_Sup_Sup @ ( B > C > $o ) @ ( image2 @ A @ ( B > C > $o ) @ B4 @ A5 ) @ B2 @ C2 ) ) ) ).
% SUP2_I
thf(fact_3417_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastAtMost
thf(fact_3418_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastAtMost
thf(fact_3419_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastLessThan
thf(fact_3420_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastLessThan
thf(fact_3421_SUP1__I,axiom,
! [A: $tType,B: $tType,A3: A,A5: set @ A,B4: A > B > $o,B2: B] :
( ( member @ A @ A3 @ A5 )
=> ( ( B4 @ A3 @ B2 )
=> ( complete_Sup_Sup @ ( B > $o ) @ ( image2 @ A @ ( B > $o ) @ B4 @ A5 ) @ B2 ) ) ) ).
% SUP1_I
thf(fact_3422_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_3423_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,C2: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A5 ) )
= C2 ) ) ) ).
% cINF_const
thf(fact_3424_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A5 ) )
= F3 ) ) ) ).
% INF_const
thf(fact_3425_finite__INT,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B )] :
( ? [X7: A] :
( ( member @ A @ X7 @ I5 )
& ( finite_finite2 @ B @ ( A5 @ X7 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) ) ) ).
% finite_INT
thf(fact_3426_INF__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Inf @ A )
=> ! [F3: C > B > A,A5: set @ C,X: B] :
( ( complete_Inf_Inf @ ( B > A ) @ ( image2 @ C @ ( B > A ) @ F3 @ A5 ) @ X )
= ( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [Y4: C] : ( F3 @ Y4 @ X )
@ A5 ) ) ) ) ).
% INF_apply
thf(fact_3427_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y4: B] :
( ( member @ B @ Y4 @ A5 )
& ( ord_less @ A @ ( F3 @ Y4 ) @ X4 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_3428_Compl__INT,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( uminus_uminus @ ( set @ A ) @ ( B4 @ X4 ) )
@ A5 ) ) ) ).
% Compl_INT
thf(fact_3429_Compl__UN,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( uminus_uminus @ ( set @ A ) @ ( B4 @ X4 ) )
@ A5 ) ) ) ).
% Compl_UN
thf(fact_3430_Inter__subset,axiom,
! [A: $tType,A5: set @ ( set @ A ),B4: set @ A] :
( ! [X14: set @ A] :
( ( member @ ( set @ A ) @ X14 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ X14 @ B4 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ B4 ) ) ) ).
% Inter_subset
thf(fact_3431_SUP1__E,axiom,
! [B: $tType,A: $tType,B4: B > A > $o,A5: set @ B,B2: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B4 @ A5 ) @ B2 )
=> ~ ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ~ ( B4 @ X3 @ B2 ) ) ) ).
% SUP1_E
thf(fact_3432_SUP2__E,axiom,
! [A: $tType,C: $tType,B: $tType,B4: C > A > B > $o,A5: set @ C,B2: A,C2: B] :
( ( complete_Sup_Sup @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B4 @ A5 ) @ B2 @ C2 )
=> ~ ! [X3: C] :
( ( member @ C @ X3 @ A5 )
=> ~ ( B4 @ X3 @ B2 @ C2 ) ) ) ).
% SUP2_E
thf(fact_3433_Inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X4: A] :
( complete_Inf_Inf @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F4: A > B] : ( F4 @ X4 )
@ A8 ) ) ) ) ) ).
% Inf_fun_def
thf(fact_3434_Some__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( some @ A @ ( complete_Inf_Inf @ A @ A5 ) )
= ( complete_Inf_Inf @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) ) ) ) ).
% Some_Inf
thf(fact_3435_Inter__lower,axiom,
! [A: $tType,B4: set @ A,A5: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B4 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ B4 ) ) ).
% Inter_lower
thf(fact_3436_Inter__greatest,axiom,
! [A: $tType,A5: set @ ( set @ A ),C4: set @ A] :
( ! [X14: set @ A] :
( ( member @ ( set @ A ) @ X14 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ X14 ) )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) ) ) ).
% Inter_greatest
thf(fact_3437_Inter__anti__mono,axiom,
! [A: $tType,B4: set @ ( set @ A ),A5: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B4 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B4 ) ) ) ).
% Inter_anti_mono
thf(fact_3438_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,X: A] :
( ! [I3: A] :
( ( member @ A @ I3 @ A5 )
=> ( ord_less_eq @ A @ X @ I3 ) )
=> ( ! [Y3: A] :
( ! [I: A] :
( ( member @ A @ I @ A5 )
=> ( ord_less_eq @ A @ Y3 @ I ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ A5 )
= X ) ) ) ) ).
% Inf_eqI
thf(fact_3439_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: set @ A,A5: set @ A] :
( ! [B5: A] :
( ( member @ A @ B5 @ B4 )
=> ? [X7: A] :
( ( member @ A @ X7 @ A5 )
& ( ord_less_eq @ A @ X7 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B4 ) ) ) ) ).
% Inf_mono
thf(fact_3440_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ X ) ) ) ).
% Inf_lower
thf(fact_3441_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A5: set @ A,V2: A] :
( ( member @ A @ U @ A5 )
=> ( ( ord_less_eq @ A @ U @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ V2 ) ) ) ) ).
% Inf_lower2
thf(fact_3442_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: A,A5: set @ A] :
( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ B2 @ X4 ) ) ) ) ) ).
% le_Inf_iff
thf(fact_3443_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,Z2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ A5 ) ) ) ) ).
% Inf_greatest
thf(fact_3444_Inter__Un__distrib,axiom,
! [A: $tType,A5: set @ ( set @ A ),B4: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B4 ) ) ) ).
% Inter_Un_distrib
thf(fact_3445_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S: set @ A,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S ) @ A3 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S )
& ( ord_less @ A @ X4 @ A3 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_3446_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z2: A,X6: set @ A] :
( ( member @ A @ Z2 @ X6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= Z2 ) ) ) ) ).
% cInf_eq_minimum
thf(fact_3447_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X6: set @ A,A3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ A3 @ X3 ) )
=> ( ! [Y3: A] :
( ! [X7: A] :
( ( member @ A @ X7 @ X6 )
=> ( ord_less_eq @ A @ Y3 @ X7 ) )
=> ( ord_less_eq @ A @ Y3 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= A3 ) ) ) ) ).
% cInf_eq
thf(fact_3448_sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B4: set @ A] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ B4 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ A @ A @ ( sup_sup @ A @ A3 ) @ B4 ) ) ) ) ).
% sup_Inf
thf(fact_3449_INF__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > C > A,B4: set @ C,A5: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ ( F3 @ I4 ) @ B4 ) )
@ A5 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [J3: C] :
( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( F3 @ I4 @ J3 )
@ A5 ) )
@ B4 ) ) ) ) ).
% INF_commute
thf(fact_3450_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B13: set @ ( set @ A ),A5: set @ A] :
( ( ( B13
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B13 ) @ A5 )
= A5 ) )
& ( ( B13
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B13 ) @ A5 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A5 )
@ B13 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_3451_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B13: set @ ( set @ A ),A5: set @ A] :
( ( ( B13
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Inf_Inf @ ( set @ A ) @ B13 ) )
= A5 ) )
& ( ( B13
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Inf_Inf @ ( set @ A ) @ B13 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 ) @ B13 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_3452_Un__Inter,axiom,
! [A: $tType,A5: set @ A,B4: set @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( complete_Inf_Inf @ ( set @ A ) @ B4 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 ) @ B4 ) ) ) ).
% Un_Inter
thf(fact_3453_INT__D,axiom,
! [A: $tType,B: $tType,B2: A,B4: B > ( set @ A ),A5: set @ B,A3: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
=> ( ( member @ B @ A3 @ A5 )
=> ( member @ A @ B2 @ ( B4 @ A3 ) ) ) ) ).
% INT_D
thf(fact_3454_INT__E,axiom,
! [A: $tType,B: $tType,B2: A,B4: B > ( set @ A ),A5: set @ B,A3: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
=> ( ~ ( member @ A @ B2 @ ( B4 @ A3 ) )
=> ~ ( member @ B @ A3 @ A5 ) ) ) ).
% INT_E
thf(fact_3455_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,A5: set @ B,G3: C > A,B4: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G3 @ B4 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A7: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [B6: C] : ( sup_sup @ A @ ( F3 @ A7 ) @ ( G3 @ B6 ) )
@ B4 ) )
@ A5 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_3456_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F3: B > A,B4: set @ B] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B4 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B6: B] : ( sup_sup @ A @ A3 @ ( F3 @ B6 ) )
@ B4 ) ) ) ) ).
% sup_INF
thf(fact_3457_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B4: set @ A,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B4 ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [B6: A] : ( sup_sup @ A @ B6 @ A3 )
@ B4 ) ) ) ) ).
% Inf_sup
thf(fact_3458_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,B4: set @ B,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B4 ) ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B6: B] : ( sup_sup @ A @ ( F3 @ B6 ) @ A3 )
@ B4 ) ) ) ) ).
% INF_sup
thf(fact_3459_Some__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A5: set @ B] :
( ( some @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( complete_Inf_Inf @ ( option @ A )
@ ( image2 @ B @ ( option @ A )
@ ^ [X4: B] : ( some @ A @ ( F3 @ X4 ) )
@ A5 ) ) ) ) ).
% Some_INF
thf(fact_3460_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ X )
= ( ! [Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less @ A @ X4 @ Y4 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_3461_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B4: set @ C,G3: C > A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ? [X7: C] :
( ( member @ C @ X7 @ B4 )
& ( ord_less_eq @ A @ ( G3 @ X7 ) @ ( F3 @ I3 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B4 )
=> ? [X7: B] :
( ( member @ B @ X7 @ A5 )
& ( ord_less_eq @ A @ ( F3 @ X7 ) @ ( G3 @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G3 @ B4 ) ) ) ) ) ) ).
% INF_eq
thf(fact_3462_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ X6 ) ) ) ) ) ).
% cInf_greatest
thf(fact_3463_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,A3: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less_eq @ A @ A3 @ X3 ) )
=> ( ! [Y3: A] :
( ! [X7: A] :
( ( member @ A @ X7 @ X6 )
=> ( ord_less_eq @ A @ Y3 @ X7 ) )
=> ( ord_less_eq @ A @ Y3 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= A3 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_3464_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,U: A] :
( ! [V4: A] :
( ( member @ A @ V4 @ A5 )
=> ( ord_less_eq @ A @ V4 @ U ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_3465_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A,X: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( member @ A @ X @ X6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X6 ) @ X ) ) ) ) ).
% cInf_le_finite
thf(fact_3466_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,Z2: A] :
( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X6 ) @ Z2 )
=> ? [X3: A] :
( ( member @ A @ X3 @ X6 )
& ( ord_less @ A @ X3 @ Z2 ) ) ) ) ) ).
% cInf_lessD
thf(fact_3467_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,X: A,A3: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( member @ A @ X @ X6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less @ A @ A3 @ X3 ) )
=> ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X6 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_3468_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B4 ) ) ) ) ).
% Inf_superset_mono
thf(fact_3469_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,X: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ( F3 @ I3 )
= X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= X ) ) ) ) ).
% INF_eq_const
thf(fact_3470_Inf__union__distrib,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B4 ) ) ) ) ).
% Inf_union_distrib
thf(fact_3471_Inter__Un__subset,axiom,
! [A: $tType,A5: set @ ( set @ A ),B4: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B4 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A5 @ B4 ) ) ) ).
% Inter_Un_subset
thf(fact_3472_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,U: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ U @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% INF_greatest
thf(fact_3473_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X4 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_3474_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A5: set @ B,F3: B > A,U: A] :
( ( member @ B @ I2 @ A5 )
=> ( ( ord_less_eq @ A @ ( F3 @ I2 ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_3475_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G3: B > A,A5: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ A5 ) ) ) ) ) ).
% INF_mono'
thf(fact_3476_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A5: set @ B,F3: B > A] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( F3 @ I2 ) ) ) ) ).
% INF_lower
thf(fact_3477_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: set @ B,A5: set @ C,F3: C > A,G3: B > A] :
( ! [M5: B] :
( ( member @ B @ M5 @ B4 )
=> ? [X7: C] :
( ( member @ C @ X7 @ A5 )
& ( ord_less_eq @ A @ ( F3 @ X7 ) @ ( G3 @ M5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B4 ) ) ) ) ) ).
% INF_mono
thf(fact_3478_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,X: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ X @ ( F3 @ I3 ) ) )
=> ( ! [Y3: A] :
( ! [I: B] :
( ( member @ B @ I @ A5 )
=> ( ord_less_eq @ A @ Y3 @ ( F3 @ I ) ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_3479_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ A3 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ord_less @ A @ ( F3 @ X4 ) @ A3 ) ) ) ) ) ).
% INF_less_iff
thf(fact_3480_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A,F3: B > A,A5: set @ B,I2: B] :
( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
=> ( ( member @ B @ I2 @ A5 )
=> ( ord_less @ A @ Y @ ( F3 @ I2 ) ) ) ) ) ).
% less_INF_D
thf(fact_3481_INF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A5: set @ B,G3: B > A] :
( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ A5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A7: B] : ( inf_inf @ A @ ( F3 @ A7 ) @ ( G3 @ A7 ) )
@ A5 ) ) ) ) ).
% INF_inf_distrib
thf(fact_3482_INF__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K: B,I5: set @ B,A5: B > A] :
( ( member @ B @ K @ I5 )
=> ( ( inf_inf @ A @ ( A5 @ K ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ A5 @ I5 ) ) )
= ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ A5 @ I5 ) ) ) ) ) ).
% INF_absorb
thf(fact_3483_INT__extend__simps_I10_J,axiom,
! [V5: $tType,U4: $tType,T: $tType,B4: U4 > ( set @ V5 ),F3: T > U4,A5: set @ T] :
( ( complete_Inf_Inf @ ( set @ V5 )
@ ( image2 @ T @ ( set @ V5 )
@ ^ [A7: T] : ( B4 @ ( F3 @ A7 ) )
@ A5 ) )
= ( complete_Inf_Inf @ ( set @ V5 ) @ ( image2 @ U4 @ ( set @ V5 ) @ B4 @ ( image2 @ T @ U4 @ F3 @ A5 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_3484_INT__lower,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ A @ A3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) @ ( B4 @ A3 ) ) ) ).
% INT_lower
thf(fact_3485_INT__greatest,axiom,
! [B: $tType,A: $tType,A5: set @ A,C4: set @ B,B4: A > ( set @ B )] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ C4 @ ( B4 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C4 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ).
% INT_greatest
thf(fact_3486_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,F3: A > ( set @ B ),G3: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ B4 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G3 @ A5 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_3487_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: B > ( set @ A ),I5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ ( A5 @ X4 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_3488_Int__Inter__image,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B4: B > ( set @ A ),C4: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( inf_inf @ ( set @ A ) @ ( A5 @ X4 ) @ ( B4 @ X4 ) )
@ C4 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ C4 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C4 ) ) ) ) ).
% Int_Inter_image
thf(fact_3489_INT__Int__distrib,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B4: B > ( set @ A ),I5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I4: B] : ( inf_inf @ ( set @ A ) @ ( A5 @ I4 ) @ ( B4 @ I4 ) )
@ I5 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ I5 ) ) ) ) ).
% INT_Int_distrib
thf(fact_3490_INT__absorb,axiom,
! [B: $tType,A: $tType,K: A,I5: set @ A,A5: A > ( set @ B )] :
( ( member @ A @ K @ I5 )
=> ( ( inf_inf @ ( set @ B ) @ ( A5 @ K ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) )
= ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) ) ) ).
% INT_absorb
thf(fact_3491_Un__INT__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B,B4: C > ( set @ A ),J5: set @ C] :
( ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B4 @ J5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I4: B] :
( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J3: C] : ( sup_sup @ ( set @ A ) @ ( A5 @ I4 ) @ ( B4 @ J3 ) )
@ J5 ) )
@ I5 ) ) ) ).
% Un_INT_distrib2
thf(fact_3492_Un__INT__distrib,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: B > ( set @ A ),I5: set @ B] :
( ( sup_sup @ ( set @ A ) @ B4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I4: B] : ( sup_sup @ ( set @ A ) @ B4 @ ( A5 @ I4 ) )
@ I5 ) ) ) ).
% Un_INT_distrib
thf(fact_3493_INT__extend__simps_I6_J,axiom,
! [L5: $tType,K6: $tType,A5: K6 > ( set @ L5 ),C4: set @ K6,B4: set @ L5] :
( ( sup_sup @ ( set @ L5 ) @ ( complete_Inf_Inf @ ( set @ L5 ) @ ( image2 @ K6 @ ( set @ L5 ) @ A5 @ C4 ) ) @ B4 )
= ( complete_Inf_Inf @ ( set @ L5 )
@ ( image2 @ K6 @ ( set @ L5 )
@ ^ [X4: K6] : ( sup_sup @ ( set @ L5 ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) ) ) ).
% INT_extend_simps(6)
thf(fact_3494_INT__extend__simps_I7_J,axiom,
! [M11: $tType,N10: $tType,A5: set @ M11,B4: N10 > ( set @ M11 ),C4: set @ N10] :
( ( sup_sup @ ( set @ M11 ) @ A5 @ ( complete_Inf_Inf @ ( set @ M11 ) @ ( image2 @ N10 @ ( set @ M11 ) @ B4 @ C4 ) ) )
= ( complete_Inf_Inf @ ( set @ M11 )
@ ( image2 @ N10 @ ( set @ M11 )
@ ^ [X4: N10] : ( sup_sup @ ( set @ M11 ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) ) ) ).
% INT_extend_simps(7)
thf(fact_3495_INT__extend__simps_I9_J,axiom,
! [S8: $tType,R8: $tType,Q9: $tType,C4: R8 > ( set @ S8 ),B4: Q9 > ( set @ R8 ),A5: set @ Q9] :
( ( complete_Inf_Inf @ ( set @ S8 )
@ ( image2 @ Q9 @ ( set @ S8 )
@ ^ [X4: Q9] : ( complete_Inf_Inf @ ( set @ S8 ) @ ( image2 @ R8 @ ( set @ S8 ) @ C4 @ ( B4 @ X4 ) ) )
@ A5 ) )
= ( complete_Inf_Inf @ ( set @ S8 ) @ ( image2 @ R8 @ ( set @ S8 ) @ C4 @ ( complete_Sup_Sup @ ( set @ R8 ) @ ( image2 @ Q9 @ ( set @ R8 ) @ B4 @ A5 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_3496_INT__extend__simps_I8_J,axiom,
! [P9: $tType,O2: $tType,B4: O2 > ( set @ P9 ),A5: set @ ( set @ O2 )] :
( ( complete_Inf_Inf @ ( set @ P9 )
@ ( image2 @ ( set @ O2 ) @ ( set @ P9 )
@ ^ [Y4: set @ O2] : ( complete_Inf_Inf @ ( set @ P9 ) @ ( image2 @ O2 @ ( set @ P9 ) @ B4 @ Y4 ) )
@ A5 ) )
= ( complete_Inf_Inf @ ( set @ P9 ) @ ( image2 @ O2 @ ( set @ P9 ) @ B4 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A5 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_3497_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ X )
= ( ! [Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ord_less @ A @ ( F3 @ X4 ) @ Y4 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_3498_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,M: A,F3: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ M @ ( F3 @ X3 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_3499_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,C2: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) )
= C2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( ( F3 @ X4 )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_3500_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A,A3: A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X6 )
=> ( ord_less @ A @ A3 @ X4 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_3501_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ).
% Inf_le_Sup
thf(fact_3502_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( member @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ A5 ) ) )
=> ( member @ A @ ( complete_Inf_Inf @ A @ A5 ) @ A5 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_3503_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,A3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S ) ) @ A3 ) ) ) ) ).
% cInf_abs_ge
thf(fact_3504_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B4 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_3505_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: set @ B,A5: set @ B,F3: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B4 @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B4 )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B4 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_3506_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,X: A,F3: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ X @ ( F3 @ I4 ) )
@ I5 ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3507_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F3: B > A,X: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ ( F3 @ I4 ) @ X )
@ I5 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I5 ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_3508_uminus__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( uminus_uminus @ A @ ( B4 @ X4 ) )
@ A5 ) ) ) ) ).
% uminus_SUP
thf(fact_3509_uminus__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B4 @ A5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( uminus_uminus @ A @ ( B4 @ X4 ) )
@ A5 ) ) ) ) ).
% uminus_INF
thf(fact_3510_INF__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M6: B > A,A5: set @ B,B4: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M6 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M6 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M6 @ B4 ) ) ) ) ) ).
% INF_union
thf(fact_3511_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ D,A5: set @ C,B4: D > ( set @ C )] :
( ( ( C4
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B4 @ C4 ) ) )
= A5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B4 @ C4 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3512_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C4: set @ A,A5: A > ( set @ B ),B4: set @ B] :
( ( ( C4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C4 ) ) @ B4 )
= B4 ) )
& ( ( C4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C4 ) ) @ B4 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3513_INT__Un,axiom,
! [A: $tType,B: $tType,M6: B > ( set @ A ),A5: set @ B,B4: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M6 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M6 @ A5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M6 @ B4 ) ) ) ) ).
% INT_Un
thf(fact_3514_UN__extend__simps_I7_J,axiom,
! [M11: $tType,N10: $tType,A5: set @ M11,B4: N10 > ( set @ M11 ),C4: set @ N10] :
( ( minus_minus @ ( set @ M11 ) @ A5 @ ( complete_Inf_Inf @ ( set @ M11 ) @ ( image2 @ N10 @ ( set @ M11 ) @ B4 @ C4 ) ) )
= ( complete_Sup_Sup @ ( set @ M11 )
@ ( image2 @ N10 @ ( set @ M11 )
@ ^ [X4: N10] : ( minus_minus @ ( set @ M11 ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) ) ) ).
% UN_extend_simps(7)
thf(fact_3515_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_3516_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S: set @ A,L: A,E3: A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S ) @ L ) ) @ E3 ) ) ) ) ).
% cInf_asclose
thf(fact_3517_INT__extend__simps_I4_J,axiom,
! [G: $tType,H10: $tType,C4: set @ H10,A5: set @ G,B4: H10 > ( set @ G )] :
( ( ( C4
= ( bot_bot @ ( set @ H10 ) ) )
=> ( ( minus_minus @ ( set @ G ) @ A5 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image2 @ H10 @ ( set @ G ) @ B4 @ C4 ) ) )
= A5 ) )
& ( ( C4
!= ( bot_bot @ ( set @ H10 ) ) )
=> ( ( minus_minus @ ( set @ G ) @ A5 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image2 @ H10 @ ( set @ G ) @ B4 @ C4 ) ) )
= ( complete_Inf_Inf @ ( set @ G )
@ ( image2 @ H10 @ ( set @ G )
@ ^ [X4: H10] : ( minus_minus @ ( set @ G ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_3518_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B4: set @ A,A3: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B4 ) @ A3 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( ( inf_inf @ A @ X4 @ A3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_3519_inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B4: set @ A] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ B4 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ A @ A @ ( inf_inf @ A @ A3 ) @ B4 ) ) ) ) ).
% inf_Sup
thf(fact_3520_SUP__inf__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,A5: set @ B,G3: C > A,B4: set @ C] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A7: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [B6: C] : ( inf_inf @ A @ ( F3 @ A7 ) @ ( G3 @ B6 ) )
@ B4 ) )
@ A5 ) ) ) ) ).
% SUP_inf_distrib2
thf(fact_3521_UN__UN__split__split__eq,axiom,
! [A: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A5: B > C > D > E > ( set @ A ),Y7: set @ ( product_prod @ D @ E ),X6: set @ ( product_prod @ B @ C )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ B @ C ) @ ( set @ A )
@ ( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X13: B,X23: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( product_prod @ D @ E ) @ ( set @ A ) @ ( product_case_prod @ D @ E @ ( set @ A ) @ ( A5 @ X13 @ X23 ) ) @ Y7 ) ) )
@ X6 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ B @ C ) @ ( set @ A )
@ ^ [X4: product_prod @ B @ C] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ D @ E ) @ ( set @ A )
@ ^ [Y4: product_prod @ D @ E] :
( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X13: B,X23: C] : ( product_case_prod @ D @ E @ ( set @ A ) @ ( A5 @ X13 @ X23 ) @ Y4 )
@ X4 )
@ Y7 ) )
@ X6 ) ) ) ).
% UN_UN_split_split_eq
thf(fact_3522_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F3: nat > ( set @ A ),S: set @ A] :
( ! [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ I3 ) @ S )
=> ( ( finite_finite2 @ A @ S )
=> ( ? [N11: nat] :
( ! [N5: nat] :
( ( ord_less_eq @ nat @ N5 @ N11 )
=> ! [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N11 )
=> ( ( ord_less @ nat @ M5 @ N5 )
=> ( ord_less @ ( set @ A ) @ ( F3 @ M5 ) @ ( F3 @ N5 ) ) ) ) )
& ! [N5: nat] :
( ( ord_less_eq @ nat @ N11 @ N5 )
=> ( ( F3 @ N11 )
= ( 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_3523_UN__constant__eq,axiom,
! [A: $tType,B: $tType,A3: A,A5: set @ A,F3: A > ( set @ B ),C2: set @ B] :
( ( member @ A @ A3 @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( F3 @ X3 )
= C2 ) )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ A5 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_3524_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A5 ) @ B4 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( finite_Fpow @ B @ A5 ) ) @ ( finite_Fpow @ A @ B4 ) ) ) ).
% image_Fpow_mono
thf(fact_3525_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L2
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K3 )
= ( sgn_sgn @ code_integer @ L2 ) )
@ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R4: code_integer,S6: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S6
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_3526_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_3527_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_3528_INF2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A5: set @ A,B4: A > B > C > $o,B2: B,C2: C] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( B4 @ X3 @ B2 @ C2 ) )
=> ( complete_Inf_Inf @ ( B > C > $o ) @ ( image2 @ A @ ( B > C > $o ) @ B4 @ A5 ) @ B2 @ C2 ) ) ).
% INF2_I
thf(fact_3529_INF1__I,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: A > B > $o,B2: B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( B4 @ X3 @ B2 ) )
=> ( complete_Inf_Inf @ ( B > $o ) @ ( image2 @ A @ ( B > $o ) @ B4 @ A5 ) @ B2 ) ) ).
% INF1_I
thf(fact_3530_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_3531_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ ( top_top @ A ) )
= A3 ) ) ).
% inf_top.right_neutral
thf(fact_3532_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B2: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ A3 @ B2 ) )
= ( ( A3
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_3533_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ A3 )
= A3 ) ) ).
% inf_top.left_neutral
thf(fact_3534_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B2: A] :
( ( ( inf_inf @ A @ A3 @ B2 )
= ( top_top @ A ) )
= ( ( A3
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_3535_top__eq__inf__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ X @ Y ) )
= ( ( X
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% top_eq_inf_iff
thf(fact_3536_inf__eq__top__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( top_top @ A ) )
= ( ( X
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% inf_eq_top_iff
thf(fact_3537_inf__top__right,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% inf_top_right
thf(fact_3538_inf__top__left,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% inf_top_left
thf(fact_3539_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_3540_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_3541_boolean__algebra_Odisj__one__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_right
thf(fact_3542_boolean__algebra_Odisj__one__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_left
thf(fact_3543_Int__UNIV,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( top_top @ ( set @ A ) ) )
= ( ( A5
= ( top_top @ ( set @ A ) ) )
& ( B4
= ( top_top @ ( set @ A ) ) ) ) ) ).
% Int_UNIV
thf(fact_3544_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_3545_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_3546_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S6: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S6: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_3547_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
~ ( P @ X4 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_3548_subset__mset_OcINF__const,axiom,
! [B: $tType,A: $tType,A5: set @ B,C2: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X4: B] : C2
@ A5 ) )
= C2 ) ) ).
% subset_mset.cINF_const
thf(fact_3549_None__in__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ ( option @ A )] :
( ( member @ ( option @ A ) @ ( none @ A ) @ A5 )
=> ( ( complete_Inf_Inf @ ( option @ A ) @ A5 )
= ( none @ A ) ) ) ) ).
% None_in_Inf
thf(fact_3550_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( ( complete_Sup_Sup @ A @ A5 )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ X4 @ ( top_top @ A ) )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A5 )
& ( ord_less @ A @ X4 @ Y4 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_3551_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_3552_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_3553_sup__compl__top__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( sup_sup @ A @ X @ Y ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left1
thf(fact_3554_sup__compl__top__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left2
thf(fact_3555_boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_left
thf(fact_3556_boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_right
thf(fact_3557_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_3558_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_3559_Diff__UNIV,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_3560_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_3561_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_3562_INF__top__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( ( top_top @ A )
= ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B4 @ A5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B4 @ X4 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(2)
thf(fact_3563_INF__top__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B4 @ A5 ) )
= ( top_top @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B4 @ X4 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(1)
thf(fact_3564_INF__top,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( top_top @ A )
@ A5 ) )
= ( top_top @ A ) ) ) ).
% INF_top
thf(fact_3565_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ X4 @ ( top_top @ A ) )
=> ? [Y4: B] :
( ( member @ B @ Y4 @ A5 )
& ( ord_less @ A @ X4 @ ( F3 @ Y4 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_3566_INT__constant,axiom,
! [B: $tType,A: $tType,A5: set @ B,C2: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] : C2
@ A5 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] : C2
@ A5 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_3567_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_3568_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C4: set @ D,A5: set @ C,B4: D > ( set @ C )] :
( ( ( C4
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) )
= ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B4 @ C4 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_3569_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ A,A5: A > ( set @ B ),B4: set @ B] :
( ( ( C4
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C4 ) ) @ B4 ) ) ) ) ).
% INT_simps(1)
thf(fact_3570_INT__simps_I3_J,axiom,
! [E: $tType,F: $tType,C4: set @ E,A5: E > ( set @ F ),B4: set @ F] :
( ( ( C4
= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E @ ( set @ F )
@ ^ [X4: E] : ( minus_minus @ ( set @ F ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E @ ( set @ F )
@ ^ [X4: E] : ( minus_minus @ ( set @ F ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E @ ( set @ F ) @ A5 @ C4 ) ) @ B4 ) ) ) ) ).
% INT_simps(3)
thf(fact_3571_INT__simps_I4_J,axiom,
! [G: $tType,H10: $tType,C4: set @ H10,A5: set @ G,B4: H10 > ( set @ G )] :
( ( ( C4
= ( bot_bot @ ( set @ H10 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G )
@ ( image2 @ H10 @ ( set @ G )
@ ^ [X4: H10] : ( minus_minus @ ( set @ G ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) )
= ( top_top @ ( set @ G ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ H10 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G )
@ ( image2 @ H10 @ ( set @ G )
@ ^ [X4: H10] : ( minus_minus @ ( set @ G ) @ A5 @ ( B4 @ X4 ) )
@ C4 ) )
= ( minus_minus @ ( set @ G ) @ A5 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image2 @ H10 @ ( set @ G ) @ B4 @ C4 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_3572_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F3: B > A,X6: set @ B,G1: A > C,G22: A > C] :
( ( ( image2 @ B @ A @ F3 @ X6 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ X6 )
=> ( ( comp @ A @ C @ B @ G1 @ F3 @ X3 )
= ( comp @ A @ C @ B @ G22 @ F3 @ X3 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_3573_Inf__nat__def1,axiom,
! [K5: set @ nat] :
( ( K5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( complete_Inf_Inf @ nat @ K5 ) @ K5 ) ) ).
% Inf_nat_def1
thf(fact_3574_rangeI,axiom,
! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_3575_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,X: B] :
( ( B2
= ( F3 @ X ) )
=> ( member @ A @ B2 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_3576_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_3577_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $true ) ) ).
% UNIV_def
thf(fact_3578_boolean__algebra_Oconj__one__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% boolean_algebra.conj_one_right
thf(fact_3579_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_3580_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_3581_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_3582_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_3583_subset__UNIV,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_3584_Int__UNIV__left,axiom,
! [A: $tType,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_3585_Int__UNIV__right,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) )
= A5 ) ).
% Int_UNIV_right
thf(fact_3586_top__option__def,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ( top_top @ ( option @ A ) )
= ( some @ A @ ( top_top @ A ) ) ) ) ).
% top_option_def
thf(fact_3587_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F3: B > A,G3: C > B,X: C,F12: D > A,G7: E > D,X10: E] :
( ( ( F3 @ ( G3 @ X ) )
= ( F12 @ ( G7 @ X10 ) ) )
=> ( ( comp @ B @ A @ C @ F3 @ G3 @ X )
= ( comp @ D @ A @ E @ F12 @ G7 @ X10 ) ) ) ).
% comp_cong
thf(fact_3588_UNIV__eq__I,axiom,
! [A: $tType,A5: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A5 )
=> ( ( top_top @ ( set @ A ) )
= A5 ) ) ).
% UNIV_eq_I
thf(fact_3589_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_3590_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X: A > B,Y: A > B,F3: C > A] :
( ( X = Y )
=> ( ( comp @ A @ B @ C @ X @ F3 )
= ( comp @ A @ B @ C @ Y @ F3 ) ) ) ).
% comp_cong_left
thf(fact_3591_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > B,Y: A > B,F3: B > C] :
( ( X = Y )
=> ( ( comp @ B @ C @ A @ F3 @ X )
= ( comp @ B @ C @ A @ F3 @ Y ) ) ) ).
% comp_cong_right
thf(fact_3592_fun__comp__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,G3: A > C,Fg: A > B] :
( ( ( comp @ C @ B @ A @ F3 @ G3 )
= Fg )
= ( ! [X4: A] :
( ( F3 @ ( G3 @ X4 ) )
= ( Fg @ X4 ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_3593_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_3594_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( A3
!= ( top_top @ A ) )
= ( ord_less @ A @ A3 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_3595_Un__UNIV__right,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_3596_Un__UNIV__left,axiom,
! [A: $tType,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B4 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_3597_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: C > B,G3: D > C,X: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apsnd @ D @ C @ A @ G3 @ X ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F3 @ G3 ) @ X ) ) ).
% apsnd_compose
thf(fact_3598_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,G3: B > C] :
( ( image2 @ B @ A
@ ^ [X4: B] : ( F3 @ ( G3 @ X4 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image2 @ C @ A @ F3 @ ( image2 @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_3599_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B2
!= ( F3 @ X3 ) ) ) ).
% rangeE
thf(fact_3600_INF2__D,axiom,
! [A: $tType,C: $tType,B: $tType,B4: C > A > B > $o,A5: set @ C,B2: A,C2: B,A3: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B4 @ A5 ) @ B2 @ C2 )
=> ( ( member @ C @ A3 @ A5 )
=> ( B4 @ A3 @ B2 @ C2 ) ) ) ).
% INF2_D
thf(fact_3601_INF2__E,axiom,
! [B: $tType,A: $tType,C: $tType,B4: C > A > B > $o,A5: set @ C,B2: A,C2: B,A3: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B4 @ A5 ) @ B2 @ C2 )
=> ( ~ ( B4 @ A3 @ B2 @ C2 )
=> ~ ( member @ C @ A3 @ A5 ) ) ) ).
% INF2_E
thf(fact_3602_INF1__D,axiom,
! [B: $tType,A: $tType,B4: B > A > $o,A5: set @ B,B2: A,A3: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B4 @ A5 ) @ B2 )
=> ( ( member @ B @ A3 @ A5 )
=> ( B4 @ A3 @ B2 ) ) ) ).
% INF1_D
thf(fact_3603_INF1__E,axiom,
! [A: $tType,B: $tType,B4: B > A > $o,A5: set @ B,B2: A,A3: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B4 @ A5 ) @ B2 )
=> ( ~ ( B4 @ A3 @ B2 )
=> ~ ( member @ B @ A3 @ A5 ) ) ) ).
% INF1_E
thf(fact_3604_UN__atMost__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atMost_UNIV
thf(fact_3605_UN__lessThan__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_lessThan_UNIV
thf(fact_3606_sup__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left1
thf(fact_3607_sup__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( sup_sup @ A @ X @ B2 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left2
thf(fact_3608_Inf__sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B4: set @ A,A3: A] :
( ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B4 ) @ A3 )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( ( sup_sup @ A @ X4 @ A3 )
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_3609_empty__in__Fpow,axiom,
! [A: $tType,A5: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A5 ) ) ).
% empty_in_Fpow
thf(fact_3610_Inf__multiset__empty,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( multiset @ A ) @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% Inf_multiset_empty
thf(fact_3611_range__subsetD,axiom,
! [B: $tType,A: $tType,F3: B > A,B4: set @ A,I2: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) @ B4 )
=> ( member @ A @ ( F3 @ I2 ) @ B4 ) ) ).
% range_subsetD
thf(fact_3612_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_3613_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H ) ) ) ).
% not_UNIV_le_Icc
thf(fact_3614_not__UNIV__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H ) ) ) ).
% not_UNIV_le_Iic
thf(fact_3615_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_3616_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_3617_INF__filter__not__bot,axiom,
! [I6: $tType,A: $tType,B4: set @ I6,F5: I6 > ( filter @ A )] :
( ! [X14: set @ I6] :
( ( ord_less_eq @ ( set @ I6 ) @ X14 @ B4 )
=> ( ( finite_finite2 @ I6 @ X14 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F5 @ X14 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F5 @ B4 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_3618_Compl__partition2,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ A5 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_3619_Compl__partition,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_3620_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_3621_SUP__INF,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y4: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [X4: C] : ( P @ X4 @ Y4 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( B > C ) @ A
@ ^ [X4: B > C] :
( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y4: B] : ( P @ ( X4 @ Y4 ) @ Y4 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% SUP_INF
thf(fact_3622_INF__SUP,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y4: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [X4: C] : ( P @ X4 @ Y4 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( B > C ) @ A
@ ^ [F4: B > C] :
( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( P @ ( F4 @ X4 ) @ X4 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% INF_SUP
thf(fact_3623_Fpow__not__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_Fpow @ A @ A5 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Fpow_not_empty
thf(fact_3624_Inter__empty,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Inter_empty
thf(fact_3625_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G3: B > A,F3: A > C] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image2 @ B @ C
@ ^ [X4: B] : ( F3 @ ( G3 @ X4 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_3626_Inf__INT__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( A > $o ) )
= ( ^ [S5: set @ ( A > $o ),X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S5 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_3627_INTER__UNIV__conv_I2_J,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B4 @ X4 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_3628_INTER__UNIV__conv_I1_J,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( ( top_top @ ( set @ A ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( B4 @ X4 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_3629_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% sup_shunt
thf(fact_3630_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A3: A,X: A,Y: A] :
( ( ( inf_inf @ A @ A3 @ X )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ X )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A3 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ Y )
= ( top_top @ A ) )
=> ( X = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_3631_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_3632_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_3633_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_3634_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,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ I4 )
@ S ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% INF_Int_eq2
thf(fact_3635_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_3636_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,C2: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y4: B] : C2
@ A5 ) )
= ( top_top @ A ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y4: B] : C2
@ A5 ) )
= C2 ) ) ) ) ).
% INF_constant
thf(fact_3637_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( image2 @ B @ A @ F3 @ ( uminus_uminus @ ( set @ B ) @ A5 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_3638_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_3639_INF__INT__eq,axiom,
! [B: $tType,A: $tType,R3: B > ( set @ A ),S: set @ B] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I4: B,X4: A] : ( member @ A @ X4 @ ( R3 @ I4 ) )
@ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R3 @ S ) ) ) ) ) ).
% INF_INT_eq
thf(fact_3640_finite__range__Some,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_range_Some
thf(fact_3641_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R3: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( R3 @ I4 ) )
@ S ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_3642_Inf__set__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X4: A] : ( complete_Inf_Inf @ $o @ ( image2 @ ( set @ A ) @ $o @ ( member @ A @ X4 ) @ A8 ) ) ) ) ) ).
% Inf_set_def
thf(fact_3643_notin__range__Some,axiom,
! [A: $tType,X: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_3644_INT__empty,axiom,
! [B: $tType,A: $tType,B4: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_3645_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
=> ( ( uminus_uminus @ A @ X )
= Y ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_3646_Fpow__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A5 ) @ ( finite_Fpow @ A @ B4 ) ) ) ).
% Fpow_mono
thf(fact_3647_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_3648_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_3649_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A8: set @ A] :
( collect @ ( set @ A )
@ ^ [X11: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X11 @ A8 )
& ( finite_finite2 @ A @ X11 ) ) ) ) ) ).
% Fpow_def
thf(fact_3650_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I5: set @ C,G3: A > B,F3: C > A] :
( ( finite_finite2 @ C @ I5 )
=> ( ! [I3: C] :
( ( member @ C @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G3 @ ( F3 @ I3 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G3 @ ( image2 @ C @ A @ F3 @ I5 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G3 @ F3 ) @ I5 ) ) ) ) ) ).
% sum_image_le
thf(fact_3651_INT__extend__simps_I3_J,axiom,
! [F: $tType,E: $tType,C4: set @ E,A5: E > ( set @ F ),B4: set @ F] :
( ( ( C4
= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E @ ( set @ F ) @ A5 @ C4 ) ) @ B4 )
= ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B4 ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E @ ( set @ F ) @ A5 @ C4 ) ) @ B4 )
= ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E @ ( set @ F )
@ ^ [X4: E] : ( minus_minus @ ( set @ F ) @ ( A5 @ X4 ) @ B4 )
@ C4 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_3652_range__mod,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( image2 @ nat @ nat
@ ^ [M3: nat] : ( modulo_modulo @ nat @ M3 @ N2 )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% range_mod
thf(fact_3653_UN__UN__finite__eq,axiom,
! [A: $tType,A5: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [N: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_3654_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_3655_UN__finite__subset,axiom,
! [A: $tType,A5: nat > ( set @ A ),C4: set @ A] :
( ! [N5: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) @ C4 ) ) ).
% UN_finite_subset
thf(fact_3656_UN__finite2__eq,axiom,
! [A: $tType,A5: nat > ( set @ A ),B4: nat > ( set @ A ),K: nat] :
( ! [N5: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N5 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B4 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_3657_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S5: set @ ( A > B > $o ),X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( 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 ) @ S5 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_3658_UN__finite2__subset,axiom,
! [A: $tType,A5: nat > ( set @ A ),B4: nat > ( set @ A ),K: nat] :
( ! [N5: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N5 @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B4 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_3659_Inf__option__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ ( option @ A ) )
= ( ^ [A8: set @ ( option @ A )] : ( if @ ( option @ A ) @ ( member @ ( option @ A ) @ ( none @ A ) @ A8 ) @ ( none @ A ) @ ( some @ A @ ( complete_Inf_Inf @ A @ ( these @ A @ A8 ) ) ) ) ) ) ) ).
% Inf_option_def
thf(fact_3660_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_3661_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N: nat,A7: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A7 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A7 @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_3662_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A7: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N: nat] : ( compow @ ( A > A ) @ N @ ( times_times @ A @ A7 ) @ ( F4 @ ( nth @ B @ Xs3 @ N ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_3663_and__not__num_Osimps_I8_J,axiom,
! [M: num,N2: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N12: num] : ( some @ num @ ( bit1 @ N12 ) )
@ ( bit_and_not_num @ M @ N2 ) ) ) ).
% and_not_num.simps(8)
thf(fact_3664_merge__true__star,axiom,
( ( times_times @ assn @ ( top_top @ assn ) @ ( top_top @ assn ) )
= ( top_top @ assn ) ) ).
% merge_true_star
thf(fact_3665_assn__basic__inequalities_I5_J,axiom,
( ( top_top @ assn )
!= ( bot_bot @ assn ) ) ).
% assn_basic_inequalities(5)
thf(fact_3666_assn__basic__inequalities_I1_J,axiom,
( ( top_top @ assn )
!= ( one_one @ assn ) ) ).
% assn_basic_inequalities(1)
thf(fact_3667_Suc__funpow,axiom,
! [N2: nat] :
( ( compow @ ( nat > nat ) @ N2 @ suc )
= ( plus_plus @ nat @ N2 ) ) ).
% Suc_funpow
thf(fact_3668_and__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% and_nonnegative_int_iff
thf(fact_3669_and__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% and_negative_int_iff
thf(fact_3670_funpow__0,axiom,
! [A: $tType,F3: A > A,X: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F3 @ X )
= X ) ).
% funpow_0
thf(fact_3671_mod__true,axiom,
! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( top_top @ assn ) @ H )
= ( in_range @ H ) ) ).
% mod_true
thf(fact_3672_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B6: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B6: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_3673_not__negative__int__iff,axiom,
! [K: int] :
( ( ord_less @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% not_negative_int_iff
thf(fact_3674_not__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% not_nonnegative_int_iff
thf(fact_3675_surj__fn,axiom,
! [A: $tType,F3: A > A,N2: nat] :
( ( ( image2 @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ ( compow @ ( A > A ) @ N2 @ F3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_3676_these__image__Some__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( these @ A @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) )
= A5 ) ).
% these_image_Some_eq
thf(fact_3677_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_3678_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(3)
thf(fact_3679_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(6)
thf(fact_3680_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ).
% and_numerals(4)
thf(fact_3681_and__minus__numerals_I4_J,axiom,
! [M: num,N2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_3682_and__minus__numerals_I8_J,axiom,
! [N2: num,M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_3683_and__minus__numerals_I3_J,axiom,
! [M: num,N2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_3684_and__minus__numerals_I7_J,axiom,
! [N2: num,M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_3685_in__Union__o__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X: A,Gset: B > ( set @ ( set @ A ) ),Gmap: C > B,A5: C] :
( ( member @ A @ X @ ( comp @ B @ ( set @ A ) @ C @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ B @ ( complete_Sup_Sup @ ( set @ A ) ) @ Gset ) @ Gmap @ A5 ) )
=> ( member @ A @ X @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( comp @ B @ ( set @ ( set @ A ) ) @ C @ Gset @ Gmap ) @ A5 ) ) ) ).
% in_Union_o_assoc
thf(fact_3686_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I5: set @ A,F5: A > ( filter @ B )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I5 )
=> ? [X7: A] :
( ( member @ A @ X7 @ I5 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X7 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ I3 ) @ ( F5 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ I5 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ I5 )
& ( ( F5 @ X4 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_3687_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: A > C,G3: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu2: C] : ( bot_bot @ ( set @ B ) )
@ F3 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image2 @ D @ B @ G3 )
@ ^ [Uu2: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_3688_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_3689_Inf__filter__not__bot,axiom,
! [A: $tType,B4: set @ ( filter @ A )] :
( ! [X14: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X14 @ B4 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X14 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X14 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B4 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_3690_comp__funpow,axiom,
! [B: $tType,A: $tType,N2: nat,F3: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N2 @ ( comp @ A @ A @ B @ F3 ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N2 @ F3 ) ) ) ).
% comp_funpow
thf(fact_3691_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_3692_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_3693_ent__true,axiom,
! [P: assn] : ( entails @ P @ ( top_top @ assn ) ) ).
% ent_true
thf(fact_3694_funpow__swap1,axiom,
! [A: $tType,F3: A > A,N2: nat,X: A] :
( ( F3 @ ( compow @ ( A > A ) @ N2 @ F3 @ X ) )
= ( compow @ ( A > A ) @ N2 @ F3 @ ( F3 @ X ) ) ) ).
% funpow_swap1
thf(fact_3695_and__not__num__eq__Some__iff,axiom,
! [M: num,N2: num,Q4: num] :
( ( ( bit_and_not_num @ M @ N2 )
= ( some @ num @ Q4 ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( numeral_numeral @ int @ Q4 ) ) ) ).
% and_not_num_eq_Some_iff
thf(fact_3696_funpow__mult,axiom,
! [A: $tType,N2: nat,M: nat,F3: A > A] :
( ( compow @ ( A > A ) @ N2 @ ( compow @ ( A > A ) @ M @ F3 ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N2 ) @ F3 ) ) ).
% funpow_mult
thf(fact_3697_int__numeral__not__and__num,axiom,
! [M: num,N2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N2 @ M ) ) ) ).
% int_numeral_not_and_num
thf(fact_3698_int__numeral__and__not__num,axiom,
! [M: num,N2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% int_numeral_and_not_num
thf(fact_3699_Union__natural,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( comp @ ( set @ ( set @ B ) ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ B ) ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F3 ) ) )
= ( comp @ ( set @ A ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( image2 @ A @ B @ F3 ) @ ( complete_Sup_Sup @ ( set @ A ) ) ) ) ).
% Union_natural
thf(fact_3700_funpow__times__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [F3: A > nat,X: A] :
( ( compow @ ( A > A ) @ ( F3 @ X ) @ ( times_times @ A @ X ) )
= ( times_times @ A @ ( power_power @ A @ X @ ( F3 @ X ) ) ) ) ) ).
% funpow_times_power
thf(fact_3701_funpow_Osimps_I2_J,axiom,
! [A: $tType,N2: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N2 ) @ F3 )
= ( comp @ A @ A @ A @ F3 @ ( compow @ ( A > A ) @ N2 @ F3 ) ) ) ).
% funpow.simps(2)
thf(fact_3702_funpow__Suc__right,axiom,
! [A: $tType,N2: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N2 ) @ F3 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N2 @ F3 ) @ F3 ) ) ).
% funpow_Suc_right
thf(fact_3703_in__these__eq,axiom,
! [A: $tType,X: A,A5: set @ ( option @ A )] :
( ( member @ A @ X @ ( these @ A @ A5 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X ) @ A5 ) ) ).
% in_these_eq
thf(fact_3704_funpow__add,axiom,
! [A: $tType,M: nat,N2: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M @ N2 ) @ F3 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M @ F3 ) @ ( compow @ ( A > A ) @ N2 @ F3 ) ) ) ).
% funpow_add
thf(fact_3705_AND__lower,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) ) ) ).
% AND_lower
thf(fact_3706_AND__upper1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ X ) ) ).
% AND_upper1
thf(fact_3707_AND__upper2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Y ) ) ).
% AND_upper2
thf(fact_3708_AND__upper1_H,axiom,
! [Y: int,Z2: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z2 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z2 ) ) ) ).
% AND_upper1'
thf(fact_3709_AND__upper2_H,axiom,
! [Y: int,Z2: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z2 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z2 ) ) ) ).
% AND_upper2'
thf(fact_3710_mod__star__trueI,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ P @ H )
=> ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H ) ) ).
% mod_star_trueI
thf(fact_3711_mod__star__trueE,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H )
=> ~ ! [H5: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ( rep_assn @ P @ H5 ) ) ).
% mod_star_trueE
thf(fact_3712_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_3713_top__assn__def,axiom,
( ( top_top @ assn )
= ( abs_assn @ in_range ) ) ).
% top_assn_def
thf(fact_3714_SUP__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: $o > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ $o @ A @ A5 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ A @ ( A5 @ $true ) @ ( A5 @ $false ) ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_3715_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N: nat,P4: A > A > $o,X4: A,Y4: A] :
? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= X4 )
& ( ( F4 @ N )
= Y4 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P4 @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_3716_INF__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: $o > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ $o @ A @ A5 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ A @ ( A5 @ $true ) @ ( A5 @ $false ) ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_3717_Un__eq__UN,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ $o @ ( set @ A )
@ ^ [B6: $o] : ( if @ ( set @ A ) @ B6 @ A8 @ B7 )
@ ( top_top @ ( set @ $o ) ) ) ) ) ) ).
% Un_eq_UN
thf(fact_3718_UN__bool__eq,axiom,
! [A: $tType,A5: $o > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ $o @ ( set @ A ) @ A5 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ ( set @ A ) @ ( A5 @ $true ) @ ( A5 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_3719_INT__bool__eq,axiom,
! [A: $tType,A5: $o > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ $o @ ( set @ A ) @ A5 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ ( set @ A ) @ ( A5 @ $true ) @ ( A5 @ $false ) ) ) ).
% INT_bool_eq
thf(fact_3720_and__less__eq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ).
% and_less_eq
thf(fact_3721_AND__upper1_H_H,axiom,
! [Y: int,Z2: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z2 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z2 ) ) ) ).
% AND_upper1''
thf(fact_3722_AND__upper2_H_H,axiom,
! [Y: int,Z2: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z2 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z2 ) ) ) ).
% AND_upper2''
thf(fact_3723_relpowp__bot,axiom,
! [A: $tType,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( A > A > $o ) @ N2 @ ( bot_bot @ ( A > A > $o ) ) )
= ( bot_bot @ ( A > A > $o ) ) ) ) ).
% relpowp_bot
thf(fact_3724_mod__h__bot__iff_I2_J,axiom,
! [H: heap_ext @ product_unit] : ( rep_assn @ ( top_top @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_h_bot_iff(2)
thf(fact_3725_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N: nat] : ( compow @ ( A > A ) @ N @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_3726_and__not__num_Osimps_I2_J,axiom,
! [N2: num] :
( ( bit_and_not_num @ one2 @ ( bit0 @ N2 ) )
= ( some @ num @ one2 ) ) ).
% and_not_num.simps(2)
thf(fact_3727_and__not__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ one2 )
= ( some @ num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(4)
thf(fact_3728_take__bit__not__mask__eq__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_not_mask_eq_0
thf(fact_3729_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C4: set @ ( set @ B ),G3: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C4 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C4 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C4 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ C4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G3 @ C4 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_3730_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_3731_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_3732_and__not__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ one2 )
= ( some @ num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(7)
thf(fact_3733_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_3734_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M: nat,N2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_3735_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_3736_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M: nat,N2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_3737_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_3738_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_3739_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C4: set @ ( set @ B ),G3: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C4 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ C4 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C4 )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ C4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G3 @ C4 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_3740_Some__image__these__eq,axiom,
! [A: $tType,A5: set @ ( option @ A )] :
( ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A5 ) )
= ( collect @ ( option @ A )
@ ^ [X4: option @ A] :
( ( member @ ( option @ A ) @ X4 @ A5 )
& ( X4
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_3741_execute__ureturn,axiom,
! [A: $tType,X: A] :
( ( heap_Time_execute @ A @ ( heap_Time_ureturn @ A @ X ) )
= ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H6: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( zero_zero @ nat ) ) ) ) ) ).
% execute_ureturn
thf(fact_3742_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_3743_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( member @ int @ K3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_3744_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_3745_insertCI,axiom,
! [A: $tType,A3: A,B4: set @ A,B2: A] :
( ( ~ ( member @ A @ A3 @ B4 )
=> ( A3 = B2 ) )
=> ( member @ A @ A3 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_3746_insert__iff,axiom,
! [A: $tType,A3: A,B2: A,A5: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B2 @ A5 ) )
= ( ( A3 = B2 )
| ( member @ A @ A3 @ A5 ) ) ) ).
% insert_iff
thf(fact_3747_insert__absorb2,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A5 ) )
= ( insert @ A @ X @ A5 ) ) ).
% insert_absorb2
thf(fact_3748_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_3749_image__insert,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: B,B4: set @ B] :
( ( image2 @ B @ A @ F3 @ ( insert @ B @ A3 @ B4 ) )
= ( insert @ A @ ( F3 @ A3 ) @ ( image2 @ B @ A @ F3 @ B4 ) ) ) ).
% image_insert
thf(fact_3750_insert__image,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F3: A > B] :
( ( member @ A @ X @ A5 )
=> ( ( insert @ B @ ( F3 @ X ) @ ( image2 @ A @ B @ F3 @ A5 ) )
= ( image2 @ A @ B @ F3 @ A5 ) ) ) ).
% insert_image
thf(fact_3751_singletonI,axiom,
! [A: $tType,A3: A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_3752_insert__subset,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B4 )
= ( ( member @ A @ X @ B4 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% insert_subset
thf(fact_3753_Int__insert__right__if1,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B4 ) )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_3754_Int__insert__right__if0,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ~ ( member @ A @ A3 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_3755_insert__inter__insert,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A5 ) @ ( insert @ A @ A3 @ B4 ) )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_3756_Int__insert__left__if1,axiom,
! [A: $tType,A3: A,C4: set @ A,B4: set @ A] :
( ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B4 ) @ C4 )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_3757_Int__insert__left__if0,axiom,
! [A: $tType,A3: A,C4: set @ A,B4: set @ A] :
( ~ ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B4 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_3758_Un__insert__right,axiom,
! [A: $tType,A5: set @ A,A3: A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B4 ) )
= ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_3759_Un__insert__left,axiom,
! [A: $tType,A3: A,B4: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ B4 ) @ C4 )
= ( insert @ A @ A3 @ ( sup_sup @ ( set @ A ) @ B4 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_3760_Diff__insert0,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B4 ) )
= ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_3761_insert__Diff1,axiom,
! [A: $tType,X: A,B4: set @ A,A5: set @ A] :
( ( member @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_3762_top2I,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% top2I
thf(fact_3763_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A3 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_3764_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ A3 ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_3765_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A3: A,A5: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A3 @ A5 ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_3766_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A3: A,A5: set @ A,B2: A] :
( ( ( insert @ A @ A3 @ A5 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_3767_cSup__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cSup_singleton
thf(fact_3768_cInf__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cInf_singleton
thf(fact_3769_insert__disjoint_I1_J,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A5 ) @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B4 )
& ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_3770_insert__disjoint_I2_J,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ A5 ) @ B4 ) )
= ( ~ ( member @ A @ A3 @ B4 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_3771_disjoint__insert_I1_J,axiom,
! [A: $tType,B4: set @ A,A3: A,A5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B4 @ ( insert @ A @ A3 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A3 @ B4 )
& ( ( inf_inf @ ( set @ A ) @ B4 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_3772_disjoint__insert_I2_J,axiom,
! [A: $tType,A5: set @ A,B2: A,B4: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ B4 ) ) )
= ( ~ ( member @ A @ B2 @ A5 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_3773_Sup__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: A,A5: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ A3 @ A5 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ).
% Sup_insert
thf(fact_3774_insert__Diff__single,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A3 @ A5 ) ) ).
% insert_Diff_single
thf(fact_3775_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( B2 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_3776_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
( ( set_or1337092689740270186AtMost @ A @ A3 @ A3 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_3777_Inf__insert,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: A,A5: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert @ A @ A3 @ A5 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A5 ) ) ) ) ).
% Inf_insert
thf(fact_3778_these__insert__Some,axiom,
! [A: $tType,X: A,A5: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X ) @ A5 ) )
= ( insert @ A @ X @ ( these @ A @ A5 ) ) ) ).
% these_insert_Some
thf(fact_3779_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ) ) ).
% prod.insert
thf(fact_3780_subset__Compl__singleton,axiom,
! [A: $tType,A5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_3781_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A] :
( ( minus_minus @ ( set @ A ) @ ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
= ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_3782_range__constant,axiom,
! [B: $tType,A: $tType,X: A] :
( ( image2 @ B @ A
@ ^ [Uu2: B] : X
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_3783_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ B,A3: A,B4: B > ( set @ A )] :
( ( ( C4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A3 @ ( B4 @ X4 ) )
@ C4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A3 @ ( B4 @ X4 ) )
@ C4 ) )
= ( insert @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C4 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_3784_UN__singleton,axiom,
! [A: $tType,A5: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X4: A] : ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= A5 ) ).
% UN_singleton
thf(fact_3785_UN__insert,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A3: B,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( insert @ B @ A3 @ A5 ) ) )
= ( sup_sup @ ( set @ A ) @ ( B4 @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% UN_insert
thf(fact_3786_INT__insert,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A3: B,A5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( insert @ B @ A3 @ A5 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B4 @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% INT_insert
thf(fact_3787_map__update__eta__repair_I2_J,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A,V2: B] :
( ( ( M @ K )
= ( none @ B ) )
=> ( ( ran @ A @ B
@ ^ [X4: A] : ( if @ ( option @ B ) @ ( X4 = K ) @ ( some @ B @ V2 ) @ ( M @ X4 ) ) )
= ( insert @ B @ V2 @ ( ran @ A @ B @ M ) ) ) ) ).
% map_update_eta_repair(2)
thf(fact_3788_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A3: A,B2: A] :
( ( ( finite_card @ A @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( A3 != B2 ) ) ).
% card_doubleton_eq_2_iff
thf(fact_3789_insert__UNIV,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% insert_UNIV
thf(fact_3790_singleton__inject,axiom,
! [A: $tType,A3: A,B2: A] :
( ( ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A3 = B2 ) ) ).
% singleton_inject
thf(fact_3791_insert__not__empty,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( insert @ A @ A3 @ A5 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_3792_doubleton__eq__iff,axiom,
! [A: $tType,A3: A,B2: A,C2: A,D3: A] :
( ( ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A3 = C2 )
& ( B2 = D3 ) )
| ( ( A3 = D3 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_3793_singleton__iff,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member @ A @ B2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A3 ) ) ).
% singleton_iff
thf(fact_3794_singletonD,axiom,
! [A: $tType,B2: A,A3: A] :
( ( member @ A @ B2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A3 ) ) ).
% singletonD
thf(fact_3795_Int__insert__right,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ( ( member @ A @ A3 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B4 ) )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) )
& ( ~ ( member @ A @ A3 @ A5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_3796_Int__insert__left,axiom,
! [A: $tType,A3: A,C4: set @ A,B4: set @ A] :
( ( ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B4 ) @ C4 )
= ( insert @ A @ A3 @ ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) )
& ( ~ ( member @ A @ A3 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A3 @ B4 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B4 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_3797_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F8: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F7 @ F8 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F8 @ F7 ) ) ) ) ).
% less_filter_def
thf(fact_3798_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A7: A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A7 )
| ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_3799_insert__Collect,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( insert @ A @ A3 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U2: A] :
( ( U2 != A3 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_3800_insert__mono,axiom,
! [A: $tType,C4: set @ A,D4: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ C4 ) @ ( insert @ A @ A3 @ D4 ) ) ) ).
% insert_mono
thf(fact_3801_subset__insert,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% subset_insert
thf(fact_3802_subset__insertI,axiom,
! [A: $tType,B4: set @ A,A3: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert @ A @ A3 @ B4 ) ) ).
% subset_insertI
thf(fact_3803_subset__insertI2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% subset_insertI2
thf(fact_3804_insert__subsetI,axiom,
! [A: $tType,X: A,A5: set @ A,X6: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X6 ) @ A5 ) ) ) ).
% insert_subsetI
thf(fact_3805_insert__Diff__if,axiom,
! [A: $tType,X: A,B4: set @ A,A5: set @ A] :
( ( ( member @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) )
& ( ~ ( member @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ B4 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_3806_insertE,axiom,
! [A: $tType,A3: A,B2: A,A5: set @ A] :
( ( member @ A @ A3 @ ( insert @ A @ B2 @ A5 ) )
=> ( ( A3 != B2 )
=> ( member @ A @ A3 @ A5 ) ) ) ).
% insertE
thf(fact_3807_insertI1,axiom,
! [A: $tType,A3: A,B4: set @ A] : ( member @ A @ A3 @ ( insert @ A @ A3 @ B4 ) ) ).
% insertI1
thf(fact_3808_insertI2,axiom,
! [A: $tType,A3: A,B4: set @ A,B2: A] :
( ( member @ A @ A3 @ B4 )
=> ( member @ A @ A3 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% insertI2
thf(fact_3809_Set_Oset__insert,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ~ ! [B11: set @ A] :
( ( A5
= ( insert @ A @ X @ B11 ) )
=> ( member @ A @ X @ B11 ) ) ) ).
% Set.set_insert
thf(fact_3810_insert__ident,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ~ ( member @ A @ X @ B4 )
=> ( ( ( insert @ A @ X @ A5 )
= ( insert @ A @ X @ B4 ) )
= ( A5 = B4 ) ) ) ) ).
% insert_ident
thf(fact_3811_insert__absorb,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ( insert @ A @ A3 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_3812_insert__eq__iff,axiom,
! [A: $tType,A3: A,A5: set @ A,B2: A,B4: set @ A] :
( ~ ( member @ A @ A3 @ A5 )
=> ( ~ ( member @ A @ B2 @ B4 )
=> ( ( ( insert @ A @ A3 @ A5 )
= ( insert @ A @ B2 @ B4 ) )
= ( ( ( A3 = B2 )
=> ( A5 = B4 ) )
& ( ( A3 != B2 )
=> ? [C8: set @ A] :
( ( A5
= ( insert @ A @ B2 @ C8 ) )
& ~ ( member @ A @ B2 @ C8 )
& ( B4
= ( insert @ A @ A3 @ C8 ) )
& ~ ( member @ A @ A3 @ C8 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_3813_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A5 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A5 ) ) ) ).
% insert_commute
thf(fact_3814_mk__disjoint__insert,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ? [B11: set @ A] :
( ( A5
= ( insert @ A @ A3 @ B11 ) )
& ~ ( member @ A @ A3 @ B11 ) ) ) ).
% mk_disjoint_insert
thf(fact_3815_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P @ X4 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A3 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_3816_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P @ X4 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A3 = X4 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_3817_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A7: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] : X4 = A7 ) ) ) ) ).
% insert_def
thf(fact_3818_finite_Ocases,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A13: set @ A] :
( ? [A6: A] :
( A3
= ( insert @ A @ A6 @ A13 ) )
=> ~ ( finite_finite2 @ A @ A13 ) ) ) ) ).
% finite.cases
thf(fact_3819_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] :
( ( A7
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,B6: A] :
( ( A7
= ( insert @ A @ B6 @ A8 ) )
& ( finite_finite2 @ A @ A8 ) ) ) ) ) ).
% finite.simps
thf(fact_3820_finite__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F9: set @ A] :
( ( finite_finite2 @ A @ F9 )
=> ( ~ ( member @ A @ X3 @ F9 )
=> ( ( P @ F9 )
=> ( P @ ( insert @ A @ X3 @ F9 ) ) ) ) )
=> ( P @ F5 ) ) ) ) ).
% finite_induct
thf(fact_3821_finite__ne__induct,axiom,
! [A: $tType,F5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] : ( P @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A,F9: set @ A] :
( ( finite_finite2 @ A @ F9 )
=> ( ( F9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X3 @ F9 )
=> ( ( P @ F9 )
=> ( P @ ( insert @ A @ X3 @ F9 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_3822_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A5: set @ A] :
( ! [A13: set @ A] :
( ~ ( finite_finite2 @ A @ A13 )
=> ( P @ A13 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,F9: set @ A] :
( ( finite_finite2 @ A @ F9 )
=> ( ~ ( member @ A @ X3 @ F9 )
=> ( ( P @ F9 )
=> ( P @ ( insert @ A @ X3 @ F9 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ).
% infinite_finite_induct
thf(fact_3823_subset__singletonD,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_3824_subset__singleton__iff,axiom,
! [A: $tType,X6: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X6
= ( bot_bot @ ( set @ A ) ) )
| ( X6
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_3825_Sup__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A3: A,A5: set @ A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ A3 @ A5 ) )
= ( sup_sup @ A @ A3 @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ).
% Sup_finite_insert
thf(fact_3826_singleton__Un__iff,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_3827_Un__singleton__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_3828_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A7: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_3829_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_3830_Diff__insert2,axiom,
! [A: $tType,A5: set @ A,A3: A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_3831_insert__Diff,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_3832_Diff__insert,axiom,
! [A: $tType,A5: set @ A,A3: A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_3833_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X6: set @ A] :
( ~ ( member @ A @ X @ X6 )
=> ( ( minus_minus @ ( set @ A ) @ X6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X6 ) ) ).
% set_minus_singleton_eq
thf(fact_3834_insert__minus__eq,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( X != Y )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% insert_minus_eq
thf(fact_3835_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_3836_Inf__finite__insert,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ! [A3: A,A5: set @ A] :
( ( complete_Inf_Inf @ A @ ( insert @ A @ A3 @ A5 ) )
= ( inf_inf @ A @ A3 @ ( complete_Inf_Inf @ A @ A5 ) ) ) ) ).
% Inf_finite_insert
thf(fact_3837_subset__Diff__insert,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X: A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ ( insert @ A @ X @ C4 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B4 @ C4 ) )
& ~ ( member @ A @ X @ A5 ) ) ) ).
% subset_Diff_insert
thf(fact_3838_card__insert__le,axiom,
! [A: $tType,A5: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ ( insert @ A @ X @ A5 ) ) ) ).
% card_insert_le
thf(fact_3839_image__constant__conv,axiom,
! [B: $tType,A: $tType,A5: set @ B,C2: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X4: B] : C2
@ A5 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_3840_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A5: set @ A,C2: B] :
( ( member @ A @ X @ A5 )
=> ( ( image2 @ A @ B
@ ^ [X4: A] : C2
@ A5 )
= ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_3841_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A5: set @ A,A3: B,B4: A > ( set @ B )] :
( ( member @ A @ U @ A5 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( insert @ B @ A3 @ ( B4 @ X4 ) )
@ A5 ) )
= ( insert @ B @ A3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_3842_INT__extend__simps_I5_J,axiom,
! [I6: $tType,J4: $tType,A3: I6,B4: J4 > ( set @ I6 ),C4: set @ J4] :
( ( insert @ I6 @ A3 @ ( complete_Inf_Inf @ ( set @ I6 ) @ ( image2 @ J4 @ ( set @ I6 ) @ B4 @ C4 ) ) )
= ( complete_Inf_Inf @ ( set @ I6 )
@ ( image2 @ J4 @ ( set @ I6 )
@ ^ [X4: J4] : ( insert @ I6 @ A3 @ ( B4 @ X4 ) )
@ C4 ) ) ) ).
% INT_extend_simps(5)
thf(fact_3843_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A5: set @ A,A3: B,B4: A > ( set @ B )] :
( ( member @ A @ U @ A5 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( insert @ B @ A3 @ ( B4 @ X4 ) )
@ A5 ) )
= ( insert @ B @ A3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_3844_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 ) ) )
=> ( ! [X3: B,S9: set @ B] :
( ( finite_finite2 @ B @ S9 )
=> ( ! [Y6: B] :
( ( member @ B @ Y6 @ S9 )
=> ( ord_less_eq @ A @ ( F3 @ Y6 ) @ ( F3 @ X3 ) ) )
=> ( ( P @ S9 )
=> ( P @ ( insert @ B @ X3 @ S9 ) ) ) ) )
=> ( P @ S ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_3845_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A13: set @ A] :
( ( finite_finite2 @ A @ A13 )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ A13 )
=> ( ord_less @ A @ X7 @ B5 ) )
=> ( ( P @ A13 )
=> ( P @ ( insert @ A @ B5 @ A13 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_3846_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A13: set @ A] :
( ( finite_finite2 @ A @ A13 )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ A13 )
=> ( ord_less @ A @ B5 @ X7 ) )
=> ( ( P @ A13 )
=> ( P @ ( insert @ A @ B5 @ A13 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_3847_finite__subset__induct,axiom,
! [A: $tType,F5: set @ A,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F9: set @ A] :
( ( finite_finite2 @ A @ F9 )
=> ( ( member @ A @ A6 @ A5 )
=> ( ~ ( member @ A @ A6 @ F9 )
=> ( ( P @ F9 )
=> ( P @ ( insert @ A @ A6 @ F9 ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_3848_finite__subset__induct_H,axiom,
! [A: $tType,F5: set @ A,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A5 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A,F9: set @ A] :
( ( finite_finite2 @ A @ F9 )
=> ( ( member @ A @ A6 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F9 @ A5 )
=> ( ~ ( member @ A @ A6 @ F9 )
=> ( ( P @ F9 )
=> ( P @ ( insert @ A @ A6 @ F9 ) ) ) ) ) ) )
=> ( P @ F5 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_3849_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: A,X: B] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F3 @ X )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_3850_infinite__remove,axiom,
! [A: $tType,S: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_3851_infinite__coinduct,axiom,
! [A: $tType,X6: ( set @ A ) > $o,A5: set @ A] :
( ( X6 @ A5 )
=> ( ! [A13: set @ A] :
( ( X6 @ A13 )
=> ? [X7: A] :
( ( member @ A @ X7 @ A13 )
& ( ( X6 @ ( minus_minus @ ( set @ A ) @ A13 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A13 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% infinite_coinduct
thf(fact_3852_finite__empty__induct,axiom,
! [A: $tType,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P @ A5 )
=> ( ! [A6: A,A13: set @ A] :
( ( finite_finite2 @ A @ A13 )
=> ( ( member @ A @ A6 @ A13 )
=> ( ( P @ A13 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A13 @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_3853_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_3854_Diff__single__insert,axiom,
! [A: $tType,A5: set @ A,X: A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_3855_subset__insert__iff,axiom,
! [A: $tType,A5: set @ A,X: A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B4 ) )
= ( ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_3856_card__1__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( one_one @ nat ) )
=> ~ ! [X3: A] :
( A5
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_3857_remove__subset,axiom,
! [A: $tType,X: A,S: set @ A] :
( ( member @ A @ X @ S )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ S ) ) ).
% remove_subset
thf(fact_3858_Union__image__insert,axiom,
! [A: $tType,B: $tType,F3: B > ( set @ A ),A3: B,B4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( insert @ B @ A3 @ B4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( F3 @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ B4 ) ) ) ) ).
% Union_image_insert
thf(fact_3859_Compl__insert,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_3860_in__image__insert__iff,axiom,
! [A: $tType,B4: set @ ( set @ A ),X: A,A5: set @ A] :
( ! [C6: set @ A] :
( ( member @ ( set @ A ) @ C6 @ B4 )
=> ~ ( member @ A @ X @ C6 ) )
=> ( ( member @ ( set @ A ) @ A5 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ B4 ) )
= ( ( member @ A @ X @ A5 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_3861_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A3: B,A5: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( insert @ B @ A3 @ A5 ) ) )
= ( sup_sup @ A @ ( F3 @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% SUP_insert
thf(fact_3862_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A3: B,A5: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( insert @ B @ A3 @ A5 ) ) )
= ( inf_inf @ A @ ( F3 @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% INF_insert
thf(fact_3863_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C4: set @ B,A3: A,B4: B > ( set @ A )] :
( ( ( C4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C4 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A3 @ ( B4 @ X4 ) )
@ C4 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_3864_finite__remove__induct,axiom,
! [A: $tType,B4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A13: set @ A] :
( ( finite_finite2 @ A @ A13 )
=> ( ( A13
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A13 @ B4 )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ A13 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A13 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A13 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_3865_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B4: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B4 )
=> ( P @ B4 ) )
=> ( ! [A13: set @ A] :
( ( finite_finite2 @ A @ A13 )
=> ( ( A13
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A13 @ B4 )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ A13 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A13 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A13 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_3866_card__Suc__eq,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K ) )
= ( ? [B6: A,B7: set @ A] :
( ( A5
= ( insert @ A @ B6 @ B7 ) )
& ~ ( member @ A @ B6 @ B7 )
& ( ( finite_card @ A @ B7 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B7
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_3867_card__eq__SucD,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K ) )
=> ? [B5: A,B11: set @ A] :
( ( A5
= ( insert @ A @ B5 @ B11 ) )
& ~ ( member @ A @ B5 @ B11 )
& ( ( finite_card @ A @ B11 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B11
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_3868_card__1__singleton__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X4: A] :
( A5
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_3869_card__le__Suc__iff,axiom,
! [A: $tType,N2: nat,A5: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( finite_card @ A @ A5 ) )
= ( ? [A7: A,B7: set @ A] :
( ( A5
= ( insert @ A @ A7 @ B7 ) )
& ~ ( member @ A @ A7 @ B7 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ B7 ) )
& ( finite_finite2 @ A @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_3870_card__1__singletonI,axiom,
! [A: $tType,S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( finite_card @ A @ S )
= ( one_one @ nat ) )
=> ( ( member @ A @ X @ S )
=> ( S
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_1_singletonI
thf(fact_3871_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K: A] :
( ( ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_3872_card__Diff1__le,axiom,
! [A: $tType,A5: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ).
% card_Diff1_le
thf(fact_3873_finite__induct__select,axiom,
! [A: $tType,S: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T7: set @ A] :
( ( ord_less @ ( set @ A ) @ T7 @ S )
=> ( ( P @ T7 )
=> ? [X7: A] :
( ( member @ A @ X7 @ ( minus_minus @ ( set @ A ) @ S @ T7 ) )
& ( P @ ( insert @ A @ X7 @ T7 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_3874_psubset__insert__iff,axiom,
! [A: $tType,A5: set @ A,X: A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B4 ) )
= ( ( ( member @ A @ X @ B4 )
=> ( ord_less @ ( set @ A ) @ A5 @ B4 ) )
& ( ~ ( member @ A @ X @ B4 )
=> ( ( ( member @ A @ X @ A5 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_3875_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A7: A,B6: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A7 @ B6 ) @ ( insert @ A @ B6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_3876_sum__diff1__nat,axiom,
! [A: $tType,A3: A,A5: set @ A,F3: A > nat] :
( ( ( member @ A @ A3 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ A @ A3 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) ) ) ) ).
% sum_diff1_nat
thf(fact_3877_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [U: A] :
( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_ord_atMost @ A @ U ) ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_3878_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N2: int] :
( ( ord_less_eq @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N2 ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N2 ) )
= ( insert @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N2 ) @ ( set_or1337092689740270186AtMost @ int @ M @ N2 ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_3879_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I4: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I4 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_3880_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ ( F3 @ X4 ) @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= ( image2 @ B @ A @ F3 @ A5 ) ) ).
% UNION_singleton_eq_range
thf(fact_3881_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,X: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( plus_plus @ A @ ( G3 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_3882_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( plus_plus @ A @ ( G3 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_3883_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,A3: B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member @ B @ A3 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_3884_card__2__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X4: A,Y4: A] :
( ( S
= ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X4 != Y4 ) ) ) ) ).
% card_2_iff
thf(fact_3885_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,X: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_3886_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_3887_card_Oremove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ A5 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_3888_card_Oinsert__remove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A5 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_3889_card__Suc__Diff1,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_3890_card__insert__disjoint_H,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert @ A @ X @ A5 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_insert_disjoint'
thf(fact_3891_card__Diff1__less,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ) ) ).
% card_Diff1_less
thf(fact_3892_card__Diff2__less,axiom,
! [A: $tType,A5: set @ A,X: A,Y: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y @ A5 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_3893_card__Diff1__less__iff,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) )
= ( ( finite_finite2 @ A @ A5 )
& ( member @ A @ X @ A5 ) ) ) ).
% card_Diff1_less_iff
thf(fact_3894_card__3__iff,axiom,
! [A: $tType,S: set @ A] :
( ( ( finite_card @ A @ S )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X4: A,Y4: A,Z7: A] :
( ( S
= ( insert @ A @ X4 @ ( insert @ A @ Y4 @ ( insert @ A @ Z7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X4 != Y4 )
& ( Y4 != Z7 )
& ( X4 != Z7 ) ) ) ) ).
% card_3_iff
thf(fact_3895_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 ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_3896_card__Diff__singleton,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_3897_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_3898_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,A3: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S )
= ( plus_plus @ A @ ( B2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_3899_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,A3: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S )
= ( times_times @ A @ ( B2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3900_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I2: C,A5: set @ C,F3: C > B] :
( ( member @ C @ I2 @ A5 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A5 @ ( insert @ C @ I2 @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X3 ) ) )
=> ( ( finite_finite2 @ C @ A5 )
=> ( ord_less_eq @ B @ ( F3 @ I2 ) @ ( groups7311177749621191930dd_sum @ C @ B @ F3 @ A5 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_3901_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A5: set @ B,F3: B > A,A3: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( F3 @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A3 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( F3 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_3902_card__insert__le__m1,axiom,
! [A: $tType,N2: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X @ Y ) ) @ N2 ) ) ) ).
% card_insert_le_m1
thf(fact_3903_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [K2: int,L4: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L4 ) )
=> ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L4 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K2 @ L4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_3904_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
=> ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_3905_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% ccpo_Sup_singleton
thf(fact_3906_execute__return,axiom,
! [A: $tType,X: A] :
( ( heap_Time_execute @ A @ ( heap_Time_return @ A @ X ) )
= ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H6: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( one_one @ nat ) ) ) ) ) ).
% execute_return
thf(fact_3907_ureturn__def,axiom,
! [A: $tType] :
( ( heap_Time_ureturn @ A )
= ( ^ [X4: A] :
( heap_Time_heap @ A
@ ^ [H6: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% ureturn_def
thf(fact_3908_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_3909_subset__mset_OcInf__singleton,axiom,
! [A: $tType,X: multiset @ A] :
( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= X ) ).
% subset_mset.cInf_singleton
thf(fact_3910_subset__mset_OcSup__singleton,axiom,
! [A: $tType,X: multiset @ A] :
( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= X ) ).
% subset_mset.cSup_singleton
thf(fact_3911_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_3912_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3913_these__insert__None,axiom,
! [A: $tType,A5: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A5 ) )
= ( these @ A @ A5 ) ) ).
% these_insert_None
thf(fact_3914_singleton__None__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) @ ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) )
= ( none @ A ) ) ) ).
% singleton_None_Sup
thf(fact_3915_UNIV__bool,axiom,
( ( top_top @ ( set @ $o ) )
= ( insert @ $o @ $false @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% UNIV_bool
thf(fact_3916_Union__insert,axiom,
! [A: $tType,A3: set @ A,B4: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( insert @ ( set @ A ) @ A3 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) ) ) ).
% Union_insert
thf(fact_3917_Inter__insert,axiom,
! [A: $tType,A3: set @ A,B4: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( insert @ ( set @ A ) @ A3 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B4 ) ) ) ).
% Inter_insert
thf(fact_3918_insert__partition,axiom,
! [A: $tType,X: set @ A,F5: set @ ( set @ A )] :
( ~ ( member @ ( set @ A ) @ X @ F5 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ ( insert @ ( set @ A ) @ X @ F5 ) )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ ( insert @ ( set @ A ) @ X @ F5 ) )
=> ( ( X3 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X3 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ X @ ( complete_Sup_Sup @ ( set @ A ) @ F5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_partition
thf(fact_3919_atLeastAtMost__insertL,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) )
= ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ).
% atLeastAtMost_insertL
thf(fact_3920_atLeastAtMostSuc__conv,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N2 ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N2 ) )
= ( insert @ nat @ ( suc @ N2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_3921_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ N2 )
= ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N2 ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_3922_return__def,axiom,
! [A: $tType] :
( ( heap_Time_return @ A )
= ( ^ [X4: A] :
( heap_Time_heap @ A
@ ^ [H6: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( one_one @ nat ) ) ) ) ) ) ).
% return_def
thf(fact_3923_atLeastLessThanSuc,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) )
= ( insert @ nat @ N2 @ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N2 )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_3924_these__not__empty__eq,axiom,
! [A: $tType,B4: set @ ( option @ A )] :
( ( ( these @ A @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B4
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B4
!= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_3925_these__empty__eq,axiom,
! [A: $tType,B4: set @ ( option @ A )] :
( ( ( these @ A @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B4
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B4
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_3926_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B,X: A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ( F3 @ Y3 )
= ( F3 @ X ) ) )
=> ( ( the_elem @ B @ ( image2 @ A @ B @ F3 @ A5 ) )
= ( F3 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_3927_atLeast1__atMost__eq__remove0,axiom,
! [N2: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N2 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_3928_atLeast1__lessThan__eq__remove0,axiom,
! [N2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N2 )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N2 ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_3929_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( insert @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_3930_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_3931_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_3932_Sup__option__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) )
= ( ^ [A8: set @ ( option @ A )] :
( if @ ( option @ A )
@ ( ( A8
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( A8
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) )
@ ( none @ A )
@ ( some @ A @ ( complete_Sup_Sup @ A @ ( these @ A @ A8 ) ) ) ) ) ) ) ).
% Sup_option_def
thf(fact_3933_execute__heap,axiom,
! [A: $tType,F3: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( heap_Time_execute @ A @ ( heap_Time_heap @ A @ F3 ) )
= ( 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 ) ) ) @ F3 ) ) ).
% execute_heap
thf(fact_3934_heap__def,axiom,
! [A: $tType] :
( ( heap_Time_heap @ A )
= ( ^ [F4: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] : ( heap_Time_Heap2 @ A @ ( 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 ) ) ) @ F4 ) ) ) ) ).
% heap_def
thf(fact_3935_bind__return,axiom,
! [A: $tType,F3: heap_Time_Heap @ A] :
( ( heap_Time_bind @ A @ A @ F3 @ ( heap_Time_return @ A ) )
= ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
@ ^ [Uu2: product_unit] : F3 ) ) ).
% bind_return
thf(fact_3936_return__bind,axiom,
! [B: $tType,A: $tType,X: B,F3: B > ( heap_Time_Heap @ A )] :
( ( heap_Time_bind @ B @ A @ ( heap_Time_return @ B @ X ) @ F3 )
= ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
@ ^ [Uu2: product_unit] : ( F3 @ X ) ) ) ).
% return_bind
thf(fact_3937_bind__lift,axiom,
! [A: $tType,B: $tType,F3: heap_Time_Heap @ B,G3: B > A] :
( ( heap_Time_bind @ B @ A @ F3 @ ( heap_Time_lift @ B @ A @ G3 ) )
= ( heap_Time_bind @ B @ A @ F3
@ ^ [X4: B] : ( heap_Time_return @ A @ ( G3 @ X4 ) ) ) ) ).
% bind_lift
thf(fact_3938_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F5: set @ A,I5: set @ A,F3: A > B,I2: A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I5 )
& ( ( F3 @ I4 )
!= ( zero_zero @ B ) ) ) )
@ F5 )
=> ( ( ( member @ A @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) @ ( F3 @ I2 ) ) ) )
& ( ~ ( member @ A @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_3939_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P3: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_3940_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,P3: B > A,I2: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( P3 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ ( insert @ B @ I2 @ I5 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P3 @ I5 ) ) )
& ( ~ ( member @ B @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ ( insert @ B @ I2 @ I5 ) )
= ( plus_plus @ A @ ( P3 @ I2 ) @ ( groups1027152243600224163dd_sum @ B @ A @ P3 @ I5 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_3941_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,I5: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( G3 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I5 ) ) ) ).
% sum.non_neutral'
thf(fact_3942_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G3 @ I4 ) @ ( H @ I4 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H @ I5 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_3943_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T2: set @ B,G3: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T2 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_3944_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T2: set @ B,H: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_3945_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T2: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T2 )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_3946_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S: set @ B,T2: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_3947_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( G3 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( H @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G3 @ I4 ) @ ( H @ I4 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_3948_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P6: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I7 )
& ( ( P6 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P6
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I7 )
& ( ( P6 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_3949_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I5: set @ A,F3: A > B,I2: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I5 )
& ( ( F3 @ I4 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) @ ( F3 @ I2 ) ) ) )
& ( ~ ( member @ A @ I2 @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I5 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_3950_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A8: set @ A] :
( A8
= ( insert @ A @ ( the_elem @ A @ A8 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_3951_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_3952_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N2: nat,X: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N2 @ X ) )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ N2 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_3953_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_3954_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I2: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I2 )
=> ( ( list_update @ A @ Xs @ I2 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_3955_nth__list__update__eq,axiom,
! [A: $tType,I2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_3956_nth__update__invalid,axiom,
! [A: $tType,I2: nat,L: list @ A,J: nat,X: A] :
( ~ ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ J @ X ) @ I2 )
= ( nth @ A @ L @ I2 ) ) ) ).
% nth_update_invalid
thf(fact_3957_set__swap,axiom,
! [A: $tType,I2: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I2 ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_3958_distinct__swap,axiom,
! [A: $tType,I2: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I2 ) ) )
= ( distinct @ A @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_3959_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A,X: A,I2: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I2 @ X ) ) @ A5 ) ) ) ).
% set_update_subsetI
thf(fact_3960_is__singletonI_H,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( X3 = Y3 ) ) )
=> ( is_singleton @ A @ A5 ) ) ) ).
% is_singletonI'
thf(fact_3961_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I2: nat,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I2 @ X ) ) @ ( insert @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_3962_in__set__upd__eq,axiom,
! [A: $tType,I2: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ Y ) ) )
= ( ( X = Y )
| ( ( member @ A @ X @ ( set2 @ A @ L ) )
& ! [Y4: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ Y4 ) ) ) ) ) ) ) ).
% in_set_upd_eq
thf(fact_3963_in__set__upd__cases,axiom,
! [A: $tType,X: A,L: list @ A,I2: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ Y ) ) )
=> ( ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( X != Y ) )
=> ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_set_upd_cases
thf(fact_3964_in__set__upd__eq__aux,axiom,
! [A: $tType,I2: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ Y ) ) )
= ( ( X = Y )
| ! [Y4: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ Y4 ) ) ) ) ) ) ).
% in_set_upd_eq_aux
thf(fact_3965_set__update__memI,axiom,
! [A: $tType,N2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs @ N2 @ X ) ) ) ) ).
% set_update_memI
thf(fact_3966_nth__list__update,axiom,
! [A: $tType,I2: nat,Xs: list @ A,J: nat,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I2 = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ J )
= X ) )
& ( ( I2 != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_3967_list__update__same__conv,axiom,
! [A: $tType,I2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I2 @ X )
= Xs )
= ( ( nth @ A @ Xs @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_3968_nth__list__update_H,axiom,
! [A: $tType,I2: nat,J: nat,L: list @ A,X: A] :
( ( ( ( I2 = J )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I2 @ X ) @ J )
= X ) )
& ( ~ ( ( I2 = J )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I2 @ X ) @ J )
= ( nth @ A @ L @ J ) ) ) ) ).
% nth_list_update'
thf(fact_3969_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A8: set @ A] :
? [X4: A] :
( A8
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_3970_is__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( is_singleton @ A @ A5 )
=> ~ ! [X3: A] :
( A5
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_3971_insert__swap__set__eq,axiom,
! [A: $tType,I2: nat,L: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( insert @ A @ ( nth @ A @ L @ I2 ) @ ( set2 @ A @ ( list_update @ A @ L @ I2 @ X ) ) )
= ( insert @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% insert_swap_set_eq
thf(fact_3972_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A3: A,I2: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ I2 ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I2 @ A3 ) ) ) ) ).
% distinct_list_update
thf(fact_3973_lookup__chain,axiom,
! [B: $tType,A: $tType] :
( ( heap @ B )
=> ! [R3: ref @ B,F3: heap_Time_Heap @ A] :
( ( heap_Time_bind @ B @ A @ ( ref_lookup @ B @ R3 )
@ ^ [Uu2: B] : F3 )
= ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
@ ^ [Uu2: product_unit] : F3 ) ) ) ).
% lookup_chain
thf(fact_3974_and__not__num_Oelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N5: num] :
( Xa
= ( bit0 @ N5 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N5: num] :
( Xa
= ( bit1 @ N5 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N12: num] : ( some @ num @ ( bit1 @ N12 ) )
@ ( bit_and_not_num @ M5 @ N5 ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_3975_set__remove1__eq,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( set2 @ A @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_remove1_eq
thf(fact_3976_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A5: B > ( set @ A ),I2: B,B4: set @ A,J5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A5 @ I2 @ B4 ) @ J5 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ B ) @ J5 @ ( insert @ B @ I2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I2 @ J5 ) @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_3977_image__update,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,F3: A > B,N2: B] :
( ~ ( member @ A @ X @ A5 )
=> ( ( image2 @ A @ B @ ( fun_upd @ A @ B @ F3 @ X @ N2 ) @ A5 )
= ( image2 @ A @ B @ F3 @ A5 ) ) ) ).
% image_update
thf(fact_3978_map__option__eq__Some,axiom,
! [A: $tType,B: $tType,F3: B > A,Xo: option @ B,Y: A] :
( ( ( map_option @ B @ A @ F3 @ Xo )
= ( some @ A @ Y ) )
= ( ? [Z7: B] :
( ( Xo
= ( some @ B @ Z7 ) )
& ( ( F3 @ Z7 )
= Y ) ) ) ) ).
% map_option_eq_Some
thf(fact_3979_None__eq__map__option__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,X: option @ B] :
( ( ( none @ A )
= ( map_option @ B @ A @ F3 @ X ) )
= ( X
= ( none @ B ) ) ) ).
% None_eq_map_option_iff
thf(fact_3980_map__option__is__None,axiom,
! [A: $tType,B: $tType,F3: B > A,Opt: option @ B] :
( ( ( map_option @ B @ A @ F3 @ Opt )
= ( none @ A ) )
= ( Opt
= ( none @ B ) ) ) ).
% map_option_is_None
thf(fact_3981_option_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: option @ A] :
( ( ( map_option @ A @ B @ F3 @ A3 )
= ( none @ B ) )
= ( A3
= ( none @ A ) ) ) ).
% option.map_disc_iff
thf(fact_3982_map__option__o__empty,axiom,
! [C: $tType,B: $tType,A: $tType,F3: C > B] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F3 )
@ ^ [X4: A] : ( none @ C ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% map_option_o_empty
thf(fact_3983_case__map__option,axiom,
! [B: $tType,A: $tType,C: $tType,G3: A,H: B > A,F3: C > B,X: option @ C] :
( ( case_option @ A @ B @ G3 @ H @ ( map_option @ C @ B @ F3 @ X ) )
= ( case_option @ A @ C @ G3 @ ( comp @ B @ A @ C @ H @ F3 ) @ X ) ) ).
% case_map_option
thf(fact_3984_option_Omap__ident,axiom,
! [A: $tType,T6: option @ A] :
( ( map_option @ A @ A
@ ^ [X4: A] : X4
@ T6 )
= T6 ) ).
% option.map_ident
thf(fact_3985_option_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F3: A > B,X2: A] :
( ( map_option @ A @ B @ F3 @ ( some @ A @ X2 ) )
= ( some @ B @ ( F3 @ X2 ) ) ) ).
% option.simps(9)
thf(fact_3986_map__option__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F3: A > B,G3: A > B] :
( ( X = Y )
=> ( ! [A6: A] :
( ( Y
= ( some @ A @ A6 ) )
=> ( ( F3 @ A6 )
= ( G3 @ A6 ) ) )
=> ( ( map_option @ A @ B @ F3 @ X )
= ( map_option @ A @ B @ G3 @ Y ) ) ) ) ).
% map_option_cong
thf(fact_3987_option_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F3: A > B] :
( ( map_option @ A @ B @ F3 @ ( none @ A ) )
= ( none @ B ) ) ).
% option.simps(8)
thf(fact_3988_map__option_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F3: B > C,G3: A > B,Option: option @ A] :
( ( map_option @ B @ C @ F3 @ ( map_option @ A @ B @ G3 @ Option ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F3 @ G3 ) @ Option ) ) ).
% map_option.compositionality
thf(fact_3989_option_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G3: B > C,F3: A > B,V2: option @ A] :
( ( map_option @ B @ C @ G3 @ ( map_option @ A @ B @ F3 @ V2 ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ G3 @ F3 ) @ V2 ) ) ).
% option.map_comp
thf(fact_3990_map__option_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C,G3: A > B] :
( ( comp @ ( option @ B ) @ ( option @ C ) @ ( option @ A ) @ ( map_option @ B @ C @ F3 ) @ ( map_option @ A @ B @ G3 ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F3 @ G3 ) ) ) ).
% map_option.comp
thf(fact_3991_option_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F3: B > nat,G3: A > B] :
( ( comp @ ( option @ B ) @ nat @ ( option @ A ) @ ( size_option @ B @ F3 ) @ ( map_option @ A @ B @ G3 ) )
= ( size_option @ A @ ( comp @ B @ nat @ A @ F3 @ G3 ) ) ) ).
% option.size_gen_o_map
thf(fact_3992_finite__update__induct,axiom,
! [B: $tType,A: $tType,F3: A > B,C2: B,P: ( A > B ) > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A7: A] :
( ( F3 @ A7 )
!= C2 ) ) )
=> ( ( P
@ ^ [A7: A] : C2 )
=> ( ! [A6: A,B5: B,F2: A > B] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [C5: A] :
( ( F2 @ C5 )
!= C2 ) ) )
=> ( ( ( F2 @ A6 )
= C2 )
=> ( ( B5 != C2 )
=> ( ( P @ F2 )
=> ( P @ ( fun_upd @ A @ B @ F2 @ A6 @ B5 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_update_induct
thf(fact_3993_set__remove1__subset,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_remove1_subset
thf(fact_3994_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A3 @ Xs ) ) ) ) ).
% sorted_remove1
thf(fact_3995_ran__map__option,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,M: B > ( option @ C )] :
( ( ran @ B @ A
@ ^ [X4: B] : ( map_option @ C @ A @ F3 @ ( M @ X4 ) ) )
= ( image2 @ C @ A @ F3 @ ( ran @ B @ C @ M ) ) ) ).
% ran_map_option
thf(fact_3996_map__option__case,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F4: B > A] :
( case_option @ ( option @ A ) @ B @ ( none @ A )
@ ^ [X4: B] : ( some @ A @ ( F4 @ X4 ) ) ) ) ) ).
% map_option_case
thf(fact_3997_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A5: set @ B,F3: B > A,Y: A] :
( ( ( member @ B @ X @ A5 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F3 @ X @ Y ) @ A5 )
= ( insert @ A @ Y @ ( image2 @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F3 @ X @ Y ) @ A5 )
= ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_3998_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_3999_and__not__num_Opelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N5 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N5 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N12: num] : ( some @ num @ ( bit1 @ N12 ) )
@ ( bit_and_not_num @ M5 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_4000_and__num_Oelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N5: num] :
( Xa
= ( bit0 @ N5 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N5: num] :
( Xa
= ( bit1 @ N5 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ? [M5: num] :
( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) ) ) )
=> ( ( ? [M5: num] :
( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N12: num] : ( some @ num @ ( bit1 @ N12 ) )
@ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_4001_xor__num_Oelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa )
= Y )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X = one2 )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( Y
!= ( some @ num @ ( bit1 @ N5 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( Y
!= ( some @ num @ ( bit0 @ N5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit1 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( Y
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( Y
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
thf(fact_4002_execute__lookup,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( ref_lookup @ A @ R3 ) @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( ref_get @ A @ H @ R3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_lookup
thf(fact_4003_empty__upd__none,axiom,
! [B: $tType,A: $tType,X: A] :
( ( fun_upd @ A @ ( option @ B )
@ ^ [X4: A] : ( none @ B )
@ X
@ ( none @ B ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% empty_upd_none
thf(fact_4004_image__map__upd,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,M: A > ( option @ B ),Y: B] :
( ~ ( member @ A @ X @ A5 )
=> ( ( image2 @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( some @ B @ Y ) ) @ A5 )
= ( image2 @ A @ ( option @ B ) @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_4005_map__option__o__map__upd,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B,M: A > ( option @ C ),A3: A,B2: C] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F3 ) @ ( fun_upd @ A @ ( option @ C ) @ M @ A3 @ ( some @ C @ B2 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F3 ) @ M ) @ A3 @ ( some @ B @ ( F3 @ B2 ) ) ) ) ).
% map_option_o_map_upd
thf(fact_4006_ran__map__upd,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A3: B,B2: A] :
( ( ( M @ A3 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ A3 @ ( some @ A @ B2 ) ) )
= ( insert @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ) ).
% ran_map_upd
thf(fact_4007_map__upd__eqD1,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),A3: A,X: B,N2: A > ( option @ B ),Y: B] :
( ( ( fun_upd @ A @ ( option @ B ) @ M @ A3 @ ( some @ B @ X ) )
= ( fun_upd @ A @ ( option @ B ) @ N2 @ A3 @ ( some @ B @ Y ) ) )
=> ( X = Y ) ) ).
% map_upd_eqD1
thf(fact_4008_map__upd__triv,axiom,
! [A: $tType,B: $tType,T6: B > ( option @ A ),K: B,X: A] :
( ( ( T6 @ K )
= ( some @ A @ X ) )
=> ( ( fun_upd @ B @ ( option @ A ) @ T6 @ K @ ( some @ A @ X ) )
= T6 ) ) ).
% map_upd_triv
thf(fact_4009_map__upd__Some__unfold,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),A3: B,B2: A,X: B,Y: A] :
( ( ( fun_upd @ B @ ( option @ A ) @ M @ A3 @ ( some @ A @ B2 ) @ X )
= ( some @ A @ Y ) )
= ( ( ( X = A3 )
& ( B2 = Y ) )
| ( ( X != A3 )
& ( ( M @ X )
= ( some @ A @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_4010_map__upd__nonempty,axiom,
! [B: $tType,A: $tType,T6: A > ( option @ B ),K: A,X: B] :
( ( fun_upd @ A @ ( option @ B ) @ T6 @ K @ ( some @ B @ X ) )
!= ( ^ [X4: A] : ( none @ B ) ) ) ).
% map_upd_nonempty
thf(fact_4011_and__num_Osimps_I1_J,axiom,
( ( bit_un7362597486090784418nd_num @ one2 @ one2 )
= ( some @ num @ one2 ) ) ).
% and_num.simps(1)
thf(fact_4012_and__num_Osimps_I3_J,axiom,
! [N2: num] :
( ( bit_un7362597486090784418nd_num @ one2 @ ( bit1 @ N2 ) )
= ( some @ num @ one2 ) ) ).
% and_num.simps(3)
thf(fact_4013_and__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one2 )
= ( some @ num @ one2 ) ) ).
% and_num.simps(7)
thf(fact_4014_and__num__eq__Some__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N2: num,Q4: num] :
( ( ( bit_un7362597486090784418nd_num @ M @ N2 )
= ( some @ num @ Q4 ) )
= ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( numeral_numeral @ A @ Q4 ) ) ) ) ).
% and_num_eq_Some_iff
thf(fact_4015_xor__num__eq__Some__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N2: num,Q4: num] :
( ( ( bit_un2480387367778600638or_num @ M @ N2 )
= ( some @ num @ Q4 ) )
= ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( numeral_numeral @ A @ Q4 ) ) ) ) ).
% xor_num_eq_Some_iff
thf(fact_4016_finite__range__updI,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),A3: B,B2: A] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F3 @ A3 @ ( some @ A @ B2 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_updI
thf(fact_4017_lookup__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_lookup @ A )
= ( ^ [R4: ref @ A] :
( heap_Time_tap @ A
@ ^ [H6: heap_ext @ product_unit] : ( ref_get @ A @ H6 @ R4 ) ) ) ) ) ).
% lookup_def
thf(fact_4018_xor__num_Osimps_I2_J,axiom,
! [N2: num] :
( ( bit_un2480387367778600638or_num @ one2 @ ( bit0 @ N2 ) )
= ( some @ num @ ( bit1 @ N2 ) ) ) ).
% xor_num.simps(2)
thf(fact_4019_xor__num_Osimps_I3_J,axiom,
! [N2: num] :
( ( bit_un2480387367778600638or_num @ one2 @ ( bit1 @ N2 ) )
= ( some @ num @ ( bit0 @ N2 ) ) ) ).
% xor_num.simps(3)
thf(fact_4020_xor__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one2 )
= ( some @ num @ ( bit1 @ M ) ) ) ).
% xor_num.simps(4)
thf(fact_4021_xor__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one2 )
= ( some @ num @ ( bit0 @ M ) ) ) ).
% xor_num.simps(7)
thf(fact_4022_numeral__and__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) ).
% numeral_and_num
thf(fact_4023_numeral__xor__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N2 ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).
% numeral_xor_num
thf(fact_4024_and__num_Osimps_I9_J,axiom,
! [M: num,N2: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N12: num] : ( some @ num @ ( bit1 @ N12 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% and_num.simps(9)
thf(fact_4025_xor__num_Osimps_I6_J,axiom,
! [M: num,N2: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).
% xor_num.simps(6)
thf(fact_4026_xor__num_Osimps_I8_J,axiom,
! [M: num,N2: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).
% xor_num.simps(8)
thf(fact_4027_and__num_Opelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N5 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N5 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N12: num] : ( some @ num @ ( bit1 @ N12 ) )
@ ( bit_un7362597486090784418nd_num @ M5 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_4028_xor__num_Opelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( ( Y
= ( some @ num @ ( bit1 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N5 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( ( Y
= ( some @ num @ ( bit0 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N5 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit1 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( ( Y
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit0 @ N5 ) )
=> ( ( Y
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N5 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N5: num] :
( ( Xa
= ( bit1 @ N5 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_4029_sngr__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As4
= ( insert @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ).
% sngr_assn_raw.elims(3)
thf(fact_4030_sngr__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ~ ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As4
= ( insert @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ).
% sngr_assn_raw.elims(2)
thf(fact_4031_relH__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [As: set @ nat,H: heap_ext @ product_unit,H3: heap_ext @ product_unit,R3: ref @ A] :
( ( relH @ As @ H @ H3 )
=> ( ( member @ nat @ ( addr_of_ref @ A @ R3 ) @ As )
=> ( ( ref_get @ A @ H @ R3 )
= ( ref_get @ A @ H3 @ R3 ) ) ) ) ) ).
% relH_ref
thf(fact_4032_sngr__assn__raw_Osimps,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,X: A,H: heap_ext @ product_unit,As: set @ nat] :
( ( sngr_assn_raw @ A @ R3 @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
= ( ( ( ref_get @ A @ H @ R3 )
= X )
& ( As
= ( insert @ nat @ ( addr_of_ref @ A @ R3 ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ R3 ) @ ( lim @ product_unit @ H ) ) ) ) ) ).
% sngr_assn_raw.simps
thf(fact_4033_sngr__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
= Y )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( Y
= ( ~ ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As4
= ( insert @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.elims(1)
thf(fact_4034_sngr__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( Y
= ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As4
= ( insert @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(1)
thf(fact_4035_sngr__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) )
=> ~ ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As4
= ( insert @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(2)
thf(fact_4036_sngr__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H4: heap_ext @ product_unit,As4: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As4 ) ) ) )
=> ( ( ( ref_get @ A @ H4 @ X )
= Xa )
& ( As4
= ( insert @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(3)
thf(fact_4037_option_Orec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,G3: C,Ga: B > C,F3: A > B] :
( ( comp @ ( option @ B ) @ C @ ( option @ A ) @ ( rec_option @ C @ B @ G3 @ Ga ) @ ( map_option @ A @ B @ F3 ) )
= ( rec_option @ C @ A @ G3
@ ^ [X4: A] : ( Ga @ ( F3 @ X4 ) ) ) ) ).
% option.rec_o_map
thf(fact_4038_option_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F22: A > C,X2: A] :
( ( rec_option @ C @ A @ F1 @ F22 @ ( some @ A @ X2 ) )
= ( F22 @ X2 ) ) ).
% option.simps(7)
thf(fact_4039_option_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F22: A > C] :
( ( rec_option @ C @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(6)
thf(fact_4040_subset__mset_Osum__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ ( multiset @ A ),X: multiset @ A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ ( multiset @ A ) ) @ Xs ) )
=> ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( list_update @ ( multiset @ A ) @ Xs @ K @ X ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ X ) @ ( nth @ ( multiset @ A ) @ Xs @ K ) ) ) ) ).
% subset_mset.sum_list_update
thf(fact_4041_times__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y4 @ V3 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V3 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) )
@ Xa
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_4042_set__removeAll,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( set2 @ A @ ( removeAll @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_removeAll
thf(fact_4043_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 ) ) )
=> ~ ? [X7: A] :
( ( member @ A @ X7 @ S )
& ( ord_less @ B @ ( F3 @ X7 ) @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4044_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X3: nat,Y3: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X3 @ Y3 ) ) ) ).
% eq_Abs_Integ
thf(fact_4045_length__removeAll__less__eq,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_4046_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_4047_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_4048_length__removeAll__less,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_4049_uminus__int_Oabs__eq,axiom,
! [X: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X4 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_4050_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_4051_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_4052_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: product_prod @ nat @ nat] :
( ( ring_1_of_int @ A @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ X ) ) ) ).
% of_int.abs_eq
thf(fact_4053_less__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) )
@ Xa
@ X ) ) ).
% less_int.abs_eq
thf(fact_4054_less__eq__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) )
@ Xa
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_4055_plus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y4 @ V3 ) ) )
@ Xa
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_4056_minus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) )
@ Xa
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_4057_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_4058_subset__mset_Oelem__le__sum__list,axiom,
! [A: $tType,K: nat,Ns: list @ ( multiset @ A )] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ ( multiset @ A ) ) @ Ns ) )
=> ( subseteq_mset @ A @ ( nth @ ( multiset @ A ) @ Ns @ K ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Ns ) ) ) ).
% subset_mset.elem_le_sum_list
thf(fact_4059_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_4060_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_4061_execute__change,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [F3: A > A,R3: ref @ A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( ref_change @ A @ F3 @ R3 ) @ H )
= ( 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 @ H @ R3 ) ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R3 @ ( F3 @ ( ref_get @ A @ H @ R3 ) ) @ H ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% execute_change
thf(fact_4062_lim__set,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,V2: A,H: heap_ext @ product_unit] :
( ( lim @ product_unit @ ( ref_set @ A @ R3 @ V2 @ H ) )
= ( lim @ product_unit @ H ) ) ) ).
% lim_set
thf(fact_4063_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_4064_mset__le__subtract,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,C4: multiset @ A] :
( ( subseteq_mset @ A @ A5 @ B4 )
=> ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A5 @ C4 ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ C4 ) ) ) ).
% mset_le_subtract
thf(fact_4065_subset__mset_Ofinite__has__minimal,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ? [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ A5 )
& ! [Xa2: multiset @ A] :
( ( member @ ( multiset @ A ) @ Xa2 @ A5 )
=> ( ( subseteq_mset @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal
thf(fact_4066_subset__mset_Ofinite__has__maximal,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ? [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ A5 )
& ! [Xa2: multiset @ A] :
( ( member @ ( multiset @ A ) @ Xa2 @ A5 )
=> ( ( subseteq_mset @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal
thf(fact_4067_mset__le__addE,axiom,
! [A: $tType,Xs: multiset @ A,Ys3: multiset @ A] :
( ( subseteq_mset @ A @ Xs @ Ys3 )
=> ~ ! [Zs2: multiset @ A] :
( Ys3
!= ( plus_plus @ ( multiset @ A ) @ Xs @ Zs2 ) ) ) ).
% mset_le_addE
thf(fact_4068_mset__le__distrib,axiom,
! [A: $tType,X6: multiset @ A,A5: multiset @ A,B4: multiset @ A] :
( ( subseteq_mset @ A @ X6 @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) )
=> ~ ! [Xa5: multiset @ A,Xb4: multiset @ A] :
( ( X6
= ( plus_plus @ ( multiset @ A ) @ Xa5 @ Xb4 ) )
=> ( ( subseteq_mset @ A @ Xa5 @ A5 )
=> ~ ( subseteq_mset @ A @ Xb4 @ B4 ) ) ) ) ).
% mset_le_distrib
thf(fact_4069_mset__union__subset,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,C4: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) @ C4 )
=> ( ( subseteq_mset @ A @ A5 @ C4 )
& ( subseteq_mset @ A @ B4 @ C4 ) ) ) ).
% mset_union_subset
thf(fact_4070_mset__le__decr__left1,axiom,
! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ B2 )
=> ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% mset_le_decr_left1
thf(fact_4071_mset__le__decr__left2,axiom,
! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ B2 )
=> ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% mset_le_decr_left2
thf(fact_4072_mset__le__incr__right1,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% mset_le_incr_right1
thf(fact_4073_mset__le__incr__right2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% mset_le_incr_right2
thf(fact_4074_insort__left__comm,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Xs: list @ A] :
( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( 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
@ X
@ Xs ) ) ) ) ).
% insort_left_comm
thf(fact_4075_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
thf(fact_4076_subset__mset_Olift__Suc__mono__le,axiom,
! [A: $tType,F3: nat > ( multiset @ A ),N2: nat,N6: nat] :
( ! [N5: nat] : ( subseteq_mset @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
=> ( ( ord_less_eq @ nat @ N2 @ N6 )
=> ( subseteq_mset @ A @ ( F3 @ N2 ) @ ( F3 @ N6 ) ) ) ) ).
% subset_mset.lift_Suc_mono_le
thf(fact_4077_subset__mset_Olift__Suc__antimono__le,axiom,
! [A: $tType,F3: nat > ( multiset @ A ),N2: nat,N6: nat] :
( ! [N5: nat] : ( subseteq_mset @ A @ ( F3 @ ( suc @ N5 ) ) @ ( F3 @ N5 ) )
=> ( ( ord_less_eq @ nat @ N2 @ N6 )
=> ( subseteq_mset @ A @ ( F3 @ N6 ) @ ( F3 @ N2 ) ) ) ) ).
% subset_mset.lift_Suc_antimono_le
thf(fact_4078_mset__le__subtract__left,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,X6: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) @ X6 )
=> ( ( subseteq_mset @ A @ B4 @ ( minus_minus @ ( multiset @ A ) @ X6 @ A5 ) )
& ( subseteq_mset @ A @ A5 @ X6 ) ) ) ).
% mset_le_subtract_left
thf(fact_4079_mset__le__subtract__right,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,X6: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) @ X6 )
=> ( ( subseteq_mset @ A @ A5 @ ( minus_minus @ ( multiset @ A ) @ X6 @ B4 ) )
& ( subseteq_mset @ A @ B4 @ X6 ) ) ) ).
% mset_le_subtract_right
thf(fact_4080_size__mset__mono,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A] :
( ( subseteq_mset @ A @ A5 @ B4 )
=> ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ A5 ) @ ( size_size @ ( multiset @ A ) @ B4 ) ) ) ).
% size_mset_mono
thf(fact_4081_subset__mset_OcInf__eq__non__empty,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A3: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X6 )
=> ( subseteq_mset @ A @ A3 @ X3 ) )
=> ( ! [Y3: multiset @ A] :
( ! [X7: multiset @ A] :
( ( member @ ( multiset @ A ) @ X7 @ X6 )
=> ( subseteq_mset @ A @ Y3 @ X7 ) )
=> ( subseteq_mset @ A @ Y3 @ A3 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ X6 )
= A3 ) ) ) ) ).
% subset_mset.cInf_eq_non_empty
thf(fact_4082_subset__mset_OcInf__greatest,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),Z2: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X6 )
=> ( subseteq_mset @ A @ Z2 @ X3 ) )
=> ( subseteq_mset @ A @ Z2 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X6 ) ) ) ) ).
% subset_mset.cInf_greatest
thf(fact_4083_subset__mset_OcSup__least,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),Z2: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X6 )
=> ( subseteq_mset @ A @ X3 @ Z2 ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ X6 ) @ Z2 ) ) ) ).
% subset_mset.cSup_least
thf(fact_4084_subset__mset_OcSup__eq__non__empty,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A3: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ X6 )
=> ( subseteq_mset @ A @ X3 @ A3 ) )
=> ( ! [Y3: multiset @ A] :
( ! [X7: multiset @ A] :
( ( member @ ( multiset @ A ) @ X7 @ X6 )
=> ( subseteq_mset @ A @ X7 @ Y3 ) )
=> ( subseteq_mset @ A @ A3 @ Y3 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ X6 )
= A3 ) ) ) ) ).
% subset_mset.cSup_eq_non_empty
thf(fact_4085_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_4086_subset__mset_OcINF__greatest,axiom,
! [A: $tType,B: $tType,A5: set @ B,M: multiset @ A,F3: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( subseteq_mset @ A @ M @ ( F3 @ X3 ) ) )
=> ( subseteq_mset @ A @ M @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) ) ) ) ).
% subset_mset.cINF_greatest
thf(fact_4087_subset__mset_OcSUP__least,axiom,
! [B: $tType,A: $tType,A5: set @ B,F3: B > ( multiset @ A ),M6: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( subseteq_mset @ A @ ( F3 @ X3 ) @ M6 ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) @ M6 ) ) ) ).
% subset_mset.cSUP_least
thf(fact_4088_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,Xs: list @ A] :
( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A3
@ ( remove1 @ A @ A3 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_4089_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_4090_relH__set__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,As: set @ nat,H: heap_ext @ product_unit,X: A] :
( ~ ( member @ nat @ ( addr_of_ref @ A @ R3 ) @ As )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As ) )
=> ( relH @ As @ H @ ( ref_set @ A @ R3 @ X @ H ) ) ) ) ) ).
% relH_set_ref
thf(fact_4091_change__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_change @ A )
= ( ^ [F4: A > A,R4: ref @ A] :
( heap_Time_bind @ A @ A @ ( ref_lookup @ A @ R4 )
@ ^ [X4: A] :
( heap_Time_bind @ product_unit @ A @ ( ref_update @ A @ R4 @ ( F4 @ X4 ) )
@ ^ [Uu2: product_unit] : ( heap_Time_return @ A @ ( F4 @ X4 ) ) ) ) ) ) ) ).
% change_def
thf(fact_4092_subset__mset_Osum__mono2,axiom,
! [A: $tType,B: $tType,B4: set @ B,A5: set @ B,F3: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ B4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B4 @ A5 ) )
=> ( 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 @ A5 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ B4 ) ) ) ) ) ).
% subset_mset.sum_mono2
thf(fact_4093_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X4: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y4: nat,Z7: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Z7 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_4094_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X4: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y4: nat,Z7: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Z7 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_4095_subset__mset_Osum__mono,axiom,
! [A: $tType,B: $tType,K5: set @ B,F3: B > ( multiset @ A ),G3: B > ( multiset @ A )] :
( ! [I3: B] :
( ( member @ B @ I3 @ K5 )
=> ( subseteq_mset @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ K5 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G3 @ K5 ) ) ) ).
% subset_mset.sum_mono
thf(fact_4096_subset__mset_Osum__nonneg__0,axiom,
! [B: $tType,A: $tType,S2: set @ B,F3: B > ( multiset @ A ),I2: B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S2 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ S2 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( member @ B @ I2 @ S2 )
=> ( ( F3 @ I2 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.sum_nonneg_0
thf(fact_4097_subset__mset_Osum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType,S2: set @ B,F3: B > ( multiset @ A ),B4: multiset @ A,I2: B] :
( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S2 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ S2 )
= B4 )
=> ( ( member @ B @ I2 @ S2 )
=> ( subseteq_mset @ A @ ( F3 @ I2 ) @ B4 ) ) ) ) ) ).
% subset_mset.sum_nonneg_leq_bound
thf(fact_4098_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,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ ( rep_Integ @ X4 ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_4099_mset__size2elem,axiom,
! [A: $tType,P: multiset @ A,Q4: A,Q8: 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 @ Q8 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ P )
=> ( P
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q8 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% mset_size2elem
thf(fact_4100_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 )
@ ^ [A7: A,B6: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A10: A,B14: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A10 ) @ Ra )
| ( ( A7 = A10 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B6 @ B14 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_4101_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,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X4 ) ) ) ) ).
% uminus_int_def
thf(fact_4102_pred__nat__def,axiom,
( pred_nat
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [M3: nat,N: nat] :
( N
= ( suc @ M3 ) ) ) ) ) ).
% pred_nat_def
thf(fact_4103_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ A4 @ B3 ) ) @ ( lex_prod @ A @ B @ R3 @ S2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A4 ) @ R3 )
| ( ( A3 = A4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_4104_mset__union__2__elem,axiom,
! [A: $tType,A3: A,B2: A,C2: A,M6: multiset @ A] :
( ( ( add_mset @ A @ A3 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( add_mset @ A @ C2 @ M6 ) )
=> ( ( ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
= M6 )
& ( B2 = C2 ) )
| ( ( A3 = C2 )
& ( ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
= M6 ) ) ) ) ).
% mset_union_2_elem
thf(fact_4105_mset__le__add__mset__decr__left1,axiom,
! [A: $tType,C2: A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A3 ) @ B2 )
=> ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% mset_le_add_mset_decr_left1
thf(fact_4106_mset__le__add__mset,axiom,
! [A: $tType,X: A,B4: multiset @ A,C4: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B4 ) @ C4 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ C4 )
& ( subseteq_mset @ A @ B4 @ C4 ) ) ) ).
% mset_le_add_mset
thf(fact_4107_mset__le__single__cases,axiom,
! [A: $tType,M6: multiset @ A,A3: A] :
( ( subseteq_mset @ A @ M6 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( ( M6
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( M6
= ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_le_single_cases
thf(fact_4108_mset__le__add__mset__decr__left2,axiom,
! [A: $tType,C2: A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A3 ) @ B2 )
=> ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) ) @ B2 ) ) ).
% mset_le_add_mset_decr_left2
thf(fact_4109_mset__le__subtract__add__mset__left,axiom,
! [A: $tType,X: A,B4: multiset @ A,X6: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B4 ) @ X6 )
=> ( ( subseteq_mset @ A @ B4 @ ( minus_minus @ ( multiset @ A ) @ X6 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
& ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ X6 ) ) ) ).
% mset_le_subtract_add_mset_left
thf(fact_4110_mset__le__subtract__add__mset__right,axiom,
! [A: $tType,X: A,B4: multiset @ A,X6: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B4 ) @ X6 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ X6 @ B4 ) )
& ( subseteq_mset @ A @ B4 @ X6 ) ) ) ).
% mset_le_subtract_add_mset_right
thf(fact_4111_mset__single__cases,axiom,
! [A: $tType,S2: A,C2: multiset @ A,R7: A,C9: multiset @ A] :
( ( ( add_mset @ A @ S2 @ C2 )
= ( add_mset @ A @ R7 @ C9 ) )
=> ( ( ( S2 = R7 )
=> ( C2 != C9 ) )
=> ~ ( ( C9
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases
thf(fact_4112_mset__single__cases_H,axiom,
! [A: $tType,S2: A,C2: multiset @ A,R7: A,C9: multiset @ A] :
( ( ( add_mset @ A @ S2 @ C2 )
= ( add_mset @ A @ R7 @ C9 ) )
=> ( ( ( S2 = R7 )
=> ( C2 != C9 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C9
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases'
thf(fact_4113_mset__single__cases2,axiom,
! [A: $tType,S2: A,C2: multiset @ A,R7: A,C9: multiset @ A] :
( ( ( add_mset @ A @ S2 @ C2 )
= ( add_mset @ A @ R7 @ C9 ) )
=> ( ( ( S2 = R7 )
=> ( C2 != C9 ) )
=> ~ ( ( C9
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases2
thf(fact_4114_mset__single__cases2_H,axiom,
! [A: $tType,S2: A,C2: multiset @ A,R7: A,C9: multiset @ A] :
( ( ( add_mset @ A @ S2 @ C2 )
= ( add_mset @ A @ R7 @ C9 ) )
=> ( ( ( S2 = R7 )
=> ( C2 != C9 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C9
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C9 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R7 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases2'
thf(fact_4115_mset__unplusm__dist__cases,axiom,
! [A: $tType,S2: A,A5: multiset @ A,B4: multiset @ A,C4: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A5 )
= ( plus_plus @ ( multiset @ A ) @ B4 @ C4 ) )
=> ( ( ( B4
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C4 ) ) )
=> ~ ( ( C4
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ B4 @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases
thf(fact_4116_mset__unplusm__dist__cases2,axiom,
! [A: $tType,B4: multiset @ A,C4: multiset @ A,S2: A,A5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ B4 @ C4 )
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A5 ) )
=> ( ( ( B4
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C4 ) ) )
=> ~ ( ( C4
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ B4 @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ S2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases2
thf(fact_4117_size__Diff1__le,axiom,
! [A: $tType,M6: multiset @ A,X: A] : ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M6 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( size_size @ ( multiset @ A ) @ M6 ) ) ).
% size_Diff1_le
thf(fact_4118_mset__size__le1__cases,axiom,
! [A: $tType,M6: multiset @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ M6 ) @ ( suc @ ( zero_zero @ nat ) ) )
=> ( ( M6
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [M5: A] :
( M6
!= ( add_mset @ A @ M5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size_le1_cases
thf(fact_4119_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y4 @ V3 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V3 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_4120_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_4121_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y4 @ V3 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_4122_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P4: A > $o,R6: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X8: A,Y8: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y4: B] :
( ( X8 = X4 )
& ( P4 @ X4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y4 ) @ ( R6 @ X4 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_4123_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X: A,Y9: B,Y: B,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y9 @ Y ) @ ( R2 @ X ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y9 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_4124_sorted__list__of__multiset__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,M6: multiset @ A] :
( ( linord6283353356039996273ltiset @ A @ ( add_mset @ A @ X @ M6 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord6283353356039996273ltiset @ A @ M6 ) ) ) ) ).
% sorted_list_of_multiset_insert
thf(fact_4125_merge__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L1: list @ A,L22: list @ A] :
( ( ( distinct @ A @ L1 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L1 ) )
=> ( ( ( distinct @ A @ L22 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L22 ) )
=> ( ( distinct @ A @ ( merge @ A @ L1 @ L22 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge @ A @ L1 @ L22 ) )
& ( ( set2 @ A @ ( merge @ A @ L1 @ L22 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ L1 ) @ ( set2 @ A @ L22 ) ) ) ) ) ) ) ).
% merge_correct
thf(fact_4126_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S: list @ A,I5: ( set @ A ) > B > $o,Sigma_0: B,F3: B > A > B] :
( ( distinct @ A @ S )
=> ( ( I5 @ ( set2 @ A @ S ) @ Sigma_0 )
=> ( ! [X3: A,It: set @ A,Sigma: B] :
( ( member @ A @ X3 @ It )
=> ( ( ord_less_eq @ ( set @ A ) @ It @ ( set2 @ A @ S ) )
=> ( ( I5 @ It @ Sigma )
=> ( I5 @ ( minus_minus @ ( set @ A ) @ It @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( F3 @ Sigma @ X3 ) ) ) ) )
=> ( I5 @ ( bot_bot @ ( set @ A ) ) @ ( foldl @ B @ A @ F3 @ Sigma_0 @ S ) ) ) ) ) ).
% distinct_foldl_invar
thf(fact_4127_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,P3: B > A,I2: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( P3 @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I2 @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P3 @ ( insert @ B @ I2 @ I5 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P3 @ I5 ) ) )
& ( ~ ( member @ B @ I2 @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P3 @ ( insert @ B @ I2 @ I5 ) )
= ( times_times @ A @ ( P3 @ I2 ) @ ( groups1962203154675924110t_prod @ B @ A @ P3 @ I5 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_4128_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P3: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P3 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_4129_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_4130_foldl__A1__eq,axiom,
! [A: $tType,F3: A > A > A,N2: A,I2: A,Ww: list @ A] :
( ! [E2: A] :
( ( F3 @ N2 @ E2 )
= E2 )
=> ( ! [E2: A] :
( ( F3 @ E2 @ N2 )
= E2 )
=> ( ! [A6: A,B5: A,C3: A] :
( ( F3 @ A6 @ ( F3 @ B5 @ C3 ) )
= ( F3 @ ( F3 @ A6 @ B5 ) @ C3 ) )
=> ( ( foldl @ A @ A @ F3 @ I2 @ Ww )
= ( F3 @ I2 @ ( foldl @ A @ A @ F3 @ N2 @ Ww ) ) ) ) ) ) ).
% foldl_A1_eq
thf(fact_4131_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A,I5: set @ B] :
( ( groups1962203154675924110t_prod @ B @ A @ G3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( G3 @ X4 )
!= ( one_one @ A ) ) ) ) )
= ( groups1962203154675924110t_prod @ B @ A @ G3 @ I5 ) ) ) ).
% prod.non_neutral'
thf(fact_4132_foldl__absorb1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Zs: list @ A] :
( ( times_times @ A @ X @ ( foldl @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) @ Zs ) )
= ( foldl @ A @ A @ ( times_times @ A ) @ X @ Zs ) ) ) ).
% foldl_absorb1
thf(fact_4133_foldl__un__empty__eq,axiom,
! [A: $tType,I2: set @ A,Ww: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ I2 @ Ww )
= ( sup_sup @ ( set @ A ) @ I2 @ ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ Ww ) ) ) ).
% foldl_un_empty_eq
thf(fact_4134_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I4: B] : ( times_times @ A @ ( G3 @ I4 ) @ ( H @ I4 ) )
@ I5 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G3 @ I5 ) @ ( groups1962203154675924110t_prod @ B @ A @ H @ I5 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_4135_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T2: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ S )
= ( groups1962203154675924110t_prod @ B @ A @ G3 @ T2 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_4136_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T2: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ T2 )
= ( groups1962203154675924110t_prod @ B @ A @ G3 @ S ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_4137_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T2: set @ B,H: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( H @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ S )
= ( groups1962203154675924110t_prod @ B @ A @ H @ T2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_4138_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S: set @ B,T2: set @ B,G3: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T2 @ S ) )
=> ( ( G3 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ( ( G3 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G3 @ T2 )
= ( groups1962203154675924110t_prod @ B @ A @ H @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_4139_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_4140_foldl__length__aux,axiom,
! [A: $tType,A3: nat,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I4: nat,X4: A] : ( suc @ I4 )
@ A3
@ L )
= ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldl_length_aux
thf(fact_4141_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,G3: B > A,H: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( G3 @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ( H @ X4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I4: B] : ( times_times @ A @ ( G3 @ I4 ) @ ( H @ I4 ) )
@ I5 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G3 @ I5 ) @ ( groups1962203154675924110t_prod @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_4142_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P6: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I7 )
& ( ( P6 @ X4 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P6
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I7 )
& ( ( P6 @ X4 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_4143_set__n__lists,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N2 @ Xs ) )
= ( collect @ ( list @ A )
@ ^ [Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys2 )
= N2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys2 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_4144_Gcd__remove0__nat,axiom,
! [M6: set @ nat] :
( ( finite_finite2 @ nat @ M6 )
=> ( ( gcd_Gcd @ nat @ M6 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M6 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_4145_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_4146_num__of__nat_Osimps_I2_J,axiom,
! [N2: nat] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( suc @ N2 ) )
= ( inc @ ( num_of_nat @ N2 ) ) ) )
& ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( num_of_nat @ ( suc @ N2 ) )
= one2 ) ) ) ).
% num_of_nat.simps(2)
thf(fact_4147_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_4148_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( gcd_Gcd @ A @ A5 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_4149_Gcd__mono,axiom,
! [A: $tType,B: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( dvd_dvd @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( dvd_dvd @ A @ ( gcd_Gcd @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( gcd_Gcd @ A @ ( image2 @ B @ A @ G3 @ A5 ) ) ) ) ) ).
% Gcd_mono
thf(fact_4150_subset__subseqs,axiom,
! [A: $tType,X6: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ ( set2 @ A @ Xs ) )
=> ( member @ ( set @ A ) @ X6 @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_4151_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_4152_Gcd__eq__Max,axiom,
! [M6: set @ nat] :
( ( finite_finite2 @ nat @ M6 )
=> ( ( M6
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M6 )
=> ( ( gcd_Gcd @ nat @ M6 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M3: nat] :
( collect @ nat
@ ^ [D6: nat] : ( dvd_dvd @ nat @ D6 @ M3 ) )
@ M6 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_4153_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu3: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N: nat] :
( ( semiring_1_of_nat @ A @ N )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_4154_Gcd__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Gcd @ int @ ( image2 @ int @ int @ ( abs_abs @ int ) @ K5 ) )
= ( gcd_Gcd @ int @ K5 ) ) ).
% Gcd_abs_eq
thf(fact_4155_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_4156_drop__upd__irrelevant,axiom,
! [A: $tType,M: nat,N2: nat,L: list @ A,X: A] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( drop @ A @ N2 @ ( list_update @ A @ L @ M @ X ) )
= ( drop @ A @ N2 @ L ) ) ) ).
% drop_upd_irrelevant
thf(fact_4157_drop__update__cancel,axiom,
! [A: $tType,N2: nat,M: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N2 @ M )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N2 @ X ) )
= ( drop @ A @ M @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_4158_Gcd__nat__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Gcd @ nat
@ ( image2 @ int @ nat
@ ^ [K3: int] : ( nat2 @ ( abs_abs @ int @ K3 ) )
@ K5 ) )
= ( nat2 @ ( gcd_Gcd @ int @ K5 ) ) ) ).
% Gcd_nat_abs_eq
thf(fact_4159_Max__divisors__self__nat,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D6: nat] : ( dvd_dvd @ nat @ D6 @ N2 ) ) )
= N2 ) ) ).
% Max_divisors_self_nat
thf(fact_4160_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_4161_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_4162_Gcd__int__eq,axiom,
! [N7: set @ nat] :
( ( gcd_Gcd @ int @ ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ N7 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N7 ) ) ) ).
% Gcd_int_eq
thf(fact_4163_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ B,C2: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [Uu2: B] : C2
@ A5 ) )
= C2 ) ) ) ) ).
% Max_const
thf(fact_4164_nth__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A,I2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N2 @ Xs ) @ I2 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N2 @ I2 ) ) ) ) ).
% nth_drop
thf(fact_4165_set__drop__subset,axiom,
! [A: $tType,N2: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ N2 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_drop_subset
thf(fact_4166_sorted__drop,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N2: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N2 @ Xs ) ) ) ) ).
% sorted_drop
thf(fact_4167_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max_ge
thf(fact_4168_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( member @ A @ X @ A5 )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= X ) ) ) ) ) ).
% Max_eqI
thf(fact_4169_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ B4 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B4 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ A5 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( lattic643756798349783984er_Max @ A @ B4 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_4170_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ord_less_eq @ A @ A3 @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_4171_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ A5 ) ) ) ) ).
% Max_in
thf(fact_4172_Gcd__int__greater__eq__0,axiom,
! [K5: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K5 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_4173_set__drop__subset__set__drop,axiom,
! [A: $tType,N2: nat,M: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N2 @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_4174_drop__update__swap,axiom,
! [A: $tType,M: nat,N2: nat,Xs: list @ A,X: A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N2 @ X ) )
= ( list_update @ A @ ( drop @ A @ M @ Xs ) @ ( minus_minus @ nat @ N2 @ M ) @ X ) ) ) ).
% drop_update_swap
thf(fact_4175_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A5 )
= M )
= ( ( member @ A @ M @ A5 )
& ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_4176_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_4177_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ( member @ A @ M @ A5 )
& ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_4178_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
=> ! [A15: A] :
( ( member @ A @ A15 @ A5 )
=> ( ord_less_eq @ A @ A15 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_4179_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A5 )
=> ( ord_less_eq @ A @ A6 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_4180_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_4181_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A5 )
=> ( ord_less_eq @ A @ B5 @ A3 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A3 @ A5 ) )
= A3 ) ) ) ) ).
% Max_insert2
thf(fact_4182_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= ( lattic643756798349783984er_Max @ A @ X6 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_4183_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% Max_Sup
thf(fact_4184_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X11: set @ nat] :
( if @ nat
@ ( X11
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X11 ) ) ) ) ).
% Sup_nat_def
thf(fact_4185_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M6: set @ A,N7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M6 @ N7 )
=> ( ( M6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M6 ) @ ( lattic643756798349783984er_Max @ A @ N7 ) ) ) ) ) ) ).
% Max_mono
thf(fact_4186_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ ( lattic643756798349783984er_Max @ A @ B4 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_4187_card__le__Suc__Max,axiom,
! [S: set @ nat] :
( ( finite_finite2 @ nat @ S )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S ) ) ) ) ).
% card_le_Suc_Max
thf(fact_4188_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F3: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F3 @ S ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_4189_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M3: nat,N: nat] :
( if @ nat
@ ( N
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N ) @ M3 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_4190_in__set__drop__conv__nth,axiom,
! [A: $tType,X: A,N2: nat,L: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( drop @ A @ N2 @ L ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq @ nat @ N2 @ I4 )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) )
& ( X
= ( nth @ A @ L @ I4 ) ) ) ) ) ).
% in_set_drop_conv_nth
thf(fact_4191_Gcd__int__def,axiom,
( ( gcd_Gcd @ int )
= ( ^ [K7: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image2 @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K7 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_4192_sum__le__card__Max,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( times_times @ nat @ ( finite_card @ A @ A5 ) @ ( lattic643756798349783984er_Max @ nat @ ( image2 @ A @ nat @ F3 @ A5 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_4193_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_4194_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_4195_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_4196_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_4197_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K: A] :
( ( member @ A @ I2 @ ( set_ord_greaterThan @ A @ K ) )
= ( ord_less @ A @ K @ I2 ) ) ) ).
% greaterThan_iff
thf(fact_4198_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_4199_restrict__restrict,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),A5: set @ A,B4: set @ A] :
( ( restrict_map @ A @ B @ ( restrict_map @ A @ B @ M @ A5 ) @ B4 )
= ( restrict_map @ A @ B @ M @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% restrict_restrict
thf(fact_4200_restrict__map__empty,axiom,
! [B: $tType,A: $tType,D4: set @ A] :
( ( restrict_map @ A @ B
@ ^ [X4: A] : ( none @ B )
@ D4 )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% restrict_map_empty
thf(fact_4201_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% greaterThan_subset_iff
thf(fact_4202_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_4203_Max__divisors__self__int,axiom,
! [N2: int] :
( ( N2
!= ( zero_zero @ int ) )
=> ( ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D6: int] : ( dvd_dvd @ int @ D6 @ N2 ) ) )
= ( abs_abs @ int @ N2 ) ) ) ).
% Max_divisors_self_int
thf(fact_4204_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_4205_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D4: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_4206_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X: A,D4: set @ A,M: A > ( option @ B ),Y: option @ B] :
( ( ( member @ A @ X @ D4 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ Y ) @ D4 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) )
& ( ~ ( member @ A @ X @ D4 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ Y ) @ D4 )
= ( restrict_map @ A @ B @ M @ D4 ) ) ) ) ).
% restrict_fun_upd
thf(fact_4207_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D4: set @ A,M: A > ( option @ B )] :
( ( ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ D4 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_4208_restrict__map__eq_I2_J,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A5: set @ B,K: B,V2: A] :
( ( ( restrict_map @ B @ A @ M @ A5 @ K )
= ( some @ A @ V2 ) )
= ( ( ( M @ K )
= ( some @ A @ V2 ) )
& ( member @ B @ K @ A5 ) ) ) ).
% restrict_map_eq(2)
thf(fact_4209_linorder_OMin_Ocong,axiom,
! [A: $tType] :
( ( lattices_Min @ A )
= ( lattices_Min @ A ) ) ).
% linorder.Min.cong
thf(fact_4210_le__map__restrict,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M: A > ( option @ B ),X6: set @ A] : ( ord_less_eq @ ( A > ( option @ B ) ) @ ( restrict_map @ A @ B @ M @ X6 ) @ M ) ) ).
% le_map_restrict
thf(fact_4211_restrict__map__subset__eq,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),R2: set @ A,M8: A > ( option @ B ),R9: set @ A] :
( ( ( restrict_map @ A @ B @ M @ R2 )
= M8 )
=> ( ( ord_less_eq @ ( set @ A ) @ R9 @ R2 )
=> ( ( restrict_map @ A @ B @ M @ R9 )
= ( restrict_map @ A @ B @ M8 @ R9 ) ) ) ) ).
% restrict_map_subset_eq
thf(fact_4212_greaterThan__non__empty,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
( ( set_ord_greaterThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% greaterThan_non_empty
thf(fact_4213_foldl__foldl__conv__concat,axiom,
! [A: $tType,B: $tType,F3: A > B > A,A3: A,Xs: list @ ( list @ B )] :
( ( foldl @ A @ ( list @ B ) @ ( foldl @ A @ B @ F3 ) @ A3 @ Xs )
= ( foldl @ A @ B @ F3 @ A3 @ ( concat @ B @ Xs ) ) ) ).
% foldl_foldl_conv_concat
thf(fact_4214_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_4215_ran__restrictD,axiom,
! [B: $tType,A: $tType,Y: A,M: B > ( option @ A ),A5: set @ B] :
( ( member @ A @ Y @ ( ran @ B @ A @ ( restrict_map @ B @ A @ M @ A5 ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( ( M @ X3 )
= ( some @ A @ Y ) ) ) ) ).
% ran_restrictD
thf(fact_4216_map__restrict__insert__none__simp,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),X: B,S2: set @ B] :
( ( ( M @ X )
= ( none @ A ) )
=> ( ( restrict_map @ B @ A @ M @ ( uminus_uminus @ ( set @ B ) @ ( insert @ B @ X @ S2 ) ) )
= ( restrict_map @ B @ A @ M @ ( uminus_uminus @ ( set @ B ) @ S2 ) ) ) ) ).
% map_restrict_insert_none_simp
thf(fact_4217_restrict__map__upd,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),S: set @ A,K: A,V2: B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F3 @ S ) @ K @ ( some @ B @ V2 ) )
= ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ K @ ( some @ B @ V2 ) ) @ ( insert @ A @ K @ S ) ) ) ).
% restrict_map_upd
thf(fact_4218_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_4219_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),D4: set @ A,X: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D4 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ).
% fun_upd_restrict
thf(fact_4220_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X: A] :
( ( restrict_map @ A @ B @ F3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_4221_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X: A] :
( ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% map_upd_eq_restrict
thf(fact_4222_distinct__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A ) @ Xs )
=> ( ! [Ys4: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys4 ) )
=> ( ! [Ys4: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys4 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_4223_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_4224_length__remdups__concat,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss ) ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xss ) ) ) ) ) ).
% length_remdups_concat
thf(fact_4225_restrict__map__upds,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B,D4: set @ A,M: A > ( option @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ D4 )
=> ( ( restrict_map @ A @ B @ ( map_upds @ A @ B @ M @ Xs @ Ys3 ) @ D4 )
= ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D4 @ ( set2 @ A @ Xs ) ) ) @ Xs @ Ys3 ) ) ) ) ).
% restrict_map_upds
thf(fact_4226_distinct__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ A @ ( concat @ A @ Xs ) )
= ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs ) )
& ! [Ys2: list @ A] :
( ( member @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys2 ) )
& ! [Ys2: list @ A,Zs3: list @ A] :
( ( ( member @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( member @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( Ys2 != Zs3 ) )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys2 ) @ ( set2 @ A @ Zs3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% distinct_concat_iff
thf(fact_4227_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 ) ) ) ) )
=> ? [N13: nat] :
( ! [N9: nat] :
( ( ord_less_eq @ nat @ N9 @ N13 )
=> ! [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N13 )
=> ( ( ord_less @ nat @ M4 @ N9 )
=> ( ord_less @ A @ ( F3 @ M4 ) @ ( F3 @ N9 ) ) ) ) )
& ! [N9: nat] :
( ( ord_less_eq @ nat @ N13 @ N9 )
=> ( ( F3 @ N13 )
= ( F3 @ N9 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_4228_slice__eq__bounds__empty,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( slice @ A @ I2 @ I2 @ Xs )
= ( nil @ A ) ) ).
% slice_eq_bounds_empty
thf(fact_4229_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] :
( ( slice @ A @ Begin @ End @ ( nil @ A ) )
= ( nil @ A ) ) ).
% slice_Nil
thf(fact_4230_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_4231_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_4232_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_4233_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_4234_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_4235_drop__eq__Nil2,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( drop @ A @ N2 @ Xs ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% drop_eq_Nil2
thf(fact_4236_drop__eq__Nil,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( drop @ A @ N2 @ Xs )
= ( nil @ A ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% drop_eq_Nil
thf(fact_4237_drop__all,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 )
=> ( ( drop @ A @ N2 @ Xs )
= ( nil @ A ) ) ) ).
% drop_all
thf(fact_4238_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( linord4507533701916653071of_set @ A @ A5 )
= ( nil @ A ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4239_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I2: nat,M: A > ( option @ B ),Ys3: list @ B,Y: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I2 )
=> ( ( map_upds @ A @ B @ M @ Xs @ ( list_update @ B @ Ys3 @ I2 @ Y ) )
= ( map_upds @ A @ B @ M @ Xs @ Ys3 ) ) ) ).
% map_upds_list_update2_drop
thf(fact_4240_map__upds__twist,axiom,
! [A: $tType,B: $tType,A3: A,As: list @ A,M: A > ( option @ B ),B2: B,Bs: list @ B] :
( ~ ( member @ A @ A3 @ ( set2 @ A @ As ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ A3 @ ( some @ B @ B2 ) ) @ As @ Bs )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ As @ Bs ) @ A3 @ ( some @ B @ B2 ) ) ) ) ).
% map_upds_twist
thf(fact_4241_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_4242_merge_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L22: list @ A] :
( ( merge @ A @ ( nil @ A ) @ L22 )
= L22 ) ) ).
% merge.simps(1)
thf(fact_4243_shuffles_Osimps_I1_J,axiom,
! [A: $tType,Ys3: list @ A] :
( ( shuffles @ A @ ( nil @ A ) @ Ys3 )
= ( insert @ ( list @ A ) @ Ys3 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(1)
thf(fact_4244_shuffles_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(2)
thf(fact_4245_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_4246_mono__pow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,N2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N2 @ F3 ) ) ) ) ).
% mono_pow
thf(fact_4247_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% monoD
thf(fact_4248_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% monoE
thf(fact_4249_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( order_mono @ A @ B @ F3 ) ) ) ).
% monoI
thf(fact_4250_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F4: A > B] :
! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ) ) ).
% mono_def
thf(fact_4251_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_4252_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% mono_strict_invE
thf(fact_4253_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mono_invE
thf(fact_4254_mono__iff__le__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N: nat] : ( ord_less_eq @ A @ ( F4 @ N ) @ ( F4 @ ( suc @ N ) ) ) ) ) ) ).
% mono_iff_le_Suc
thf(fact_4255_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F3: A > B,A5: A,B4: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( inf_inf @ A @ A5 @ B4 ) ) @ ( inf_inf @ B @ ( F3 @ A5 ) @ ( F3 @ B4 ) ) ) ) ) ).
% mono_inf
thf(fact_4256_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F3: A > B,A5: A,B4: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F3 @ A5 ) @ ( F3 @ B4 ) ) @ ( F3 @ ( sup_sup @ A @ A5 @ B4 ) ) ) ) ) ).
% mono_sup
thf(fact_4257_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,A5: A,B4: A,N2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N2 @ F3 @ A5 ) @ ( compow @ ( A > A ) @ N2 @ F3 @ B4 ) ) ) ) ) ).
% funpow_mono
thf(fact_4258_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_4259_len__greater__imp__nonempty,axiom,
! [A: $tType,X: nat,L: list @ A] :
( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ L ) )
=> ( L
!= ( nil @ A ) ) ) ).
% len_greater_imp_nonempty
thf(fact_4260_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_4261_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_4262_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_4263_Rings_Omono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A3 ) ) ) ) ).
% Rings.mono_mult
thf(fact_4264_mono__times__nat,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( order_mono @ nat @ nat @ ( times_times @ nat @ N2 ) ) ) ).
% mono_times_nat
thf(fact_4265_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F3: A > B,M: A,N2: A,M8: B,N6: B] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ( image2 @ A @ B @ F3 @ ( set_or7035219750837199246ssThan @ A @ M @ N2 ) )
= ( set_or7035219750837199246ssThan @ B @ M8 @ N6 ) )
=> ( ( ord_less @ A @ M @ N2 )
=> ( ( F3 @ M )
= M8 ) ) ) ) ) ).
% mono_image_least
thf(fact_4266_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F3: A > A,P3: A,K: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ P3 ) @ P3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F3 @ ( bot_bot @ A ) ) @ P3 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_4267_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F3: A > A,P3: A,K: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ P3 @ ( F3 @ P3 ) )
=> ( ord_less_eq @ A @ P3 @ ( compow @ ( A > A ) @ K @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_4268_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,I2: nat,J: nat,X: A,Y: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ ( F3 @ X ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I2 @ F3 @ X ) @ ( compow @ ( A > A ) @ J @ F3 @ Y ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_4269_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% mono_Sup
thf(fact_4270_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A5: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A5 @ X4 ) )
@ I5 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A5 @ I5 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_4271_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A5: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A5 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A5 @ X4 ) )
@ I5 ) ) ) ) ) ).
% mono_INF
thf(fact_4272_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A5 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ) ) ).
% mono_Inf
thf(fact_4273_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_4274_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M: nat,N2: nat,F3: A > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M @ F3 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N2 @ F3 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_4275_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M: nat,N2: nat,F3: A > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N2 @ F3 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_4276_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F3 @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_4277_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( order_mono @ nat @ nat
@ ^ [M3: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M3 ) @ M3 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_4278_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys3 )
= ( ? [F4: nat > nat] :
( ( order_mono @ nat @ nat @ F4 )
& ( ( image2 @ nat @ nat @ F4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys3 ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys3 @ ( F4 @ 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 ) ) ) )
= ( ( F4 @ I4 )
= ( F4 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_4279_distinct__concat_H,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] :
( Ys2
!= ( nil @ A ) )
@ Xs ) )
=> ( ! [Ys4: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys4 ) )
=> ( ! [Ys4: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys4 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat'
thf(fact_4280_fold__union__pair,axiom,
! [B: $tType,A: $tType,B4: set @ A,X: B,A5: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B4 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y4: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B4 ) )
@ A5 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y4: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y4 ) )
@ A5
@ B4 ) ) ) ).
% fold_union_pair
thf(fact_4281_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_4282_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_4283_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_4284_take__update,axiom,
! [A: $tType,N2: nat,L: list @ A,I2: nat,X: A] :
( ( take @ A @ N2 @ ( list_update @ A @ L @ I2 @ X ) )
= ( list_update @ A @ ( take @ A @ N2 @ L ) @ I2 @ X ) ) ).
% take_update
thf(fact_4285_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_4286_take__all,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 )
=> ( ( take @ A @ N2 @ Xs )
= Xs ) ) ).
% take_all
thf(fact_4287_take__all__iff,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( take @ A @ N2 @ Xs )
= Xs )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) ) ).
% take_all_iff
thf(fact_4288_nth__take,axiom,
! [A: $tType,I2: nat,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ N2 )
=> ( ( nth @ A @ ( take @ A @ N2 @ Xs ) @ I2 )
= ( nth @ A @ Xs @ I2 ) ) ) ).
% nth_take
thf(fact_4289_take__update__cancel,axiom,
! [A: $tType,N2: nat,M: nat,Xs: list @ A,Y: A] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( take @ A @ N2 @ ( list_update @ A @ Xs @ M @ Y ) )
= ( take @ A @ N2 @ Xs ) ) ) ).
% take_update_cancel
thf(fact_4290_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_4291_removeAll__filter__not__eq,axiom,
! [A: $tType] :
( ( removeAll @ A )
= ( ^ [X4: A] :
( filter2 @ A
@ ^ [Y4: A] : X4 != Y4 ) ) ) ).
% removeAll_filter_not_eq
thf(fact_4292_filter__nth__ex__nth,axiom,
! [A: $tType,N2: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) )
=> ? [M5: nat] :
( ( ord_less_eq @ nat @ N2 @ M5 )
& ( ord_less @ nat @ M5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ ( filter2 @ A @ P @ Xs ) @ N2 )
= ( nth @ A @ Xs @ M5 ) )
& ( ( filter2 @ A @ P @ ( take @ A @ M5 @ Xs ) )
= ( take @ A @ N2 @ ( filter2 @ A @ P @ Xs ) ) ) ) ) ).
% filter_nth_ex_nth
thf(fact_4293_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_4294_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_4295_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_4296_mono__Int,axiom,
! [B: $tType,A: $tType,F3: ( set @ A ) > ( set @ B ),A5: set @ A,B4: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) @ ( inf_inf @ ( set @ B ) @ ( F3 @ A5 ) @ ( F3 @ B4 ) ) ) ) ).
% mono_Int
thf(fact_4297_mono__Un,axiom,
! [B: $tType,A: $tType,F3: ( set @ A ) > ( set @ B ),A5: set @ A,B4: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F3 @ A5 ) @ ( F3 @ B4 ) ) @ ( F3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% mono_Un
thf(fact_4298_set__take__subset,axiom,
! [A: $tType,N2: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ N2 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_take_subset
thf(fact_4299_sorted__take,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N2: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N2 @ Xs ) ) ) ) ).
% sorted_take
thf(fact_4300_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_4301_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_4302_inter__set__filter,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( set2 @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_4303_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G3: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X4: A] :
( X4
= ( G3 @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_4304_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_4305_concat__filter__neq__Nil,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( concat @ A
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] :
( Ys2
!= ( nil @ A ) )
@ Xs ) )
= ( concat @ A @ Xs ) ) ).
% concat_filter_neq_Nil
thf(fact_4306_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_4307_set__take__subset__set__take,axiom,
! [A: $tType,M: nat,N2: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N2 @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_4308_union__fold__insert,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= ( finite_fold @ A @ ( set @ A ) @ ( insert @ A ) @ B4 @ A5 ) ) ) ).
% union_fold_insert
thf(fact_4309_sup__Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A5 ) @ B4 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ B4 @ A5 ) ) ) ) ).
% sup_Sup_fold_sup
thf(fact_4310_inf__Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B4: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A5 ) @ B4 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ B4 @ A5 ) ) ) ) ).
% inf_Inf_fold_inf
thf(fact_4311_slice__def,axiom,
! [A: $tType] :
( ( slice @ A )
= ( ^ [From2: nat,To2: nat,List: list @ A] : ( take @ A @ ( minus_minus @ nat @ To2 @ From2 ) @ ( drop @ A @ From2 @ List ) ) ) ) ).
% slice_def
thf(fact_4312_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list @ A,Ys3: list @ A] :
( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys3 @ I3 ) ) )
=> ( ( take @ A @ K @ Xs )
= ( take @ A @ K @ Ys3 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_4313_set__minus__filter__out,axiom,
! [A: $tType,Xs: list @ A,Y: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : X4 != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_4314_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( complete_Sup_Sup @ A @ A5 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A5 ) ) ) ) ).
% Sup_fold_sup
thf(fact_4315_Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( complete_Inf_Inf @ A @ A5 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ A5 ) ) ) ) ).
% Inf_fold_inf
thf(fact_4316_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys3 ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
@ Zs )
= Ys3 ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_4317_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys3 ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_4318_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys3 ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Ys3 ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_4319_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys3 ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Ys3 ) )
@ Zs )
= Ys3 ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_4320_length__filter__conv__card,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P3 @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P3 @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_4321_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_4322_remdups__adj__adjacent,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I2 )
!= ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I2 ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_4323_image__fold__insert,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( image2 @ A @ B @ F3 @ A5 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K3: A] : ( insert @ B @ ( F3 @ K3 ) )
@ ( bot_bot @ ( set @ B ) )
@ A5 ) ) ) ).
% image_fold_insert
thf(fact_4324_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_4325_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X: A,Ys3: list @ B,Xs: list @ A,F3: A > ( option @ B ),Y: B] :
( ( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys3 ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ ( some @ B @ Y ) ) @ Xs @ Ys3 )
= ( map_upds @ A @ B @ F3 @ Xs @ Ys3 ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys3 ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ ( some @ B @ Y ) ) @ Xs @ Ys3 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F3 @ Xs @ Ys3 ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_4326_Union__take__drop__id,axiom,
! [A: $tType,N2: nat,L: list @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ N2 @ L ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( take @ ( set @ A ) @ N2 @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ L ) ) ) ).
% Union_take_drop_id
thf(fact_4327_sup__SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B4: A,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( sup_sup @ A @ B4 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ B4 @ A5 ) ) ) ) ).
% sup_SUP_fold_sup
thf(fact_4328_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B4: A,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( inf_inf @ A @ B4 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F3 ) @ B4 @ A5 ) ) ) ) ).
% inf_INF_fold_inf
thf(fact_4329_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_4330_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list @ A,I2: nat,J: nat] :
( ( distinct @ A @ Vs )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I2 @ Vs ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_4331_foldl__list__update,axiom,
! [B: $tType,A: $tType,N2: nat,Xs: list @ A,F3: B > A > B,A3: B,X: A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( foldl @ B @ A @ F3 @ A3 @ ( list_update @ A @ Xs @ N2 @ X ) )
= ( foldl @ B @ A @ F3 @ ( F3 @ ( foldl @ B @ A @ F3 @ A3 @ ( take @ A @ N2 @ Xs ) ) @ X ) @ ( drop @ A @ ( suc @ N2 ) @ Xs ) ) ) ) ).
% foldl_list_update
thf(fact_4332_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ ( bot_bot @ A ) @ A5 ) ) ) ) ).
% SUP_fold_sup
thf(fact_4333_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F3 ) @ ( top_top @ A ) @ A5 ) ) ) ) ).
% INF_fold_inf
thf(fact_4334_listset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( listset @ A @ ( nil @ ( set @ A ) ) )
= ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listset.simps(1)
thf(fact_4335_Set__filter__fold,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( filter3 @ A @ P @ A5 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X4: A,A16: set @ A] : ( if @ ( set @ A ) @ ( P @ X4 ) @ ( insert @ A @ X4 @ A16 ) @ A16 )
@ ( bot_bot @ ( set @ A ) )
@ A5 ) ) ) ).
% Set_filter_fold
thf(fact_4336_graph__map__upd,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),K: A,V2: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V2 ) ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_4337_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 ) )
= ( insert @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(2)
thf(fact_4338_member__filter,axiom,
! [A: $tType,X: A,P: A > $o,A5: set @ A] :
( ( member @ A @ X @ ( filter3 @ A @ P @ A5 ) )
= ( ( member @ A @ X @ A5 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_4339_graph__empty,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B
@ ^ [X4: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% graph_empty
thf(fact_4340_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P4: A > $o,A8: set @ A] :
( collect @ A
@ ^ [A7: A] :
( ( member @ A @ A7 @ A8 )
& ( P4 @ A7 ) ) ) ) ) ).
% Set.filter_def
thf(fact_4341_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V2: B,M: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ M ) )
=> ( ( M @ K )
= ( some @ B @ V2 ) ) ) ).
% in_graphD
thf(fact_4342_in__graphI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),K: B,V2: A] :
( ( ( M @ K )
= ( some @ A @ V2 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V2 ) @ ( graph @ B @ A @ M ) ) ) ).
% in_graphI
thf(fact_4343_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V2: B,M: A > ( option @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A5 ) ) )
=> ( member @ A @ K @ A5 ) ) ).
% graph_restrictD(1)
thf(fact_4344_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu2: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_4345_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_4346_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V2: B,M: A > ( option @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A5 ) ) )
=> ( ( M @ K )
= ( some @ B @ V2 ) ) ) ).
% graph_restrictD(2)
thf(fact_4347_inter__Set__filter,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( filter3 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 )
@ B4 ) ) ) ).
% inter_Set_filter
thf(fact_4348_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ( ( shuffles @ A @ ( nil @ A ) @ Ys3 )
= ( insert @ ( list @ A ) @ Ys3 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(1)
thf(fact_4349_coinduct3__mono__lemma,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [F3: A > ( set @ B ),X6: set @ B,B4: set @ B] :
( ( order_mono @ A @ ( set @ B ) @ F3 )
=> ( order_mono @ A @ ( set @ B )
@ ^ [X4: A] : ( sup_sup @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F3 @ X4 ) @ X6 ) @ B4 ) ) ) ) ).
% coinduct3_mono_lemma
thf(fact_4350_Pow__fold,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( pow @ A @ A5 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A8: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A8 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X4 ) @ A8 ) )
@ ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A5 ) ) ) ).
% Pow_fold
thf(fact_4351_shuffles_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( insert @ ( 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
= ( insert @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( Y
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y3 @ Ys4 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs2 ) @ Ys4 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_4352_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lex @ A @ R3 ) )
=> ~ ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Ys3 ) )
=> ( ( ( take @ A @ I3 @ Xs )
= ( take @ A @ I3 @ Ys3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Ys3 @ I3 ) ) @ R3 ) ) ) ) ) ).
% lex_take_index
thf(fact_4353_Pow__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( member @ ( set @ A ) @ A5 @ ( pow @ A @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% Pow_iff
thf(fact_4354_PowI,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( member @ ( set @ A ) @ A5 @ ( pow @ A @ B4 ) ) ) ).
% PowI
thf(fact_4355_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_4356_Pow__Int__eq,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( pow @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( pow @ A @ B4 ) ) ) ).
% Pow_Int_eq
thf(fact_4357_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F3: B > A,A3: A,X: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ ( cons @ B @ X @ Xs ) )
= ( plus_plus @ A @ ( F3 @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_4358_enumerate__simps_I2_J,axiom,
! [B: $tType,N2: nat,X: B,Xs: list @ B] :
( ( enumerate @ B @ N2 @ ( cons @ B @ X @ Xs ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N2 @ X ) @ ( enumerate @ B @ ( suc @ N2 ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_4359_map__upds__Cons,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),A3: A,As: list @ A,B2: B,Bs: list @ B] :
( ( map_upds @ A @ B @ M @ ( cons @ A @ A3 @ As ) @ ( cons @ B @ B2 @ Bs ) )
= ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ A3 @ ( some @ B @ B2 ) ) @ As @ Bs ) ) ).
% map_upds_Cons
thf(fact_4360_Pow__singleton__iff,axiom,
! [A: $tType,X6: set @ A,Y7: set @ A] :
( ( ( pow @ A @ X6 )
= ( insert @ ( set @ A ) @ Y7 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X6
= ( bot_bot @ ( set @ A ) ) )
& ( Y7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_4361_Pow__empty,axiom,
! [A: $tType] :
( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_4362_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys3 ) ) @ ( lex @ A @ R3 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lex @ A @ R3 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_4363_nth__Cons__pos,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_4364_neq__NilE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( L
!= ( cons @ A @ X3 @ Xs2 ) ) ) ).
% neq_NilE
thf(fact_4365_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_4366_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X12: A,X22: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X12 @ ( cons @ A @ X22 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_4367_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R2: A > B > $o,Xs: list @ A,Ys3: list @ B] :
( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ B ) )
=> ( ! [X_1: list @ B] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys4: list @ B] :
( ( R2 @ X3 @ Y3 )
=> ( ( P @ Xs2 @ ( cons @ B @ Y3 @ Ys4 ) )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) ) ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys4: list @ B] :
( ~ ( R2 @ X3 @ Y3 )
=> ( ( P @ ( cons @ A @ X3 @ Xs2 ) @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_4368_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P2: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( nil @ A ) ) )
=> ( ! [P2: A > A > $o,X3: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [P2: A > A > $o,X3: A,Y3: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_4369_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F2: A > B,X3: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ( ! [F2: A > B,X3: A,Y3: A,Zs2: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) ) )
=> ~ ! [A6: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A6 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_4370_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P2: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( nil @ A ) ) )
=> ~ ! [P2: A > A > $o,X3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P2 @ ( cons @ A @ X3 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_4371_Cons__shuffles__subset2,axiom,
! [A: $tType,Y: A,Xs: list @ A,Ys3: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ Xs @ Ys3 ) ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys3 ) ) ) ).
% Cons_shuffles_subset2
thf(fact_4372_Cons__shuffles__subset1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys3: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ Ys3 ) ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys3 ) ) ).
% Cons_shuffles_subset1
thf(fact_4373_PowD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( member @ ( set @ A ) @ A5 @ ( pow @ A @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% PowD
thf(fact_4374_Pow__top,axiom,
! [A: $tType,A5: set @ A] : ( member @ ( set @ A ) @ A5 @ ( pow @ A @ A5 ) ) ).
% Pow_top
thf(fact_4375_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_4376_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys3: list @ A] :
( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys3 ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys3 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys3 ) ) ) ) ).
% shuffles.simps(3)
thf(fact_4377_Pow__bottom,axiom,
! [A: $tType,B4: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B4 ) ) ).
% Pow_bottom
thf(fact_4378_Pow__not__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( pow @ A @ A5 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_4379_Pow__def,axiom,
! [A: $tType] :
( ( pow @ A )
= ( ^ [A8: set @ A] :
( collect @ ( set @ A )
@ ^ [B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% Pow_def
thf(fact_4380_drop__Cons,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( drop @ A @ N2 @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ X @ Xs )
@ ^ [M3: nat] : ( drop @ A @ M3 @ Xs )
@ N2 ) ) ).
% drop_Cons
thf(fact_4381_list__update_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I2: nat,V2: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ I2 @ V2 )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V2 @ Xs )
@ ^ [J3: nat] : ( cons @ A @ X @ ( list_update @ A @ Xs @ J3 @ V2 ) )
@ I2 ) ) ).
% list_update.simps(2)
thf(fact_4382_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_4383_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
=> ( Xs
!= ( cons @ A @ X @ Ys3 ) ) ) ).
% impossible_Cons
thf(fact_4384_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Zs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted2
thf(fact_4385_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ~ ! [X3: A,Xs2: list @ A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Ys4 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_4386_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F2: A > B > C] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [F2: A > B > C,A6: A,As4: list @ A,B5: B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As4 ) @ ( cons @ B @ B5 @ Bs2 ) ) ) )
=> ( ! [A6: A > B > C,V4: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [A6: A > B > C,V4: B,Va: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ).
% zipf.cases
thf(fact_4387_merge_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [L23: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L23 ) )
=> ( ! [V4: A,Va: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ A ) ) )
=> ~ ! [X12: A,L12: list @ A,X22: A,L23: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ L12 ) @ ( cons @ A @ X22 @ L23 ) ) ) ) ) ) ).
% merge.cases
thf(fact_4388_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Xs2: list @ A,Y3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_4389_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [P2: A > B > $o] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [P2: A > B > $o,A6: A,As4: list @ A,B5: B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As4 ) @ ( cons @ B @ B5 @ Bs2 ) ) ) )
=> ( ! [P2: A > B > $o,V4: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [P2: A > B > $o,V4: B,Va: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ).
% list_all_zip.cases
thf(fact_4390_partition__rev_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
( ! [P2: A > $o,Yes: list @ A,No: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P2 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) )
=> ~ ! [P2: A > $o,Yes: list @ A,No: list @ A,X3: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P2 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X3 @ Xs2 ) ) ) ) ) ).
% partition_rev.cases
thf(fact_4391_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F2: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F2: A > B,A6: A,As4: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As4 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_4392_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R10: A > A > $o,Sl: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R10 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ~ ! [R10: A > A > $o,Sl: list @ A,X3: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R10 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X3 @ Xs2 ) ) ) ) ) ).
% quicksort_by_rel.cases
thf(fact_4393_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R10: A > A > $o,X3: A,Xs2: list @ A,Y3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R10 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) )
=> ( ! [R10: A > A > $o,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R10 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) ) )
=> ~ ! [R10: A > A > $o,V4: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R10 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V4 @ Va ) ) ) ) ) ) ).
% mergesort_by_rel_merge.cases
thf(fact_4394_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
( ! [Xs1: list @ A,Xs22: list @ A] :
( X
!= ( 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: A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [Xs1: list @ A,Xs22: list @ A,X12: A,X22: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X12 @ ( cons @ A @ X22 @ Xs2 ) ) ) ) ) ) ).
% mergesort_by_rel_split.cases
thf(fact_4395_drop__eq__ConsD,axiom,
! [A: $tType,N2: nat,Xs: list @ A,X: A,Xs4: list @ A] :
( ( ( drop @ A @ N2 @ Xs )
= ( cons @ A @ X @ Xs4 ) )
=> ( ( drop @ A @ ( suc @ N2 ) @ Xs )
= Xs4 ) ) ).
% drop_eq_ConsD
thf(fact_4396_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Y: B,Ys3: list @ B] :
( ( ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y @ Ys3 ) )
= ( cons @ B @ X @ ( cons @ B @ Y @ Ys3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y @ Ys3 ) )
= ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F3 @ X @ Ys3 ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_4397_merge_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X1: A,X2: A,L1: list @ A,L22: list @ A] :
( ( ( ord_less @ A @ X1 @ X2 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L22 ) )
= ( cons @ A @ X1 @ ( merge @ A @ L1 @ ( cons @ A @ X2 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X2 )
=> ( ( ( X1 = X2 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L22 ) )
= ( cons @ A @ X1 @ ( merge @ A @ L1 @ L22 ) ) ) )
& ( ( X1 != X2 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L22 ) )
= ( cons @ A @ X2 @ ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ).
% merge.simps(3)
thf(fact_4398_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_4399_Pow__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( pow @ A @ B4 ) ) ) ).
% Pow_mono
thf(fact_4400_subset__Pow__Union,axiom,
! [A: $tType,A5: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A5 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) ) ) ).
% subset_Pow_Union
thf(fact_4401_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B,B4: set @ A] :
( ( ( image2 @ B @ A @ F3 @ A5 )
= B4 )
=> ( ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( pow @ B @ A5 ) )
= ( pow @ A @ B4 ) ) ) ).
% image_Pow_surj
thf(fact_4402_Pow__set_I2_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( pow @ B @ ( set2 @ B @ ( cons @ B @ X @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow @ B @ ( set2 @ B @ Xs ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert @ B @ X ) @ ( pow @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_4403_take__Cons,axiom,
! [A: $tType,N2: nat,X: A,Xs: list @ A] :
( ( take @ A @ N2 @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( nil @ A )
@ ^ [M3: nat] : ( cons @ A @ X @ ( take @ A @ M3 @ Xs ) )
@ N2 ) ) ).
% take_Cons
thf(fact_4404_Pow__INT__eq,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( pow @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ ( set @ A ) )
@ ( image2 @ B @ ( set @ ( set @ A ) )
@ ^ [X4: B] : ( pow @ A @ ( B4 @ X4 ) )
@ A5 ) ) ) ).
% Pow_INT_eq
thf(fact_4405_nth__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,N2: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N2 )
= ( case_nat @ A @ X @ ( nth @ A @ Xs ) @ N2 ) ) ).
% nth_Cons
thf(fact_4406_Fpow__subset__Pow,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A5 ) @ ( pow @ A @ A5 ) ) ).
% Fpow_subset_Pow
thf(fact_4407_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R3 ) ) ).
% Nil2_notin_lex
thf(fact_4408_Nil__notin__lex,axiom,
! [A: $tType,Ys3: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) @ ( lex @ A @ R3 ) ) ).
% Nil_notin_lex
thf(fact_4409_length__compl__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [E2: A,L4: list @ A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L4 ) )
=> ( P @ Ll ) )
=> ( P @ ( cons @ A @ E2 @ L4 ) ) )
=> ( P @ L ) ) ) ).
% length_compl_induct
thf(fact_4410_Suc__le__length__iff,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X4: A,Ys2: list @ A] :
( ( Xs
= ( cons @ A @ X4 @ Ys2 ) )
& ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_4411_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_4412_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ Ys3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys3 ) )
=> ( ord_less_eq @ A @ X @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys3 ) ) ) ) ).
% sorted_simps(2)
thf(fact_4413_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_4414_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys3 ) )
=> ( ord_less @ A @ X @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys3 ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_4415_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X11: $o > A,Y10: $o > A] :
( ( ord_less_eq @ A @ ( X11 @ $false ) @ ( Y10 @ $false ) )
& ( ord_less_eq @ A @ ( X11 @ $true ) @ ( Y10 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_4416_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F3: B > A,A3: B] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ord_less_eq @ A @ ( F3 @ A3 ) @ ( F3 @ X3 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A3 @ Xs )
= ( cons @ B @ A3 @ Xs ) ) ) ) ).
% insort_is_Cons
thf(fact_4417_merge_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( merge @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ( ! [V4: A,Va: list @ A] :
( ( X
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V4 @ Va ) ) ) )
=> ~ ! [X12: A,L12: list @ A] :
( ( X
= ( cons @ A @ X12 @ L12 ) )
=> ! [X22: A,L23: list @ A] :
( ( Xa
= ( cons @ A @ X22 @ L23 ) )
=> ~ ( ( ( ord_less @ A @ X12 @ X22 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L12 @ ( cons @ A @ X22 @ L23 ) ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X22 )
=> ( ( ( X12 = X22 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L12 @ L23 ) ) ) )
& ( ( X12 != X22 )
=> ( Y
= ( cons @ A @ X22 @ ( merge @ A @ ( cons @ A @ X12 @ L12 ) @ L23 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge.elims
thf(fact_4418_Un__Pow__subset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( pow @ A @ B4 ) ) @ ( pow @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% Un_Pow_subset
thf(fact_4419_UN__Pow__subset,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ord_less_eq @ ( set @ ( set @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ B @ ( set @ ( set @ A ) )
@ ^ [X4: B] : ( pow @ A @ ( B4 @ X4 ) )
@ A5 ) )
@ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% UN_Pow_subset
thf(fact_4420_Pow__insert,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( pow @ A @ ( insert @ A @ A3 @ A5 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow @ A @ A5 ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ A3 ) @ ( pow @ A @ A5 ) ) ) ) ).
% Pow_insert
thf(fact_4421_Fpow__Pow__finite,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A8: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow @ A @ A8 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_4422_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A5 ) @ B4 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( pow @ B @ A5 ) ) @ ( pow @ A @ B4 ) ) ) ).
% image_Pow_mono
thf(fact_4423_shuffles_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P @ ( nil @ A ) @ Ys4 ) )
=> ( ! [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 ) ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ( P @ Xs2 @ ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( P @ ( cons @ A @ X3 @ Xs2 ) @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_4424_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys3 ) ) )
=> ( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys3 ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys3 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys3 ) ) ) ) ) ).
% shuffles.psimps(3)
thf(fact_4425_shuffles_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( insert @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( insert @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( Y
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y3 @ Ys4 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X3 @ Xs2 ) @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_4426_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_4427_binomial__def,axiom,
( binomial
= ( ^ [N: nat,K3: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K7: set @ nat] :
( ( member @ ( set @ nat ) @ K7 @ ( pow @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
& ( ( finite_card @ nat @ K7 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_4428_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N2: nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N2 )
= X )
= ( N2
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_4429_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A,N2: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N2 )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_4430_Cons__nth__drop__Suc,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( cons @ A @ ( nth @ A @ Xs @ I2 ) @ ( drop @ A @ ( suc @ I2 ) @ Xs ) )
= ( drop @ A @ I2 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_4431_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X: A,Xs: list @ A] :
( ( ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( slice @ A @ Begin @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) )
& ( ~ ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% slice_Cons
thf(fact_4432_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_4433_set__Cons__sing__Nil,axiom,
! [A: $tType,A5: set @ A] :
( ( set_Cons @ A @ A5 @ ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image2 @ A @ ( list @ A )
@ ^ [X4: A] : ( cons @ A @ X4 @ ( nil @ A ) )
@ A5 ) ) ).
% set_Cons_sing_Nil
thf(fact_4434_merge_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( merge @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ( ! [V4: A,Va: list @ A] :
( ( X
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( cons @ A @ V4 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X12: A,L12: list @ A] :
( ( X
= ( cons @ A @ X12 @ L12 ) )
=> ! [X22: A,L23: list @ A] :
( ( Xa
= ( cons @ A @ X22 @ L23 ) )
=> ( ( ( ( ord_less @ A @ X12 @ X22 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L12 @ ( cons @ A @ X22 @ L23 ) ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X22 )
=> ( ( ( X12 = X22 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L12 @ L23 ) ) ) )
& ( ( X12 != X22 )
=> ( Y
= ( cons @ A @ X22 @ ( merge @ A @ ( cons @ A @ X12 @ L12 ) @ L23 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ L12 ) @ ( cons @ A @ X22 @ L23 ) ) ) ) ) ) ) ) ) ) ) ).
% merge.pelims
thf(fact_4435_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys3: list @ B,M: A > ( option @ B ),X: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys3 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ Xs @ Ys3 ) @ X @ ( some @ B @ ( nth @ B @ Ys3 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_4436_empty__append__eq__id,axiom,
! [A: $tType] :
( ( append @ A @ ( nil @ A ) )
= ( ^ [X4: list @ A] : X4 ) ) ).
% empty_append_eq_id
thf(fact_4437_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E22: A,L1: list @ A,E12: A,E23: A,L22: list @ A] :
( ( ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) )
= ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L22 ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_4438_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L1: list @ A,E12: A,E23: A,L22: list @ A,E1: A,E22: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L22 ) ) )
= ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_4439_list__se__match_I1_J,axiom,
! [A: $tType,L1: list @ A,L22: list @ A,A3: A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L22 )
= ( cons @ A @ A3 @ ( nil @ A ) ) )
= ( ( L1
= ( cons @ A @ A3 @ ( nil @ A ) ) )
& ( L22
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_4440_list__se__match_I2_J,axiom,
! [A: $tType,L22: list @ A,L1: list @ A,A3: A] :
( ( L22
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L22 )
= ( cons @ A @ A3 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( L22
= ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_4441_list__se__match_I3_J,axiom,
! [A: $tType,L1: list @ A,A3: A,L22: list @ A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A3 @ ( nil @ A ) )
= ( append @ A @ L1 @ L22 ) )
= ( ( L1
= ( cons @ A @ A3 @ ( nil @ A ) ) )
& ( L22
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_4442_list__se__match_I4_J,axiom,
! [A: $tType,L22: list @ A,A3: A,L1: list @ A] :
( ( L22
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A3 @ ( nil @ A ) )
= ( append @ A @ L1 @ L22 ) )
= ( ( L1
= ( nil @ A ) )
& ( L22
= ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_4443_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E3: A,L1: list @ A,E4: A,L22: list @ A] :
( ( ( cons @ A @ E3 @ ( nil @ A ) )
= ( append @ A @ L1 @ ( cons @ A @ E4 @ L22 ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E4 = E3 )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_4444_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L1: list @ A,E4: A,L22: list @ A,E3: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E4 @ L22 ) )
= ( cons @ A @ E3 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E4 = E3 )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_4445_op__conc__empty__img__id,axiom,
! [A: $tType,L6: set @ ( list @ A )] :
( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( nil @ A ) ) @ L6 )
= L6 ) ).
% op_conc_empty_img_id
thf(fact_4446_nth__append__first,axiom,
! [A: $tType,I2: nat,L: list @ A,L3: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( append @ A @ L @ L3 ) @ I2 )
= ( nth @ A @ L @ I2 ) ) ) ).
% nth_append_first
thf(fact_4447_distinct__append,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A] :
( ( distinct @ A @ ( append @ A @ Xs @ Ys3 ) )
= ( ( distinct @ A @ Xs )
& ( distinct @ A @ Ys3 )
& ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% distinct_append
thf(fact_4448_merge__list_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) )] :
( ( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [L4: list @ A] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L4 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A ),L4: list @ A] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L4 @ ( nil @ ( list @ A ) ) ) ) )
=> ~ ! [Acc22: list @ ( list @ A ),L12: list @ A,L23: list @ A,Ls: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) ) ) ) ) ) ) ) ).
% merge_list.cases
thf(fact_4449_list__match__lel__lel,axiom,
! [A: $tType,C12: list @ A,Qs: A,C23: list @ A,C13: list @ A,Qs2: A,C24: list @ A] :
( ( ( append @ A @ C12 @ ( cons @ A @ Qs @ C23 ) )
= ( append @ A @ C13 @ ( cons @ A @ Qs2 @ C24 ) ) )
=> ( ! [C21: list @ A] :
( ( C12
= ( append @ A @ C13 @ ( cons @ A @ Qs2 @ C21 ) ) )
=> ( C24
!= ( append @ A @ C21 @ ( cons @ A @ Qs @ C23 ) ) ) )
=> ( ( ( C13 = C12 )
=> ( ( Qs2 = Qs )
=> ( C24 != C23 ) ) )
=> ~ ! [C212: list @ A] :
( ( C13
= ( append @ A @ C12 @ ( cons @ A @ Qs @ C212 ) ) )
=> ( C23
!= ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C24 ) ) ) ) ) ) ) ).
% list_match_lel_lel
thf(fact_4450_foldl__conc__empty__eq,axiom,
! [A: $tType,I2: list @ A,Ww: list @ ( list @ A )] :
( ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ I2 @ Ww )
= ( append @ A @ I2 @ ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ ( nil @ A ) @ Ww ) ) ) ).
% foldl_conc_empty_eq
thf(fact_4451_mergesort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
~ ! [R10: A > A > $o,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R10 @ Xs2 ) ) ).
% mergesort_by_rel.cases
thf(fact_4452_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys3: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys3 @ Zs )
= ( cons @ A @ X @ Xs ) )
=> ( ( ( Ys3
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys5: list @ A] :
( ( Ys3
= ( cons @ A @ X @ Ys5 ) )
=> ( ( append @ A @ Ys5 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_4453_list__Cons__eq__append__cases,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys3: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys3 @ Zs ) )
=> ( ( ( Ys3
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys5: list @ A] :
( ( Ys3
= ( cons @ A @ X @ Ys5 ) )
=> ( ( append @ A @ Ys5 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_4454_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys3
!= ( nil @ B ) )
=> ( ! [X3: A,Y3: B] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( cons @ B @ Y3 @ ( nil @ B ) ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B] :
( ( Xs2
!= ( nil @ A ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) )
=> ( ! [X3: A,Y3: B,Ys4: list @ B] :
( ( Ys4
!= ( nil @ B ) )
=> ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys4: list @ B] :
( ( P @ Xs2 @ Ys4 )
=> ( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys4
!= ( nil @ B ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_4455_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_4456_rev__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys3: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys4: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys4: list @ B] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ) ).
% rev_induct2'
thf(fact_4457_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_4458_in__set__list__format,axiom,
! [A: $tType,E3: A,L: list @ A] :
( ( member @ A @ E3 @ ( set2 @ A @ L ) )
=> ~ ! [L12: list @ A,L23: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ E3 @ L23 ) ) ) ) ).
% in_set_list_format
thf(fact_4459_xy__in__set__cases,axiom,
! [A: $tType,X: A,L: list @ A,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ L ) )
=> ( ( member @ A @ Y @ ( set2 @ A @ L ) )
=> ( ( ( X = Y )
=> ! [L12: list @ A,L23: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y @ L23 ) ) ) )
=> ( ( ( X != Y )
=> ! [L12: list @ A,L23: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ X @ ( append @ A @ L23 @ ( cons @ A @ Y @ L32 ) ) ) ) ) )
=> ~ ( ( X != Y )
=> ! [L12: list @ A,L23: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y @ ( append @ A @ L23 @ ( cons @ A @ X @ L32 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_4460_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_4461_set__union__code,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) )
= ( set2 @ A @ ( append @ A @ Xs @ Ys3 ) ) ) ).
% set_union_code
thf(fact_4462_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_4463_lex__append__leftI,axiom,
! [A: $tType,Ys3: list @ A,Zs: list @ A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys3 @ Zs ) @ ( lex @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys3 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R3 ) ) ) ).
% lex_append_leftI
thf(fact_4464_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys3 ) )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys3 )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ ( set2 @ A @ Ys3 ) )
=> ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ) ) ) ).
% sorted_append
thf(fact_4465_list__update__append1,axiom,
! [A: $tType,I2: nat,Xs: list @ A,Ys3: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys3 ) @ I2 @ X )
= ( append @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ Ys3 ) ) ) ).
% list_update_append1
thf(fact_4466_foldl__rule__aux__P,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I5 @ Sigma_0 @ L0 )
=> ( ! [L12: list @ A,L23: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L12 @ ( cons @ A @ X3 @ L23 ) ) )
=> ( ( I5 @ Sigma @ ( cons @ A @ X3 @ L23 ) )
=> ( I5 @ ( F3 @ Sigma @ X3 ) @ L23 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I5 @ Sigma @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_aux_P
thf(fact_4467_foldl__rule__aux,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2] :
( ( I5 @ Sigma_0 @ L0 )
=> ( ! [L12: list @ A,L23: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L12 @ ( cons @ A @ X3 @ L23 ) ) )
=> ( ( I5 @ Sigma @ ( cons @ A @ X3 @ L23 ) )
=> ( I5 @ ( F3 @ Sigma @ X3 ) @ L23 ) ) )
=> ( I5 @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) @ ( nil @ A ) ) ) ) ).
% foldl_rule_aux
thf(fact_4468_foldl__rule__P,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I5 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L12: list @ A,L23: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L12 @ ( cons @ A @ X3 @ L23 ) ) )
=> ( ( I5 @ Sigma @ L12 @ ( cons @ A @ X3 @ L23 ) )
=> ( I5 @ ( F3 @ Sigma @ X3 ) @ ( append @ A @ L12 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ L23 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I5 @ Sigma @ L0 @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_P
thf(fact_4469_foldl__rule,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F3: Sigma2 > A > Sigma2] :
( ( I5 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L12: list @ A,L23: list @ A,X3: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L12 @ ( cons @ A @ X3 @ L23 ) ) )
=> ( ( I5 @ Sigma @ L12 @ ( cons @ A @ X3 @ L23 ) )
=> ( I5 @ ( F3 @ Sigma @ X3 ) @ ( append @ A @ L12 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ L23 ) ) )
=> ( I5 @ ( foldl @ Sigma2 @ A @ F3 @ Sigma_0 @ L0 ) @ L0 @ ( nil @ A ) ) ) ) ).
% foldl_rule
thf(fact_4470_drop__take__drop__unsplit,axiom,
! [A: $tType,I2: nat,J: nat,L: list @ A] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( append @ A @ ( drop @ A @ I2 @ ( take @ A @ J @ L ) ) @ ( drop @ A @ J @ L ) )
= ( drop @ A @ I2 @ L ) ) ) ).
% drop_take_drop_unsplit
thf(fact_4471_lex__append__left__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys3: list @ A,Zs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys3 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R3 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys3 @ Zs ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_left_iff
thf(fact_4472_lex__append__leftD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys3: list @ A,Zs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys3 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys3 @ Zs ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_leftD
thf(fact_4473_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lex @ A @ R3 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys3 @ Vs ) ) @ ( lex @ A @ R3 ) ) ) ) ).
% lex_append_rightI
thf(fact_4474_length__Suc__rev__conv,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N2 ) )
= ( ? [Ys2: list @ A,Y4: A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_rev_conv
thf(fact_4475_length__compl__rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [L4: list @ A,E2: A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L4 ) )
=> ( P @ Ll ) )
=> ( P @ ( append @ A @ L4 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) ) )
=> ( P @ L ) ) ) ).
% length_compl_rev_induct
thf(fact_4476_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ~ ( distinct @ A @ Xs )
=> ~ ! [Y3: A,Ys4: list @ A] :
( ( distinct @ A @ Ys4 )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Ys4 ) )
=> ! [Zs2: list @ A] :
( Xs
!= ( append @ A @ Ys4 @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_4477_filter__eq__snocD,axiom,
! [A: $tType,P: A > $o,L: list @ A,L3: list @ A,X: A] :
( ( ( filter2 @ A @ P @ L )
= ( append @ A @ L3 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ L ) )
& ( P @ X ) ) ) ).
% filter_eq_snocD
thf(fact_4478_nth__append,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys3: list @ A] :
( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys3 ) @ N2 )
= ( nth @ A @ Xs @ N2 ) ) )
& ( ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys3 ) @ N2 )
= ( nth @ A @ Ys3 @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_4479_list__update__append,axiom,
! [A: $tType,N2: nat,Xs: list @ A,Ys3: list @ A,X: A] :
( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys3 ) @ N2 @ X )
= ( append @ A @ ( list_update @ A @ Xs @ N2 @ X ) @ Ys3 ) ) )
& ( ~ ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys3 ) @ N2 @ X )
= ( append @ A @ Xs @ ( list_update @ A @ Ys3 @ ( minus_minus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_4480_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_4481_slice__prepend,axiom,
! [A: $tType,I2: nat,K: nat,Xs: list @ A,Ys3: list @ A] :
( ( ord_less_eq @ nat @ I2 @ K )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( slice @ A @ I2 @ K @ Xs )
= ( slice @ A @ ( plus_plus @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) ) @ ( plus_plus @ nat @ K @ ( size_size @ ( list @ A ) @ Ys3 ) ) @ ( append @ A @ Ys3 @ Xs ) ) ) ) ) ).
% slice_prepend
thf(fact_4482_sorted__append__bigger,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ Y ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_append_bigger
thf(fact_4483_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F3: B > A,A3: A,Xs: list @ B,Ys3: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ ( append @ B @ Xs @ Ys3 ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A3 @ Ys3 ) ) ) ) ) ).
% horner_sum_append
thf(fact_4484_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ A3 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A3
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_4485_take__Suc__conv__app__nth,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ ( suc @ I2 ) @ Xs )
= ( append @ A @ ( take @ A @ I2 @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I2 ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_4486_take__update__last,axiom,
! [A: $tType,N2: nat,List2: list @ A,X: A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ List2 ) )
=> ( ( list_update @ A @ ( take @ A @ ( suc @ N2 ) @ List2 ) @ N2 @ X )
= ( append @ A @ ( take @ A @ N2 @ List2 ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% take_update_last
thf(fact_4487_id__take__nth__drop,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( Xs
= ( append @ A @ ( take @ A @ I2 @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I2 ) @ ( drop @ A @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_4488_upd__conv__take__nth__drop,axiom,
! [A: $tType,I2: nat,Xs: list @ A,A3: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I2 @ A3 )
= ( append @ A @ ( take @ A @ I2 @ Xs ) @ ( cons @ A @ A3 @ ( drop @ A @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_4489_butlast__upd__last__eq,axiom,
! [A: $tType,L: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( list_update @ A @ ( butlast @ A @ L ) @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ X )
= ( append @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ L ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% butlast_upd_last_eq
thf(fact_4490_sort__key__by__quicksort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F4: B > A,Xs3: list @ B] :
( append @ B
@ ( linorder_sort_key @ B @ A @ F4
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F4 @ X4 ) @ ( F4 @ ( nth @ B @ Xs3 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs3 ) )
@ ( append @ B
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F4 @ X4 )
= ( F4 @ ( nth @ B @ Xs3 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs3 )
@ ( linorder_sort_key @ B @ A @ F4
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F4 @ ( nth @ B @ Xs3 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( F4 @ X4 ) )
@ Xs3 ) ) ) ) ) ) ) ).
% sort_key_by_quicksort
thf(fact_4491_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_4492_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_4493_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_4494_last__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( last @ A @ ( drop @ A @ N2 @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_drop
thf(fact_4495_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_4496_snoc__eq__iff__butlast_H,axiom,
! [A: $tType,Ys3: list @ A,Xs: list @ A,X: A] :
( ( Ys3
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( ( Ys3
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys3 )
= Xs )
& ( ( last @ A @ Ys3 )
= X ) ) ) ).
% snoc_eq_iff_butlast'
thf(fact_4497_distinct__butlast__swap,axiom,
! [A: $tType,Pq: list @ A,I2: nat] :
( ( distinct @ A @ Pq )
=> ( distinct @ A @ ( butlast @ A @ ( list_update @ A @ Pq @ I2 @ ( last @ A @ Pq ) ) ) ) ) ).
% distinct_butlast_swap
thf(fact_4498_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_4499_butlast__update_H,axiom,
! [A: $tType,L: list @ A,I2: nat,X: A] :
( ( list_update @ A @ ( butlast @ A @ L ) @ I2 @ X )
= ( butlast @ A @ ( list_update @ A @ L @ I2 @ X ) ) ) ).
% butlast_update'
thf(fact_4500_sort__key__stable,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,K: B,Xs: list @ A] :
( ( filter2 @ A
@ ^ [Y4: A] :
( ( F3 @ Y4 )
= K )
@ ( linorder_sort_key @ A @ B @ F3 @ Xs ) )
= ( filter2 @ A
@ ^ [Y4: A] :
( ( F3 @ Y4 )
= K )
@ Xs ) ) ) ).
% sort_key_stable
thf(fact_4501_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_4502_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_4503_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_4504_butlast__subset,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( butlast @ A @ Xs ) ) @ A5 ) ) ) ).
% butlast_subset
thf(fact_4505_butlast__eq__cons__conv,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
= ( ? [Xl: A] :
( L
= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl @ ( nil @ A ) ) ) ) ) ) ) ).
% butlast_eq_cons_conv
thf(fact_4506_butlast__eq__consE,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
=> ~ ! [Xl2: A] :
( L
!= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl2 @ ( nil @ A ) ) ) ) ) ) ).
% butlast_eq_consE
thf(fact_4507_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_4508_nth__butlast,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ N2 ) ) ) ).
% nth_butlast
thf(fact_4509_take__butlast,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ N2 @ ( butlast @ A @ Xs ) )
= ( take @ A @ N2 @ Xs ) ) ) ).
% take_butlast
thf(fact_4510_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_4511_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_4512_butlast__take,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( butlast @ A @ ( take @ A @ N2 @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_take
thf(fact_4513_take__minus__one__conv__butlast,axiom,
! [A: $tType,N2: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( take @ A @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( butlast @ A @ ( take @ A @ N2 @ L ) ) ) ) ).
% take_minus_one_conv_butlast
thf(fact_4514_last__take__nth__conv,axiom,
! [A: $tType,N2: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( take @ A @ N2 @ L ) )
= ( nth @ A @ L @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).
% last_take_nth_conv
thf(fact_4515_sort__key__by__quicksort__code,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F4: B > A,Xs3: list @ B] :
( case_list @ ( list @ B ) @ B @ ( nil @ B )
@ ^ [X4: B] :
( case_list @ ( list @ B ) @ B @ Xs3
@ ^ [Y4: B] :
( case_list @ ( list @ B ) @ B @ ( if @ ( list @ B ) @ ( ord_less_eq @ A @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) @ Xs3 @ ( cons @ B @ Y4 @ ( cons @ B @ X4 @ ( nil @ B ) ) ) )
@ ^ [Ab: B,List: list @ B] :
( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( list @ B )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( list @ B )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( append @ B @ ( linorder_sort_key @ B @ A @ F4 @ Lts ) @ ( append @ B @ Eqs @ ( linorder_sort_key @ B @ A @ F4 @ Gts ) ) ) )
@ ( linorder_part @ B @ A @ F4 @ ( F4 @ ( 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_4516_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_4517_quicksort_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( linorder_quicksort @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y4: A] :
~ ( ord_less_eq @ A @ X @ Y4 )
@ Xs ) )
@ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X ) @ Xs ) ) ) ) ) ) ).
% quicksort.simps(2)
thf(fact_4518_quicksort_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Y: list @ A] :
( ( ( linorder_quicksort @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs2 ) )
=> ( Y
!= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y4: A] :
~ ( ord_less_eq @ A @ X3 @ Y4 )
@ Xs2 ) )
@ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X3 ) @ Xs2 ) ) ) ) ) ) ) ) ) ).
% quicksort.elims
thf(fact_4519_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: B > C,F1: B,F22: A > ( list @ A ) > B,List2: list @ A] :
( ( H @ ( case_list @ B @ A @ F1 @ F22 @ List2 ) )
= ( case_list @ C @ A @ ( H @ F1 )
@ ^ [X13: A,X23: list @ A] : ( H @ ( F22 @ X13 @ X23 ) )
@ List2 ) ) ).
% list.case_distrib
thf(fact_4520_sort__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4 )
= ( linorder_quicksort @ A ) ) ) ).
% sort_quicksort
thf(fact_4521_sorted__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linorder_quicksort @ A @ Xs ) ) ) ).
% sorted_quicksort
thf(fact_4522_remdups__adj__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( case_list @ ( list @ A ) @ A @ ( cons @ A @ X @ ( nil @ A ) )
@ ^ [Y4: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( X = Y4 ) @ ( cons @ A @ Y4 @ Xs3 ) @ ( cons @ A @ X @ ( cons @ A @ Y4 @ Xs3 ) ) )
@ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_4523_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_4524_part__code_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Pivot: A,X: B,Xs: list @ B] :
( ( linorder_part @ B @ A @ F3 @ Pivot @ ( cons @ B @ X @ Xs ) )
= ( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ ( F3 @ X ) @ Pivot ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( cons @ B @ X @ Lts ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ Gts ) ) @ ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ Pivot @ ( F3 @ X ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ ( cons @ B @ X @ Gts ) ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( cons @ B @ X @ Eqs ) @ Gts ) ) ) ) )
@ ( linorder_part @ B @ A @ F3 @ Pivot @ Xs ) ) ) ) ).
% part_code(2)
thf(fact_4525_part__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_part @ B @ A )
= ( ^ [F4: B > A,Pivot2: A,Xs3: list @ B] :
( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) )
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F4 @ X4 ) @ Pivot2 )
@ Xs3 )
@ ( product_Pair @ ( list @ B ) @ ( list @ B )
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F4 @ X4 )
= Pivot2 )
@ Xs3 )
@ ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ Pivot2 @ ( F4 @ X4 ) )
@ Xs3 ) ) ) ) ) ) ).
% part_def
thf(fact_4526_quicksort_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Y: list @ A] :
( ( ( linorder_quicksort @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( nil @ A ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs2 ) )
=> ( ( Y
= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y4: A] :
~ ( ord_less_eq @ A @ X3 @ Y4 )
@ Xs2 ) )
@ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X3 ) @ Xs2 ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( cons @ A @ X3 @ Xs2 ) ) ) ) ) ) ) ) ).
% quicksort.pelims
thf(fact_4527_upto_Opsimps,axiom,
! [I2: int,J: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J ) )
=> ( ( ( ord_less_eq @ int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( cons @ int @ I2 @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J ) ) ) )
& ( ~ ( ord_less_eq @ int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_4528_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_4529_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_4530_sort__upto,axiom,
! [I2: int,J: int] :
( ( linorder_sort_key @ int @ int
@ ^ [X4: int] : X4
@ ( upto @ I2 @ J ) )
= ( upto @ I2 @ J ) ) ).
% sort_upto
thf(fact_4531_upto__Nil,axiom,
! [I2: int,J: int] :
( ( ( upto @ I2 @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I2 ) ) ).
% upto_Nil
thf(fact_4532_upto__Nil2,axiom,
! [I2: int,J: int] :
( ( ( nil @ int )
= ( upto @ I2 @ J ) )
= ( ord_less @ int @ J @ I2 ) ) ).
% upto_Nil2
thf(fact_4533_upto__empty,axiom,
! [J: int,I2: int] :
( ( ord_less @ int @ J @ I2 )
=> ( ( upto @ I2 @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_4534_nth__upto,axiom,
! [I2: int,K: nat,J: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I2 @ ( semiring_1_of_nat @ int @ K ) ) @ J )
=> ( ( nth @ int @ ( upto @ I2 @ J ) @ K )
= ( plus_plus @ int @ I2 @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% nth_upto
thf(fact_4535_upto__rec__numeral_I1_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N2 ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_4536_upto__rec__numeral_I4_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_4537_upto__rec__numeral_I3_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N2 ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N2 ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_4538_upto__rec__numeral_I2_J,axiom,
! [M: num,N2: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N2 ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_4539_sorted__upto,axiom,
! [M: int,N2: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N2 ) ) ).
% sorted_upto
thf(fact_4540_sorted__wrt__upto,axiom,
! [I2: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I2 @ J ) ) ).
% sorted_wrt_upto
thf(fact_4541_list_Odisc__eq__case_I2_J,axiom,
! [A: $tType,List2: list @ A] :
( ( List2
!= ( nil @ A ) )
= ( case_list @ $o @ A @ $false
@ ^ [Uu2: A,Uv: list @ A] : $true
@ List2 ) ) ).
% list.disc_eq_case(2)
thf(fact_4542_list_Odisc__eq__case_I1_J,axiom,
! [A: $tType,List2: list @ A] :
( ( List2
= ( nil @ A ) )
= ( case_list @ $o @ A @ $true
@ ^ [Uu2: A,Uv: list @ A] : $false
@ List2 ) ) ).
% list.disc_eq_case(1)
thf(fact_4543_upto__split2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_eq @ int @ I2 @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I2 @ K )
= ( append @ int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_4544_upto__split1,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_eq @ int @ I2 @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I2 @ K )
= ( append @ int @ ( upto @ I2 @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_4545_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_4546_upto_Osimps,axiom,
( upto
= ( ^ [I4: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I4 @ J3 ) @ ( cons @ int @ I4 @ ( upto @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_4547_upto__rec1,axiom,
! [I2: int,J: int] :
( ( ord_less_eq @ int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( cons @ int @ I2 @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_4548_upto__rec2,axiom,
! [I2: int,J: int] :
( ( ord_less_eq @ int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( append @ int @ ( upto @ I2 @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_4549_upto__split3,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_eq @ int @ I2 @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I2 @ K )
= ( append @ int @ ( upto @ I2 @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_4550_extract__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys3: 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 ) ) @ Ys3 @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
= ( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys3 ) )
& ( P @ X4 ) ) ) ) ).
% extract_Some_iff
thf(fact_4551_extract__SomeE,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys3: 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 ) ) @ Ys3 @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
=> ( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ Ys3 ) )
& ( P @ X7 ) ) ) ) ).
% extract_SomeE
thf(fact_4552_splice_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P @ ( nil @ A ) @ Ys4 ) )
=> ( ! [X3: A,Xs2: list @ A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Ys4 ) )
=> ( ( P @ Ys4 @ Xs2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ Ys4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_4553_extract__Cons__code,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X @ Xs ) ) ) ) )
& ( ~ ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Ys2: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y4: A,Zs3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X @ Ys2 ) @ ( product_Pair @ A @ ( list @ A ) @ Y4 @ Zs3 ) ) ) ) )
@ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_4554_subset__eq__mset__impl_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
( ( ( subset_eq_mset_impl @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( some @ $o
@ ( Xa
!= ( nil @ A ) ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ 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 )
@ ^ [Y4: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X3 )
@ Xa ) ) ) ) ) ) ).
% subset_eq_mset_impl.elims
thf(fact_4555_splice_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs2 ) )
=> ( ( Y
= ( cons @ A @ X3 @ ( splice @ A @ Xa @ Xs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_4556_Id__on__fold,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( id_on @ A @ A5 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A5 ) ) ) ).
% Id_on_fold
thf(fact_4557_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A8: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A8 ) ) ) ) ).
% Id_on_def
thf(fact_4558_Id__onI,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A5 ) ) ) ).
% Id_onI
thf(fact_4559_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_4560_Id__on__iff,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A5 ) )
= ( ( X = Y )
& ( member @ A @ X @ A5 ) ) ) ).
% Id_on_iff
thf(fact_4561_Id__on__eqI,axiom,
! [A: $tType,A3: A,B2: A,A5: set @ A] :
( ( A3 = B2 )
=> ( ( member @ A @ A3 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id_on @ A @ A5 ) ) ) ) ).
% Id_on_eqI
thf(fact_4562_Id__onE,axiom,
! [A: $tType,C2: product_prod @ A @ A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A5 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( C2
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% Id_onE
thf(fact_4563_subset__eq__mset__impl_Osimps_I1_J,axiom,
! [A: $tType,Ys3: list @ A] :
( ( subset_eq_mset_impl @ A @ ( nil @ A ) @ Ys3 )
= ( some @ $o
@ ( Ys3
!= ( nil @ A ) ) ) ) ).
% subset_eq_mset_impl.simps(1)
thf(fact_4564_Id__on__def_H,axiom,
! [A: $tType,A5: A > $o] :
( ( id_on @ A @ ( collect @ A @ A5 ) )
= ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] :
( ( X4 = Y4 )
& ( A5 @ X4 ) ) ) ) ) ).
% Id_on_def'
thf(fact_4565_subset__eq__mset__impl_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys3: list @ A] :
( ( subset_eq_mset_impl @ A @ ( cons @ A @ X @ Xs ) @ Ys3 )
= ( 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 )
@ ^ [X4: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X )
@ Ys3 ) ) ) ).
% subset_eq_mset_impl.simps(2)
thf(fact_4566_splice_Opsimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys3 ) )
=> ( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys3 )
= ( cons @ A @ X @ ( splice @ A @ Ys3 @ Xs ) ) ) ) ).
% splice.psimps(2)
thf(fact_4567_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ( ( splice @ A @ ( nil @ A ) @ Ys3 )
= Ys3 ) ) ).
% splice.psimps(1)
thf(fact_4568_subset__eq__mset__impl_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
( ( ( subset_eq_mset_impl @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( some @ $o
@ ( Xa
!= ( nil @ A ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ 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 )
@ ^ [Y4: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X3 )
@ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% subset_eq_mset_impl.pelims
thf(fact_4569_merge__list__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Ls2: list @ ( list @ A ),As: list @ ( list @ A )] :
( ! [L4: list @ A] :
( ( member @ ( list @ A ) @ L4 @ ( set2 @ ( list @ A ) @ Ls2 ) )
=> ( ( distinct @ A @ L4 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L4 ) ) )
=> ( ! [L4: list @ A] :
( ( member @ ( list @ A ) @ L4 @ ( set2 @ ( list @ A ) @ As ) )
=> ( ( distinct @ A @ L4 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L4 ) ) )
=> ( ( distinct @ A @ ( merge_list @ A @ As @ Ls2 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge_list @ A @ As @ Ls2 ) )
& ( ( set2 @ A @ ( merge_list @ A @ As @ Ls2 ) )
= ( set2 @ A @ ( concat @ A @ ( append @ ( list @ A ) @ As @ Ls2 ) ) ) ) ) ) ) ) ).
% merge_list_correct
thf(fact_4570_take__hd__drop,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( append @ A @ ( take @ A @ N2 @ Xs ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N2 @ Xs ) ) @ ( nil @ A ) ) )
= ( take @ A @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_4571_Succ__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Succ @ A )
= ( ^ [Kl: set @ ( list @ A ),Kl2: list @ A] :
( collect @ A
@ ^ [K3: A] : ( member @ ( list @ A ) @ ( append @ A @ Kl2 @ ( cons @ A @ K3 @ ( nil @ A ) ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_4572_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_4573_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_4574_merge__list_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc23: list @ ( list @ A ),L: list @ A] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) ) ).
% merge_list.simps(4)
thf(fact_4575_merge__list_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc23: list @ ( list @ A )] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) ).
% merge_list.simps(3)
thf(fact_4576_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_4577_merge__list_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc23: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( ( merge_list @ A @ Acc23 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ Acc23 ) @ Ls2 ) ) ) ).
% merge_list.simps(5)
thf(fact_4578_sorted__hd__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ! [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( hd @ A @ Xs ) @ X7 ) ) ) ) ) ).
% sorted_hd_min
thf(fact_4579_hd__drop__conv__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( drop @ A @ N2 @ Xs ) )
= ( nth @ A @ Xs @ N2 ) ) ) ).
% hd_drop_conv_nth
thf(fact_4580_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_4581_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_4582_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_4583_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_4584_merge__list_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ A ) ) ) )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ! [L4: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L4 @ ( nil @ ( list @ A ) ) ) )
=> ( Y != L4 ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc22 ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc22 ) )
=> ! [L4: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L4 @ ( nil @ ( list @ A ) ) ) )
=> ( Y
!= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L4 @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ) )
=> ~ ! [L12: list @ A,L23: list @ A,Ls: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) )
=> ( Y
!= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ X ) @ Ls ) ) ) ) ) ) ) ) ) ).
% merge_list.elims
thf(fact_4585_merge__list_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ! [L4: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L4 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y = L4 )
=> ~ ( 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 ) @ L4 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc22 ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc22 ) )
=> ! [L4: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L4 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L4 @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L4 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ~ ! [L12: list @ A,L23: list @ A,Ls: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) )
=> ( ( Y
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ X ) @ Ls ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
thf(fact_4586_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,A5: set @ A,B4: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( finite_fold @ A @ B @ F3 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A5 ) @ B4 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_4587_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,A5: set @ A,X: A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ A5 )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_4588_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,G3: A > nat] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( finite4664212375090638736ute_on @ A @ B @ S
@ ^ [X4: A] : ( compow @ ( B > B ) @ ( G3 @ X4 ) @ ( F3 @ X4 ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on_funpow
thf(fact_4589_Finite__Set_Ofold__cong,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,G3: A > B > B,A5: set @ A,S2: B,T6: B,B4: set @ A] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( finite4664212375090638736ute_on @ A @ B @ S @ G3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( F3 @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( S2 = T6 )
=> ( ( A5 = B4 )
=> ( ( finite_fold @ A @ B @ F3 @ S2 @ A5 )
= ( finite_fold @ A @ B @ G3 @ T6 @ B4 ) ) ) ) ) ) ) ) ) ).
% Finite_Set.fold_cong
thf(fact_4590_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A5: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z2 @ A5 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z2 ) @ A5 ) ) ) ) ) ).
% comp_fun_commute_on.fold_fun_left_comm
thf(fact_4591_comp__fun__commute__on_Ofold__insert2,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A5: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z2 ) @ A5 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert2
thf(fact_4592_comp__fun__commute__on_Ofold__insert,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A5: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z2 @ A5 ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert
thf(fact_4593_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [B: $tType,A: $tType,C: $tType,S: set @ A,F3: A > B > B,G3: C > A,R2: set @ C] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G3 @ ( top_top @ ( set @ C ) ) ) @ S )
=> ( finite4664212375090638736ute_on @ C @ B @ R2 @ ( comp @ A @ ( B > B ) @ C @ F3 @ G3 ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_4594_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_4595_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_4596_merge__list_Opsimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc23: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) ) ).
% merge_list.psimps(3)
thf(fact_4597_merge__list_Opsimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc23: list @ ( list @ A ),L: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) ) ) ).
% merge_list.psimps(4)
thf(fact_4598_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 ) ) ) )
=> ( ! [L4: 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 ) @ L4 @ ( nil @ ( list @ A ) ) ) ) )
=> ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L4 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc22: list @ ( list @ A ),L4: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L4 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L4 @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L4 @ ( nil @ ( list @ A ) ) ) ) ) )
=> ( ! [Acc22: list @ ( list @ A ),L12: list @ A,L23: list @ A,Ls: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) ) )
=> ( ( P @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ Acc22 ) @ Ls )
=> ( P @ Acc22 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ).
% merge_list.pinduct
thf(fact_4599_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc23: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc23 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) )
=> ( ( merge_list @ A @ Acc23 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ Acc23 ) @ Ls2 ) ) ) ) ).
% merge_list.psimps(5)
thf(fact_4600_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A5: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_4601_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A3: B,Xs: list @ B,F3: B > A] :
( ( member @ B @ A3 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ A3 )
= ( F3 @ X4 ) )
@ Xs ) )
= A3 )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A3 @ ( remove1 @ B @ A3 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_4602_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list @ A,N2: A,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N2 @ Ns ) ) @ ( lenlex @ A @ R3 ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N2 ) @ R3 ) )
| ( ( M = N2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R3 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_4603_sort__mergesort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4 )
= ( mergesort @ A ) ) ) ).
% sort_mergesort
thf(fact_4604_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X4: A] : X4 )
= ( ^ [Xs3: list @ A] : Xs3 ) ) ).
% map_ident
thf(fact_4605_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R3 ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_4606_nth__map,axiom,
! [B: $tType,A: $tType,N2: nat,Xs: list @ A,F3: A > B] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F3 @ Xs ) @ N2 )
= ( F3 @ ( nth @ A @ Xs @ N2 ) ) ) ) ).
% nth_map
thf(fact_4607_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_4608_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G3: A > B > A,A3: A,F3: C > B,Xs: list @ C] :
( ( foldl @ A @ B @ G3 @ A3 @ ( map @ C @ B @ F3 @ Xs ) )
= ( foldl @ A @ C
@ ^ [A7: A,X4: C] : ( G3 @ A7 @ ( F3 @ X4 ) )
@ A3
@ Xs ) ) ).
% foldl_map
thf(fact_4609_list_Omap__ident,axiom,
! [A: $tType,T6: list @ A] :
( ( map @ A @ A
@ ^ [X4: A] : X4
@ T6 )
= T6 ) ).
% list.map_ident
thf(fact_4610_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_4611_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W2: list @ A,F3: B > A,Ww: list @ B,A3: B] :
( ( W2
= ( map @ B @ A @ F3 @ Ww ) )
=> ( ( cons @ A @ ( F3 @ A3 ) @ W2 )
= ( map @ B @ A @ F3 @ ( cons @ B @ A3 @ Ww ) ) ) ) ).
% map_consI(1)
thf(fact_4612_map__eq__consE,axiom,
! [B: $tType,A: $tType,F3: B > A,Ls2: list @ B,Fa: A,Fl: list @ A] :
( ( ( map @ B @ A @ F3 @ Ls2 )
= ( cons @ A @ Fa @ Fl ) )
=> ~ ! [A6: B,L4: list @ B] :
( ( Ls2
= ( cons @ B @ A6 @ L4 ) )
=> ( ( ( F3 @ A6 )
= Fa )
=> ( ( map @ B @ A @ F3 @ L4 )
!= Fl ) ) ) ) ).
% map_eq_consE
thf(fact_4613_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,L: list @ B,L3: list @ B,I2: nat] :
( ( ( map @ B @ A @ F3 @ L )
= ( map @ B @ A @ F3 @ L3 ) )
=> ( ( F3 @ ( nth @ B @ L @ I2 ) )
= ( F3 @ ( nth @ B @ L3 @ I2 ) ) ) ) ).
% map_eq_nth_eq
thf(fact_4614_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R2: A > A > $o,F3: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ R2 @ ( map @ B @ A @ F3 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X4: B,Y4: B] : ( R2 @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_4615_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs3: list @ A,Ys2: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys2 )
@ Xs3 ) ) ) ) ).
% product_concat_map
thf(fact_4616_append__eq__mapE,axiom,
! [B: $tType,A: $tType,Fl: list @ A,Fl2: list @ A,F3: B > A,Ls2: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F3 @ Ls2 ) )
=> ~ ! [L4: list @ B,L7: list @ B] :
( ( Ls2
= ( append @ B @ L4 @ L7 ) )
=> ( ( ( map @ B @ A @ F3 @ L4 )
= Fl )
=> ( ( map @ B @ A @ F3 @ L7 )
!= Fl2 ) ) ) ) ).
% append_eq_mapE
thf(fact_4617_map__eq__appendE,axiom,
! [B: $tType,A: $tType,F3: B > A,Ls2: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F3 @ Ls2 )
= ( append @ A @ Fl @ Fl2 ) )
=> ~ ! [L4: list @ B,L7: list @ B] :
( ( Ls2
= ( append @ B @ L4 @ L7 ) )
=> ( ( ( map @ B @ A @ F3 @ L4 )
= Fl )
=> ( ( map @ B @ A @ F3 @ L7 )
!= Fl2 ) ) ) ) ).
% map_eq_appendE
thf(fact_4618_Misc_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Fl: list @ A,Fl2: list @ A,F3: B > A,Ls2: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F3 @ Ls2 ) )
= ( ? [L2: list @ B,L8: list @ B] :
( ( Ls2
= ( append @ B @ L2 @ L8 ) )
& ( ( map @ B @ A @ F3 @ L2 )
= Fl )
& ( ( map @ B @ A @ F3 @ L8 )
= Fl2 ) ) ) ) ).
% Misc.append_eq_map_conv
thf(fact_4619_Misc_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Ls2: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F3 @ Ls2 )
= ( append @ A @ Fl @ Fl2 ) )
= ( ? [L2: list @ B,L8: list @ B] :
( ( Ls2
= ( append @ B @ L2 @ L8 ) )
& ( ( map @ B @ A @ F3 @ L2 )
= Fl )
& ( ( map @ B @ A @ F3 @ L8 )
= Fl2 ) ) ) ) ).
% Misc.map_eq_append_conv
thf(fact_4620_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( n_lists @ A @ ( suc @ N2 ) @ Xs )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys2: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y4: A] : ( cons @ A @ Y4 @ Ys2 )
@ Xs )
@ ( n_lists @ A @ N2 @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_4621_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 ) )
= ( ^ [L2: list @ A] : ( image2 @ A @ B @ F3 @ ( set2 @ A @ L2 ) ) ) ) ).
% set_oo_map_alt
thf(fact_4622_map__consI_I2_J,axiom,
! [B: $tType,A: $tType,W2: list @ A,L: list @ A,F3: B > A,Ww: list @ B,A3: B] :
( ( ( append @ A @ W2 @ L )
= ( append @ A @ ( map @ B @ A @ F3 @ Ww ) @ L ) )
=> ( ( cons @ A @ ( F3 @ A3 ) @ ( append @ A @ W2 @ L ) )
= ( append @ A @ ( map @ B @ A @ F3 @ ( cons @ B @ A3 @ Ww ) ) @ L ) ) ) ).
% map_consI(2)
thf(fact_4623_distinct__map__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,L: list @ B,X: B,Y: B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ L ) )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( ( member @ B @ X @ ( set2 @ B @ L ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ L ) )
=> ( X = Y ) ) ) ) ) ).
% distinct_map_eq
thf(fact_4624_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,Y4: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y4 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_4625_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_4626_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ).
% sorted_insort_key
thf(fact_4627_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_4628_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,X: 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 @ X @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_4629_lenlex__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R3 ) ) ) ).
% lenlex_irreflexive
thf(fact_4630_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R3 ) ) ).
% Nil_lenlex_iff2
thf(fact_4631_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys3: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys3 )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys3 ) @ ( product @ A @ B @ Xs @ Ys3 ) ) ) ).
% product.simps(2)
thf(fact_4632_transpose__aux__filter__head,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H6: A,T4: list @ A] : ( cons @ A @ H6 @ ( nil @ A ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ A @ ( hd @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] :
( Ys2
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_head
thf(fact_4633_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,G3: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F3
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= ( G3 @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_4634_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_4635_distinct__idx,axiom,
! [B: $tType,A: $tType,F3: B > A,L: list @ B,I2: nat,J: nat] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ L ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ( F3 @ ( nth @ B @ L @ I2 ) )
= ( F3 @ ( nth @ B @ L @ J ) ) )
=> ( I2 = J ) ) ) ) ) ).
% distinct_idx
thf(fact_4636_map__upd__eq,axiom,
! [B: $tType,A: $tType,I2: nat,L: list @ A,F3: A > B,X: A] :
( ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F3 @ ( nth @ A @ L @ I2 ) )
= ( F3 @ X ) ) )
=> ( ( map @ A @ B @ F3 @ ( list_update @ A @ L @ I2 @ X ) )
= ( map @ A @ B @ F3 @ L ) ) ) ).
% map_upd_eq
thf(fact_4637_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,P: B > $o,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( P @ X )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F3 @ X @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ) ).
% filter_insort
thf(fact_4638_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( subseqs @ A @ Xs ) ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_4639_lenlex__length,axiom,
! [A: $tType,Ms: list @ A,Ns: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R3 ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% lenlex_length
thf(fact_4640_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs: list @ A,R2: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys3: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs @ Ys3 ) ) @ ( lenlex @ A @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_4641_map__by__foldl,axiom,
! [B: $tType,A: $tType,F3: A > B,L: list @ A] :
( ( foldl @ ( list @ B ) @ A
@ ^ [L2: list @ B,X4: A] : ( append @ B @ L2 @ ( cons @ B @ ( F3 @ X4 ) @ ( nil @ B ) ) )
@ ( nil @ B )
@ L )
= ( map @ A @ B @ F3 @ L ) ) ).
% map_by_foldl
thf(fact_4642_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_4643_map__distinct__upd__conv,axiom,
! [B: $tType,A: $tType,I2: nat,L: list @ A,F3: A > B,X: B] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( distinct @ A @ L )
=> ( ( list_update @ B @ ( map @ A @ B @ F3 @ L ) @ I2 @ X )
= ( map @ A @ B @ ( fun_upd @ A @ B @ F3 @ ( nth @ A @ L @ I2 ) @ X ) @ L ) ) ) ) ).
% map_distinct_upd_conv
thf(fact_4644_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys2 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys2 ) @ ( lex @ A @ R4 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_4645_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,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ~ ! [L4: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L4 ) )
=> ( ( ( set2 @ B @ L4 )
= A5 )
=> ( ( size_size @ ( list @ B ) @ L4 )
!= ( finite_card @ B @ A5 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_4646_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F3: nat > B,Ns: list @ nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( ( 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_4647_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_4648_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_4649_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% member_le_sum_list
thf(fact_4650_sum__list__filter__le__nat,axiom,
! [A: $tType,F3: A > nat,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ ( filter2 @ A @ P @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) ) ) ).
% sum_list_filter_le_nat
thf(fact_4651_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: B > A,G3: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ Xs ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G3 @ Xs ) ) ) ) ) ).
% sum_list_addf
thf(fact_4652_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_4653_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_4654_sum__list__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F3: B > A,G3: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ Xs ) )
= ( minus_minus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G3 @ Xs ) ) ) ) ) ).
% sum_list_subtractf
thf(fact_4655_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X: B,Xs: list @ B] :
( ( map_filter @ B @ A @ F3 @ ( cons @ B @ X @ Xs ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F3 @ Xs )
@ ^ [Y4: A] : ( cons @ A @ Y4 @ ( map_filter @ B @ A @ F3 @ Xs ) )
@ ( F3 @ X ) ) ) ).
% map_filter_simps(1)
thf(fact_4656_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_4657_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_4658_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_4659_sum__list__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [Xs: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs ) ) ) ) ).
% sum_list_abs
thf(fact_4660_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_4661_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G3: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G3 @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_4662_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F3: B > A,P: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( F3 @ X4 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_4663_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_4664_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G3: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G3 @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_4665_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [K: nat,Ns: list @ A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_4666_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R3: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X4: C] : R3
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R3 ) ) ) ).
% sum_list_triv
thf(fact_4667_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_4668_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_4669_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_4670_card__length__sum__list__rec,axiom,
! [M: nat,N7: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N7 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= ( minus_minus @ nat @ M @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N7 ) ) ) )
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L2 ) @ ( one_one @ nat ) )
= N7 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_4671_card__length__sum__list,axiom,
! [M: nat,N7: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N7 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N7 @ M ) @ ( one_one @ nat ) ) @ N7 ) ) ).
% card_length_sum_list
thf(fact_4672_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_4673_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X6: set @ A,F3: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X6 )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X4 ) @ ( F3 @ X4 ) )
@ X6 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_4674_product__code,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys3: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys3 ) )
= ( 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 ) @ Ys3 )
@ Xs ) ) ) ) ).
% product_code
thf(fact_4675_transpose_Opinduct,axiom,
! [A: $tType,A0: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
=> ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( P @ ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( P @ Xss2 )
=> ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ( ! [X3: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss2 ) )
=> ( ( P
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H6: A,T4: list @ A] : ( cons @ ( list @ A ) @ T4 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) )
=> ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss2 ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_4676_transpose_Oelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( Y
!= ( transpose @ A @ Xss2 ) ) )
=> ~ ! [X3: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss2 ) )
=> ( Y
!= ( cons @ ( list @ A )
@ ( cons @ A @ X3
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H6: A,T4: list @ A] : ( cons @ A @ H6 @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H6: A,T4: list @ A] : ( cons @ ( list @ A ) @ T4 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_4677_nth__transpose,axiom,
! [A: $tType,I2: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I2 )
= ( map @ ( list @ A ) @ A
@ ^ [Xs3: list @ A] : ( nth @ A @ Xs3 @ I2 )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys2 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_4678_transpose_Opelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( Y
= ( transpose @ A @ Xss2 ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss2 ) )
=> ( ( Y
= ( cons @ ( list @ A )
@ ( cons @ A @ X3
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H6: A,T4: list @ A] : ( cons @ A @ H6 @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H6: A,T4: list @ A] : ( cons @ ( list @ A ) @ T4 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss2 ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_4679_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H6: A,T4: list @ A] : ( cons @ A @ H6 @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H6: A,T4: list @ A] : ( cons @ ( list @ A ) @ T4 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_4680_transpose_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H6: A,T4: list @ A] : ( cons @ A @ H6 @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H6: A,T4: list @ A] : ( cons @ ( list @ A ) @ T4 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_4681_product__lists_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( product_lists @ A @ ( cons @ ( list @ A ) @ Xs @ Xss ) )
= ( concat @ ( list @ A )
@ ( map @ A @ ( list @ ( list @ A ) )
@ ^ [X4: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 ) @ ( product_lists @ A @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_4682_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,A5: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L ) )
& ( ( set2 @ B @ L )
= A5 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A5 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_4683_transpose__aux__filter__tail,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H6: A,T4: list @ A] : ( cons @ ( list @ A ) @ T4 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ ( list @ A ) @ ( tl @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] :
( Ys2
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_tail
thf(fact_4684_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list @ A,X: A] :
( ( L
!= ( nil @ A ) )
=> ( ( ( X
= ( hd @ A @ L ) )
| ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
= ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_4685_in__set__tlD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_tlD
thf(fact_4686_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_4687_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_4688_not__hd__in__tl,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( X
!= ( hd @ A @ Xs ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% not_hd_in_tl
thf(fact_4689_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_4690_tl__def,axiom,
! [A: $tType] :
( ( tl @ A )
= ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [X21: A,X222: list @ A] : X222 ) ) ).
% tl_def
thf(fact_4691_tl__append,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A] :
( ( tl @ A @ ( append @ A @ Xs @ Ys3 ) )
= ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys3 )
@ ^ [Z7: A,Zs3: list @ A] : ( append @ A @ Zs3 @ Ys3 )
@ Xs ) ) ).
% tl_append
thf(fact_4692_tl__subset,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( tl @ A @ Xs ) ) @ A5 ) ) ) ).
% tl_subset
thf(fact_4693_Misc_Onth__tl,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( suc @ N2 ) ) ) ) ).
% Misc.nth_tl
thf(fact_4694_list__take__induct__tl2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys3: list @ B,P: B > A > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ B @ Ys3 @ N5 ) @ ( nth @ A @ Xs @ N5 ) ) )
=> ! [N9: nat] :
( ( ord_less @ nat @ N9 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( P @ ( nth @ B @ ( tl @ B @ Ys3 ) @ N9 ) @ ( nth @ A @ ( tl @ A @ Xs ) @ N9 ) ) ) ) ) ).
% list_take_induct_tl2
thf(fact_4695_distinct__hd__tl,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( X
= ( hd @ A @ Xs ) )
=> ~ ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% distinct_hd_tl
thf(fact_4696_remove1__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( remove1 @ A @ ( hd @ A @ Xs ) @ Xs )
= ( tl @ A @ Xs ) ) ) ).
% remove1_tl
thf(fact_4697_List_Onth__tl,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( suc @ N2 ) ) ) ) ).
% List.nth_tl
thf(fact_4698_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_4699_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,A5: set @ B,B4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ S )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 )
= ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ B4 ) )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( A5 = B4 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
thf(fact_4700_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,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) )
= A5 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
thf(fact_4701_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,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) )
= ( finite_card @ B @ A5 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_4702_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,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( distinct @ A @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_4703_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,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_4704_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,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_4705_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,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 )
= ( nil @ B ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4706_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_4707_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,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_4708_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,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert @ B @ X @ A5 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_4709_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I2: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys2 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I2 ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I2 ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_4710_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys2: list @ A] : ( nth @ A @ Ys2 @ I2 )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys2 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I2 ) ) ) ) ).
% transpose_column
thf(fact_4711_sorted__wrt__rev__linord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ L ) ) ) ) ).
% sorted_wrt_rev_linord
thf(fact_4712_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_4713_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,Y4: A] : ( P @ Y4 @ X4 )
@ Xs ) ) ).
% sorted_wrt_rev
thf(fact_4714_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_4715_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_4716_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_4717_rev__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rev @ A @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N2 ) ) ) ) ) ).
% rev_nth
thf(fact_4718_rev__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% rev_update
thf(fact_4719_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_4720_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,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_4721_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I2: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_4722_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_4723_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys2: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys2 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I2 ) ) ) ) ) ).
% transpose_column_length
thf(fact_4724_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,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member @ B @ X @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert @ B @ X @ A5 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_4725_remove__rev__alt__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X4: A,Xs3: list @ A] :
( filter2 @ A
@ ^ [Y4: A] : Y4 != X4
@ ( rev @ A @ Xs3 ) ) ) ) ).
% remove_rev_alt_def
thf(fact_4726_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N2: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N2
= ( zero_zero @ nat ) ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I3 ) )
= N2 ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I4: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I4 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% transpose_rectangle
thf(fact_4727_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_4728_sort__upt,axiom,
! [M: nat,N2: nat] :
( ( linorder_sort_key @ nat @ nat
@ ^ [X4: nat] : X4
@ ( upt @ M @ N2 ) )
= ( upt @ M @ N2 ) ) ).
% sort_upt
thf(fact_4729_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_4730_hd__upt,axiom,
! [I2: nat,J: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( hd @ nat @ ( upt @ I2 @ J ) )
= I2 ) ) ).
% hd_upt
thf(fact_4731_upt__conv__Nil,axiom,
! [J: nat,I2: nat] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( upt @ I2 @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_4732_upt__merge,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ( ord_less_eq @ nat @ I2 @ J )
& ( ord_less_eq @ nat @ J @ K ) )
=> ( ( append @ nat @ ( upt @ I2 @ J ) @ ( upt @ J @ K ) )
= ( upt @ I2 @ K ) ) ) ).
% upt_merge
thf(fact_4733_upt__eq__Nil__conv,axiom,
! [I2: nat,J: nat] :
( ( ( upt @ I2 @ J )
= ( nil @ nat ) )
= ( ( J
= ( zero_zero @ nat ) )
| ( ord_less_eq @ nat @ J @ I2 ) ) ) ).
% upt_eq_Nil_conv
thf(fact_4734_take__upt,axiom,
! [I2: nat,M: nat,N2: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ M ) @ N2 )
=> ( ( take @ nat @ M @ ( upt @ I2 @ N2 ) )
= ( upt @ I2 @ ( plus_plus @ nat @ I2 @ M ) ) ) ) ).
% take_upt
thf(fact_4735_upt__rec__numeral,axiom,
! [M: num,N2: num] :
( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
= ( cons @ nat @ ( numeral_numeral @ nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) )
& ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N2 ) )
= ( nil @ nat ) ) ) ) ).
% upt_rec_numeral
thf(fact_4736_last__upt,axiom,
! [I2: nat,J: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( last @ nat @ ( upt @ I2 @ J ) )
= ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% last_upt
thf(fact_4737_nth__upt,axiom,
! [I2: nat,K: nat,J: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J )
=> ( ( nth @ nat @ ( upt @ I2 @ J ) @ K )
= ( plus_plus @ nat @ I2 @ K ) ) ) ).
% nth_upt
thf(fact_4738_sum__list__upt,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N2 ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ M @ N2 ) ) ) ) ).
% sum_list_upt
thf(fact_4739_map__add__upt_H,axiom,
! [Ofs: nat,A3: nat,B2: nat] :
( ( map @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ I4 @ Ofs )
@ ( upt @ A3 @ B2 ) )
= ( upt @ ( plus_plus @ nat @ A3 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_4740_upt__eq__append__conv,axiom,
! [I2: nat,J: nat,Xs: list @ nat,Ys3: list @ nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ( upt @ I2 @ J )
= ( append @ nat @ Xs @ Ys3 ) )
= ( ? [K3: nat] :
( ( ord_less_eq @ nat @ I2 @ K3 )
& ( ord_less_eq @ nat @ K3 @ J )
& ( ( upt @ I2 @ K3 )
= Xs )
& ( ( upt @ K3 @ J )
= Ys3 ) ) ) ) ) ).
% upt_eq_append_conv
thf(fact_4741_upt__append,axiom,
! [I2: nat,J: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( append @ nat @ ( upt @ ( zero_zero @ nat ) @ I2 ) @ ( upt @ I2 @ J ) )
= ( upt @ ( zero_zero @ nat ) @ J ) ) ) ).
% upt_append
thf(fact_4742_upt__rec,axiom,
( upt
= ( ^ [I4: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I4 @ J3 ) @ ( cons @ nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_4743_upt__conv__Cons,axiom,
! [I2: nat,J: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( upt @ I2 @ J )
= ( cons @ nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_4744_upt__Suc,axiom,
! [I2: nat,J: nat] :
( ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I2 @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_4745_upt__Suc__append,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I2 @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_4746_upt__add__eq__append,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( upt @ I2 @ ( plus_plus @ nat @ J @ K ) )
= ( append @ nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_4747_butlast__upt,axiom,
! [M: nat,N2: nat] :
( ( butlast @ nat @ ( upt @ M @ N2 ) )
= ( upt @ M @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% butlast_upt
thf(fact_4748_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_4749_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_4750_sorted__wrt__upt,axiom,
! [M: nat,N2: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M @ N2 ) ) ).
% sorted_wrt_upt
thf(fact_4751_map__add__upt,axiom,
! [N2: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ I4 @ N2 )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N2 @ ( plus_plus @ nat @ M @ N2 ) ) ) ).
% map_add_upt
thf(fact_4752_sorted__upt,axiom,
! [M: nat,N2: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N2 ) ) ).
% sorted_upt
thf(fact_4753_filter__upt__take__conv,axiom,
! [A: $tType,P: A > $o,M: nat,L: list @ A,N2: nat] :
( ( filter2 @ nat
@ ^ [I4: nat] : ( P @ ( nth @ A @ ( take @ A @ M @ L ) @ I4 ) )
@ ( upt @ N2 @ M ) )
= ( filter2 @ nat
@ ^ [I4: nat] : ( P @ ( nth @ A @ L @ I4 ) )
@ ( upt @ N2 @ M ) ) ) ).
% filter_upt_take_conv
thf(fact_4754_upt__eq__lel__conv,axiom,
! [L: nat,H: nat,Is1: list @ nat,I2: nat,Is2: list @ nat] :
( ( ( upt @ L @ H )
= ( append @ nat @ Is1 @ ( cons @ nat @ I2 @ Is2 ) ) )
= ( ( Is1
= ( upt @ L @ I2 ) )
& ( Is2
= ( upt @ ( suc @ I2 ) @ H ) )
& ( ord_less_eq @ nat @ L @ I2 )
& ( ord_less @ nat @ I2 @ H ) ) ) ).
% upt_eq_lel_conv
thf(fact_4755_upt__eq__Cons__conv,axiom,
! [I2: nat,J: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I2 @ J )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I2 @ J )
& ( I2 = X )
& ( ( upt @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_4756_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_4757_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_4758_upt__filter__extend,axiom,
! [U: nat,U5: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ U @ U5 )
=> ( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ U @ I3 )
& ( ord_less @ nat @ I3 @ U5 ) )
=> ~ ( P @ I3 ) )
=> ( ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
= ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U5 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_4759_map__decr__upt,axiom,
! [M: nat,N2: nat] :
( ( map @ nat @ nat
@ ^ [N: nat] : ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
= ( upt @ M @ N2 ) ) ).
% map_decr_upt
thf(fact_4760_drop__takeWhile,axiom,
! [A: $tType,I2: nat,P: A > $o,L: list @ A] :
( ( ord_less_eq @ nat @ I2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
=> ( ( drop @ A @ I2 @ ( takeWhile @ A @ P @ L ) )
= ( takeWhile @ A @ P @ ( drop @ A @ I2 @ L ) ) ) ) ).
% drop_takeWhile
thf(fact_4761_enumerate__map__upt,axiom,
! [A: $tType,N2: nat,F3: nat > A,M: nat] :
( ( enumerate @ A @ N2 @ ( map @ nat @ A @ F3 @ ( upt @ N2 @ M ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K3: nat] : ( product_Pair @ nat @ A @ K3 @ ( F3 @ K3 ) )
@ ( upt @ N2 @ M ) ) ) ).
% enumerate_map_upt
thf(fact_4762_takeWhile__not__last,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( takeWhile @ A
@ ^ [Y4: A] :
( Y4
!= ( last @ A @ Xs ) )
@ Xs )
= ( butlast @ A @ Xs ) ) ) ).
% takeWhile_not_last
thf(fact_4763_filter__upt__last,axiom,
! [A: $tType,P: A > $o,L: list @ A,Js2: list @ nat,J: nat,I2: nat] :
( ( ( filter2 @ nat
@ ^ [K3: nat] : ( P @ ( nth @ A @ L @ K3 ) )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( append @ nat @ Js2 @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
=> ( ( ord_less @ nat @ J @ I2 )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I2 ) ) ) ) ) ).
% filter_upt_last
thf(fact_4764_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_4765_eq__len__takeWhile__conv,axiom,
! [A: $tType,I2: nat,P: A > $o,L: list @ A] :
( ( I2
= ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
= ( ( ord_less_eq @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I2 )
=> ( P @ ( nth @ A @ L @ J3 ) ) )
& ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I2 ) ) ) ) ) ).
% eq_len_takeWhile_conv
thf(fact_4766_less__length__takeWhile__conv,axiom,
! [A: $tType,I2: nat,P: A > $o,L: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
= ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
& ! [J3: nat] :
( ( ord_less_eq @ nat @ J3 @ I2 )
=> ( P @ ( nth @ A @ L @ J3 ) ) ) ) ) ).
% less_length_takeWhile_conv
thf(fact_4767_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N2: nat,Xs: list @ A,P: A > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N2 )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) ) )
=> ( ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ N2 ) ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( take @ A @ N2 @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_4768_map__upt__Suc,axiom,
! [A: $tType,F3: nat > A,N2: nat] :
( ( map @ nat @ A @ F3 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) )
= ( cons @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I4: nat] : ( F3 @ ( suc @ I4 ) )
@ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% map_upt_Suc
thf(fact_4769_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_4770_nth__map__upt,axiom,
! [A: $tType,I2: nat,N2: nat,M: nat,F3: nat > A] :
( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ N2 @ M ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F3 @ ( upt @ M @ N2 ) ) @ I2 )
= ( F3 @ ( plus_plus @ nat @ M @ I2 ) ) ) ) ).
% nth_map_upt
thf(fact_4771_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N2: nat,M: nat,F3: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N2 @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( F3 @ ( plus_plus @ nat @ M @ I3 ) ) ) )
=> ( ( map @ nat @ A @ F3 @ ( upt @ M @ N2 ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_4772_map__nth__upt__drop__take__conv,axiom,
! [A: $tType,N7: nat,L: list @ A,M6: nat] :
( ( ord_less_eq @ nat @ N7 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( map @ nat @ A @ ( nth @ A @ L ) @ ( upt @ M6 @ N7 ) )
= ( drop @ A @ M6 @ ( take @ A @ N7 @ L ) ) ) ) ).
% map_nth_upt_drop_take_conv
thf(fact_4773_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,T6: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F3 @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ T6 @ ( F3 @ X4 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X4: B] : ( ord_less @ A @ T6 @ ( F3 @ X4 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_4774_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P4: 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 ) ) ) )
@ ^ [Y4: A,Ys2: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y4 @ Ys2 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% extract_def
thf(fact_4775_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F4: A > ( option @ B ),Xs3: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F4 )
@ ( filter2 @ A
@ ^ [X4: A] :
( ( F4 @ X4 )
!= ( none @ B ) )
@ Xs3 ) ) ) ) ).
% map_filter_def
thf(fact_4776_remove__rev__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X4: A] :
( filter_rev @ A
@ ( comp @ $o @ $o @ A @ (~)
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X4 ) ) ) ) ) ).
% remove_rev_def
thf(fact_4777_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_4778_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P: B > $o,Q: B > $o,G3: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X4: B] :
( ( P @ X4 )
& ( Q @ X4 ) )
@ G3 )
= ( ^ [X4: A] :
( ( comp @ B @ $o @ A @ P @ G3 @ X4 )
& ( comp @ B @ $o @ A @ Q @ G3 @ X4 ) ) ) ) ).
% conj_comp_iff
thf(fact_4779_option_Osel,axiom,
! [A: $tType,X2: A] :
( ( the2 @ A @ ( some @ A @ X2 ) )
= X2 ) ).
% option.sel
thf(fact_4780_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option2: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option2
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option2
!= ( none @ A ) )
=> ( ( the2 @ A @ Option )
= ( the2 @ A @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_4781_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_4782_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_4783_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_4784_option_Omap__sel,axiom,
! [B: $tType,A: $tType,A3: option @ A,F3: A > B] :
( ( A3
!= ( none @ A ) )
=> ( ( the2 @ B @ ( map_option @ A @ B @ F3 @ A3 ) )
= ( F3 @ ( the2 @ A @ A3 ) ) ) ) ).
% option.map_sel
thf(fact_4785_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F13: B,F23: A > B,Option3: option @ A] :
( if @ B
@ ( Option3
= ( none @ A ) )
@ F13
@ ( F23 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_4786_remdups__adj__Cons_H,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y4: A] : Y4 = X
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_4787_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A8: set @ ( option @ A )] :
( image2 @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X4: option @ A] :
( ( member @ ( option @ A ) @ X4 @ A8 )
& ( X4
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_4788_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_4789_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F1 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_4790_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_4791_length__dropWhile__takeWhile,axiom,
! [A: $tType,X: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( ord_less @ nat @ ( plus_plus @ nat @ X @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_dropWhile_takeWhile
thf(fact_4792_remdups__adj__append__dropWhile,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys3: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys3 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X4: A] : X4 = Y
@ Ys3 ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_4793_remdups__adj__append_H_H,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( remdups_adj @ A @ ( append @ A @ Xs @ Ys3 ) )
= ( append @ A @ ( remdups_adj @ A @ Xs )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y4: A] :
( Y4
= ( last @ A @ Xs ) )
@ Ys3 ) ) ) ) ) ).
% remdups_adj_append''
thf(fact_4794_tl__remdups__adj,axiom,
! [A: $tType,Ys3: list @ A] :
( ( Ys3
!= ( nil @ A ) )
=> ( ( tl @ A @ ( remdups_adj @ A @ Ys3 ) )
= ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X4: A] :
( X4
= ( hd @ A @ Ys3 ) )
@ ( tl @ A @ Ys3 ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_4795_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_4796_dropWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( dropWhile @ A
@ ^ [Y4: A] : Y4 != X
@ ( rev @ A @ Xs ) )
= ( cons @ A @ X
@ ( rev @ A
@ ( takeWhile @ A
@ ^ [Y4: A] : Y4 != X
@ Xs ) ) ) ) ) ) ).
% dropWhile_neq_rev
thf(fact_4797_takeWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( takeWhile @ A
@ ^ [Y4: A] : Y4 != X
@ ( rev @ A @ Xs ) )
= ( rev @ A
@ ( tl @ A
@ ( dropWhile @ A
@ ^ [Y4: A] : Y4 != X
@ Xs ) ) ) ) ) ) ).
% takeWhile_neq_rev
thf(fact_4798_filter__rev__alt,axiom,
! [A: $tType] :
( ( filter_rev @ A )
= ( ^ [P4: A > $o,L2: list @ A] : ( filter2 @ A @ P4 @ ( rev @ A @ L2 ) ) ) ) ).
% filter_rev_alt
thf(fact_4799_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X4: A,Xa4: list @ A] : ( some @ A @ X4 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% find_dropWhile
thf(fact_4800_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F4: A > $o,Xs3: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F4 @ Xs3 ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F4 ) @ Xs3 ) ) ) ) ).
% partition_filter_conv
thf(fact_4801_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_4802_find__SomeD_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
=> ( P @ X ) ) ).
% find_SomeD(1)
thf(fact_4803_partition__rev_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A,X: A,Xs: list @ A] :
( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ ( cons @ A @ X @ Xs ) )
= ( partition_rev @ A @ P @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes2 ) @ No2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X @ No2 ) ) ) @ Xs ) ) ).
% partition_rev.simps(2)
thf(fact_4804_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_4805_find_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( some @ A @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( find @ A @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_4806_find__SomeD_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% find_SomeD(2)
thf(fact_4807_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_4808_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 ) )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ Yes2 ) )
=> ( P @ X7 ) )
& ! [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ No2 ) )
=> ~ ( P @ X7 ) ) ) ) ).
% partition_P
thf(fact_4809_partition_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( partition @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ^ [Yes3: list @ A,No3: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes3 ) @ No3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes3 @ ( cons @ A @ X @ No3 ) ) )
@ ( partition @ A @ P @ Xs ) ) ) ).
% partition.simps(2)
thf(fact_4810_partition__rev_Oelims,axiom,
! [A: $tType,X: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( partition_rev @ A @ X @ Xa @ Xb )
= Y )
=> ( ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ( Xb
= ( nil @ A ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) ) ) )
=> ~ ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ! [X3: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X3 @ Xs2 ) )
=> ( Y
!= ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X3 @ No ) ) ) @ Xs2 ) ) ) ) ) ) ).
% partition_rev.elims
thf(fact_4811_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_4812_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I4 ) )
& ( X
= ( nth @ A @ Xs @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_4813_find__Some__iff2,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I4 ) )
& ( X
= ( nth @ A @ Xs @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_4814_partition__rev_Opelims,axiom,
! [A: $tType,X: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( partition_rev @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ( Xb
= ( nil @ A ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) ) ) ) )
=> ~ ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ! [X3: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X3 @ Xs2 ) )
=> ( ( Y
= ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X3 @ No ) ) ) @ Xs2 ) )
=> ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% partition_rev.pelims
thf(fact_4815_quicksort__by__rel_Osimps_I2_J,axiom,
! [A: $tType,R2: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
( ( quicksort_by_rel @ A @ R2 @ Sl2 @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R2 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R2 @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y4: A] : ( R2 @ Y4 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ).
% quicksort_by_rel.simps(2)
thf(fact_4816_quicksort__by__rel_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X3 @ Xs2 ) )
=> ( Y
!= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X3 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y4: A] : ( X @ Y4 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) ) ) ) ) ) ).
% quicksort_by_rel.elims
thf(fact_4817_set__quicksort__by__rel,axiom,
! [A: $tType,R2: A > A > $o,Sl2: list @ A,Xs: list @ A] :
( ( set2 @ A @ ( quicksort_by_rel @ A @ R2 @ Sl2 @ Xs ) )
= ( set2 @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% set_quicksort_by_rel
thf(fact_4818_quicksort__by__rel_Osimps_I1_J,axiom,
! [A: $tType,R2: A > A > $o,Sl2: list @ A] :
( ( quicksort_by_rel @ A @ R2 @ Sl2 @ ( nil @ A ) )
= Sl2 ) ).
% quicksort_by_rel.simps(1)
thf(fact_4819_quicksort__by__rel__remove__acc__guared,axiom,
! [A: $tType,Sl2: list @ A,R2: A > A > $o,Xs: list @ A] :
( ( Sl2
!= ( nil @ A ) )
=> ( ( quicksort_by_rel @ A @ R2 @ Sl2 @ Xs )
= ( append @ A @ ( quicksort_by_rel @ A @ R2 @ ( nil @ A ) @ Xs ) @ Sl2 ) ) ) ).
% quicksort_by_rel_remove_acc_guared
thf(fact_4820_quicksort__by__rel__remove__acc,axiom,
! [A: $tType] :
( ( quicksort_by_rel @ A )
= ( ^ [R6: A > A > $o,Sl3: list @ A,Xs3: list @ A] : ( append @ A @ ( quicksort_by_rel @ A @ R6 @ ( nil @ A ) @ Xs3 ) @ Sl3 ) ) ) ).
% quicksort_by_rel_remove_acc
thf(fact_4821_sorted__wrt__quicksort__by__rel,axiom,
! [X15: $tType,R2: X15 > X15 > $o,Xs: list @ X15] :
( ! [X3: X15,Y3: X15] :
( ( R2 @ X3 @ Y3 )
| ( R2 @ Y3 @ X3 ) )
=> ( ! [X3: X15,Y3: X15,Z3: X15] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( sorted_wrt @ X15 @ R2 @ ( quicksort_by_rel @ X15 @ R2 @ ( nil @ X15 ) @ Xs ) ) ) ) ).
% sorted_wrt_quicksort_by_rel
thf(fact_4822_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_4823_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_4824_quicksort__by__rel_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X3 @ Xs2 ) )
=> ( ( Y
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X3 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y4: A] : ( X @ Y4 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( cons @ A @ X3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% quicksort_by_rel.pelims
thf(fact_4825_quicksort__by__rel_Opsimps_I2_J,axiom,
! [A: $tType,R2: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R2 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( cons @ A @ X @ Xs ) ) ) )
=> ( ( quicksort_by_rel @ A @ R2 @ Sl2 @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R2 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R2 @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y4: A] : ( R2 @ Y4 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ) ).
% quicksort_by_rel.psimps(2)
thf(fact_4826_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 ) ) )
=> ( ! [R10: 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 ) ) @ R10 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ( P @ R10 @ Sl @ ( nil @ A ) ) )
=> ( ! [R10: A > A > $o,Sl: list @ A,X3: 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 ) ) @ R10 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X3 @ Xs2 ) ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z7: A] : ( R10 @ Z7 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
= Xa2 )
=> ( P @ R10 @ Sl @ Y6 ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z7: A] : ( R10 @ Z7 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
= Xa2 )
=> ( P @ R10 @ ( cons @ A @ X3 @ ( quicksort_by_rel @ A @ R10 @ Sl @ Y6 ) ) @ Xb2 ) ) )
=> ( P @ R10 @ Sl @ ( cons @ A @ X3 @ Xs2 ) ) ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% quicksort_by_rel.pinduct
thf(fact_4827_quicksort__by__rel_Opsimps_I1_J,axiom,
! [A: $tType,R2: 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 ) ) @ R2 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( nil @ A ) ) ) )
=> ( ( quicksort_by_rel @ A @ R2 @ Sl2 @ ( nil @ A ) )
= Sl2 ) ) ).
% quicksort_by_rel.psimps(1)
thf(fact_4828_filter__rev__aux__alt,axiom,
! [A: $tType] :
( ( filter_rev_aux @ A )
= ( ^ [A7: list @ A,P4: A > $o,L2: list @ A] : ( append @ A @ ( filter2 @ A @ P4 @ ( rev @ A @ L2 ) ) @ A7 ) ) ) ).
% filter_rev_aux_alt
thf(fact_4829_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel1 @ A @ R3 ) )
= ( ? [Y4: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ Y4 ) @ R3 )
& ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys3
= ( list_update @ A @ Xs @ N @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_4830_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 )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( F3 @ K3 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F3 ) @ ( set2 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_4831_empty__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A] : ( none @ B ) )
= ( map_of @ A @ B @ Xys ) )
= ( Xys
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% empty_eq_map_of_iff
thf(fact_4832_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys3 ) ) @ ( listrel1 @ A @ R3 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
& ( Xs = Ys3 ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_4833_listrel1__mono,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R3 ) @ ( listrel1 @ A @ S2 ) ) ) ).
% listrel1_mono
thf(fact_4834_map__of__None__filterD,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),X: B,P: ( product_prod @ B @ A ) > $o] :
( ( ( map_of @ B @ A @ Xs @ X )
= ( none @ A ) )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ P @ Xs ) @ X )
= ( none @ A ) ) ) ).
% map_of_None_filterD
thf(fact_4835_map__of__filter__in,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Z2: A,P: B > A > $o] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ Z2 ) )
=> ( ( P @ K @ Z2 )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) @ Xs ) @ K )
= ( some @ A @ Z2 ) ) ) ) ).
% map_of_filter_in
thf(fact_4836_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel1 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys3 ) ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel1I2
thf(fact_4837_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R3 ) ) ).
% not_listrel1_Nil
thf(fact_4838_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R3 ) ) ).
% not_Nil_listrel1
thf(fact_4839_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel1 @ A @ R3 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) ) ) ).
% listrel1_eq_len
thf(fact_4840_append__listrel1I,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel1 @ A @ R3 ) )
& ( Us = Vs ) )
| ( ( Xs = Ys3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R3 ) ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys3 @ Vs ) ) @ ( listrel1 @ A @ R3 ) ) ) ).
% append_listrel1I
thf(fact_4841_filter__rev__aux_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,A3: list @ A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( filter_rev_aux @ A @ A3 @ P @ ( cons @ A @ X @ Xs ) )
= ( filter_rev_aux @ A @ ( cons @ A @ X @ A3 ) @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( filter_rev_aux @ A @ A3 @ P @ ( cons @ A @ X @ Xs ) )
= ( filter_rev_aux @ A @ A3 @ P @ Xs ) ) ) ) ).
% filter_rev_aux.simps(2)
thf(fact_4842_filter__rev__aux_Osimps_I1_J,axiom,
! [A: $tType,A3: list @ A,P: A > $o] :
( ( filter_rev_aux @ A @ A3 @ P @ ( nil @ A ) )
= A3 ) ).
% filter_rev_aux.simps(1)
thf(fact_4843_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X3: B] :
( ( map_of @ A @ B @ L @ K )
= ( some @ B @ X3 ) ) ) ).
% weak_map_of_SomeI
thf(fact_4844_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Y: A] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ Y ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% map_of_SomeD
thf(fact_4845_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K: B,V2: C,Ps: list @ ( product_prod @ B @ C )] :
( ( ( L = K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps ) @ K )
= ( some @ C @ V2 ) ) )
& ( ( L != K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V2 ) @ Ps ) @ K )
= ( map_of @ B @ C @ Ps @ K ) ) ) ) ).
% map_of_Cons_code(2)
thf(fact_4846_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys3 ) ) @ ( listrel1 @ A @ R3 ) )
=> ( ! [X3: A] :
( ( Xs
= ( cons @ A @ X3 @ Ys3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R3 ) )
=> ~ ! [Zs2: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys3 ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_4847_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys3 ) @ ( listrel1 @ A @ R3 ) )
=> ( ! [Y3: A] :
( ( Ys3
= ( cons @ A @ Y3 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R3 ) )
=> ~ ! [Zs2: list @ A] :
( ( Ys3
= ( cons @ A @ X @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_4848_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel1I1
thf(fact_4849_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel1 @ A @ R3 ) )
=> ~ ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ! [Us2: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us2 @ ( cons @ A @ X3 @ Vs2 ) ) )
=> ( Ys3
!= ( append @ A @ Us2 @ ( cons @ A @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_4850_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs: list @ A,Ys3: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( Xs
= ( append @ A @ Us @ ( cons @ A @ X @ Vs ) ) )
=> ( ( Ys3
= ( append @ A @ Us @ ( cons @ A @ Y @ Vs ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% listrel1I
thf(fact_4851_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 )
@ ^ [K3: A,V3: C] : ( product_Pair @ A @ B @ K3 @ ( F3 @ V3 ) ) )
@ Xs ) )
= ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F3 ) @ ( map_of @ A @ C @ Xs ) ) ) ).
% map_of_map
thf(fact_4852_map__of__Some__split,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V2: A] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ V2 ) )
=> ? [Ys4: list @ ( product_prod @ B @ A ),Zs2: list @ ( product_prod @ B @ A )] :
( ( Xs
= ( append @ ( product_prod @ B @ A ) @ Ys4 @ ( cons @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V2 ) @ Zs2 ) ) )
& ( ( map_of @ B @ A @ Ys4 @ K )
= ( none @ A ) ) ) ) ).
% map_of_Some_split
thf(fact_4853_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys3: list @ A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R3 ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel1 @ A @ R3 ) )
& ( X = Y ) )
| ( ( Xs = Ys3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_4854_filter__rev__def,axiom,
! [A: $tType] :
( ( filter_rev @ A )
= ( filter_rev_aux @ A @ ( nil @ A ) ) ) ).
% filter_rev_def
thf(fact_4855_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R4: A > A > $o,Xs3: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys2 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_4856_map__of__distinct__upd4,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys3: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys3 ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys3 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys3 ) ) ) @ X @ ( none @ B ) ) ) ) ) ).
% map_of_distinct_upd4
thf(fact_4857_map__of__distinct__upd3,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys3: list @ ( product_prod @ A @ B ),Y: B,Y9: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys3 ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys3 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y9 ) @ Ys3 ) ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd3
thf(fact_4858_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_4859_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F3 @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_4860_img__fst,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( member @ A @ A3 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) ) ) ).
% img_fst
thf(fact_4861_map__of__rev__distinct,axiom,
! [B: $tType,A: $tType,M: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ M ) )
=> ( ( map_of @ A @ B @ ( rev @ ( product_prod @ A @ B ) @ M ) )
= ( map_of @ A @ B @ M ) ) ) ).
% map_of_rev_distinct
thf(fact_4862_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_4863_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,Y4: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ X4 ) @ ( product_fst @ A @ B @ Y4 ) )
@ 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_4864_map__fst__mk__snd,axiom,
! [B: $tType,A: $tType,K: B,L: list @ A] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ K )
@ L ) )
= L ) ).
% map_fst_mk_snd
thf(fact_4865_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,Y4: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ Y4 ) @ ( 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_4866_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
=> ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) ) ) ) ).
% map_of_is_SomeI
thf(fact_4867_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y: B,X: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( some @ B @ Y )
= ( map_of @ A @ B @ Xys @ X ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_4868_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_4869_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_4870_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X13: A,X23: B] : X13 ) ) ).
% fst_def
thf(fact_4871_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
@ ^ [A7: A,Uu2: B] : ( F3 @ A7 ) ) ) ).
% fn_fst_conv
thf(fact_4872_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_4873_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X2: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_4874_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A3 )
=> ( X = A3 ) ) ).
% fst_eqD
thf(fact_4875_fstE,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: A > $o] :
( ( X
= ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( ( P @ ( product_fst @ A @ B @ X ) )
=> ( P @ A3 ) ) ) ).
% fstE
thf(fact_4876_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X: A,S: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S ) )
=> ~ ! [Y3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ S ) ) ).
% in_fst_imageE
thf(fact_4877_fst__image__mp,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B ),B4: set @ A,X: A,Y: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 ) @ B4 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ A5 )
=> ( member @ A @ X @ B4 ) ) ) ).
% fst_image_mp
thf(fact_4878_distinct__map__fstD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),X: A,Y: B,Z2: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( Y = Z2 ) ) ) ) ).
% distinct_map_fstD
thf(fact_4879_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),K: A,V1: B,V22: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_4880_sorted__enumerate,axiom,
! [A: $tType,N2: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) ) ).
% sorted_enumerate
thf(fact_4881_graph__fun__upd__None,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [E5: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ E5 @ ( graph @ A @ B @ M ) )
& ( ( product_fst @ A @ B @ E5 )
!= K ) ) ) ) ).
% graph_fun_upd_None
thf(fact_4882_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > C > A,G3: A > B > C > B,A3: A,B2: B,Xs: list @ C] :
( ( product_fst @ A @ B
@ ( foldl @ ( product_prod @ A @ B ) @ C
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ B ) )
@ ^ [A7: A,B6: B,X4: C] : ( product_Pair @ A @ B @ ( F3 @ A7 @ X4 ) @ ( G3 @ A7 @ B6 @ X4 ) ) )
@ ( product_Pair @ A @ B @ A3 @ B2 )
@ Xs ) )
= ( foldl @ A @ C @ F3 @ A3 @ Xs ) ) ).
% fst_foldl
thf(fact_4883_map__of__map__to__set,axiom,
! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B ),M: A > ( option @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
=> ( ( ( map_of @ A @ B @ L )
= M )
= ( ( set2 @ ( product_prod @ A @ B ) @ L )
= ( map_to_set @ A @ B @ M ) ) ) ) ).
% map_of_map_to_set
thf(fact_4884_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_4885_map__of__Some__filter__not__in,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V2: A,P: ( product_prod @ B @ A ) > $o] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ V2 ) )
=> ( ~ ( P @ ( product_Pair @ B @ A @ K @ 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 ) @ K )
= ( none @ A ) ) ) ) ) ).
% map_of_Some_filter_not_in
thf(fact_4886_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
@ ^ [K3: A,V3: B] :
( ( map_of @ A @ B @ Xs @ K3 )
= ( some @ B @ V3 ) ) ) ) ) ) ).
% set_map_of_compr
thf(fact_4887_map__of__distinct__lookup,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys3: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys3 ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys3 ) ) @ X )
= ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_lookup
thf(fact_4888_map__of__distinct__upd2,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys3: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys3 ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys3 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys3 ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd2
thf(fact_4889_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_4890_bezw_Osimps,axiom,
( bezw
= ( ^ [X4: nat,Y4: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y4
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y4 @ ( modulo_modulo @ nat @ X4 @ Y4 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y4 @ ( modulo_modulo @ nat @ X4 @ Y4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y4 @ ( modulo_modulo @ nat @ X4 @ Y4 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X4 @ Y4 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_4891_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_4892_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F3 @ X ) )
= ( F3 @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_4893_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C,G3: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F3 @ X )
= ( product_apsnd @ C @ B @ A @ G3 @ X ) )
= ( ( F3 @ ( product_snd @ A @ C @ X ) )
= ( G3 @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_4894_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_4895_img__snd,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S )
=> ( member @ B @ B2 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S ) ) ) ).
% img_snd
thf(fact_4896_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_4897_map__snd__mk__fst,axiom,
! [B: $tType,A: $tType,K: B,L: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K ) @ L ) )
= L ) ).
% map_snd_mk_fst
thf(fact_4898_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_4899_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_4900_prod__eqI,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,Q4: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P3 )
= ( product_fst @ A @ B @ Q4 ) )
=> ( ( ( product_snd @ A @ B @ P3 )
= ( product_snd @ A @ B @ Q4 ) )
=> ( P3 = Q4 ) ) ) ).
% prod_eqI
thf(fact_4901_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y5: product_prod @ A @ B,Z4: product_prod @ A @ B] : Y5 = Z4 )
= ( ^ [S6: product_prod @ A @ B,T4: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S6 )
= ( product_fst @ A @ B @ T4 ) )
& ( ( product_snd @ A @ B @ S6 )
= ( product_snd @ A @ B @ T4 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_4902_Ex__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ? [A7: A,X11: B] : ( P @ A7 @ X11 ) )
= ( ? [Z7: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z7 ) @ ( product_snd @ A @ B @ Z7 ) ) ) ) ).
% Ex_prod_contract
thf(fact_4903_All__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ! [A7: A,X11: B] : ( P @ A7 @ X11 ) )
= ( ! [Z7: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z7 ) @ ( product_snd @ A @ B @ Z7 ) ) ) ) ).
% All_prod_contract
thf(fact_4904_sndE,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: B > $o] :
( ( X
= ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( ( P @ ( product_snd @ A @ B @ X ) )
=> ( P @ B2 ) ) ) ).
% sndE
thf(fact_4905_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A3 )
=> ( Y = A3 ) ) ).
% snd_eqD
thf(fact_4906_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_4907_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
@ ^ [Uu2: A] : F3 ) ) ).
% fn_snd_conv
thf(fact_4908_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X13: A,X23: B] : X23 ) ) ).
% snd_def
thf(fact_4909_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_4910_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
( ( P @ Y @ X )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_4911_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P3: A,Q: B > $o,Q4: B] :
( ( P @ P3 )
=> ( ( Q @ Q4 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P3 @ Q4 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P3 @ Q4 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_4912_surjective__pairing,axiom,
! [B: $tType,A: $tType,T6: product_prod @ A @ B] :
( T6
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T6 ) @ ( product_snd @ A @ B @ T6 ) ) ) ).
% surjective_pairing
thf(fact_4913_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_4914_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A3: product_prod @ A @ B] :
( ( P @ X @ Y )
=> ( ( A3
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_4915_split__beta,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,Prod3: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_4916_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F4: B > C > A,P6: product_prod @ B @ C] : ( F4 @ ( product_fst @ B @ C @ P6 ) @ ( product_snd @ B @ C @ P6 ) ) ) ) ).
% case_prod_beta
thf(fact_4917_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) )
=> ( A5 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_4918_in__snd__imageE,axiom,
! [A: $tType,B: $tType,Y: A,S: set @ ( product_prod @ B @ A )] :
( ( member @ A @ Y @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S ) )
=> ~ ! [X3: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y ) @ S ) ) ).
% in_snd_imageE
thf(fact_4919_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_4920_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_4921_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,P3: product_prod @ B @ A,Q: C > $o,F3: B > A > C] :
( ( P @ ( product_snd @ B @ A @ P3 ) @ ( product_fst @ B @ A @ P3 ) )
=> ( ! [X3: B,Y3: A] :
( ( P @ Y3 @ X3 )
=> ( Q @ ( F3 @ X3 @ Y3 ) ) )
=> ( Q @ ( product_case_prod @ B @ A @ C @ F3 @ P3 ) ) ) ) ).
% exE_realizer
thf(fact_4922_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F3: A > B > C,G3: D > A] :
( ( ^ [U2: product_prod @ D @ B] : ( F3 @ ( G3 @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X4: D] : ( F3 @ ( G3 @ X4 ) ) ) ) ).
% split_comp_eq
thf(fact_4923_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,X4: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ X4 ) @ ( product_snd @ A @ B @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_4924_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C5: A > B > C,P6: product_prod @ A @ B] : ( C5 @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) ) ) ).
% case_prod_unfold
thf(fact_4925_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_4926_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_4927_snd__image__mp,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ B @ A ),B4: set @ A,X: B,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A5 ) @ B4 )
=> ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A5 )
=> ( member @ A @ Y @ B4 ) ) ) ).
% snd_image_mp
thf(fact_4928_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F3: D > C > A,G3: B > D,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F3 @ G3 ) @ X )
= ( F3 @ ( G3 @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
thf(fact_4929_map__to__set__ran,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M3: B > ( option @ A )] : ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map_to_set @ B @ A @ M3 ) ) ) ) ).
% map_to_set_ran
thf(fact_4930_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X6: set @ A,A5: set @ ( product_prod @ A @ B ),Y7: set @ B,P: A > B > $o,Q: A > B > $o] :
( ( X6
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 ) )
=> ( ( Y7
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A5 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ Y7 )
=> ( ( P @ X3 @ Xa3 )
=> ( Q @ X3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_4931_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_4932_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_4933_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_4934_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P3 @ Ps ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P3 ) @ ( some @ B @ ( product_snd @ A @ B @ P3 ) ) ) ) ).
% map_of.simps(2)
thf(fact_4935_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 @ ( insert @ ( 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_4936_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_4937_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_4938_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,Y4: B] : ( product_Pair @ B @ A @ Y4 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_4939_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,Y4: A] : ( product_Pair @ A @ B @ Y4 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_4940_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F4: A > nat,G4: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F4 @ ( product_fst @ A @ B @ P6 ) ) @ ( G4 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_4941_mod__pure,axiom,
! [B2: $o,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( pure_assn @ B2 ) @ H )
= ( ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H )
= ( bot_bot @ ( set @ nat ) ) )
& B2 ) ) ).
% mod_pure
thf(fact_4942_effectE,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H: heap_ext @ product_unit,H3: heap_ext @ product_unit,R3: A,N2: nat] :
( ( heap_Time_effect @ A @ C2 @ H @ H3 @ R3 @ N2 )
=> ~ ( ( R3
= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H ) ) ) )
=> ( ( H3
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H ) ) ) ) )
=> ( ( N2
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H ) ) ) ) )
=> ~ ( heap_Time_success @ A @ C2 @ H ) ) ) ) ) ).
% effectE
thf(fact_4943_execute__bind__success,axiom,
! [B: $tType,A: $tType,F3: heap_Time_Heap @ A,H: heap_ext @ product_unit,G3: A > ( heap_Time_Heap @ B )] :
( ( heap_Time_success @ A @ F3 @ H )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F3 @ G3 ) @ H )
= ( heap_Time_timeFrame @ B @ ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F3 @ H ) ) ) ) @ ( heap_Time_execute @ B @ ( G3 @ ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F3 @ H ) ) ) ) @ ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F3 @ H ) ) ) ) ) ) ) ) ).
% execute_bind_success
thf(fact_4944_mod__emp,axiom,
! [H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( one_one @ assn ) @ H )
= ( ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H )
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_emp
thf(fact_4945_ge__eq__refl,axiom,
! [A: $tType,R2: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ R2 )
=> ( R2 @ X @ X ) ) ).
% ge_eq_refl
thf(fact_4946_refl__ge__eq,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X3: A] : ( R2 @ X3 @ X3 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ R2 ) ) ).
% refl_ge_eq
thf(fact_4947_mod__star__trueE_H,axiom,
! [P: assn,H: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ P @ ( top_top @ assn ) ) @ H )
=> ~ ! [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 ) @ H ) )
=> ( ( ord_less_eq @ ( set @ nat ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H5 ) @ ( product_snd @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H ) )
=> ~ ( rep_assn @ P @ H5 ) ) ) ) ).
% mod_star_trueE'
thf(fact_4948_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_4949_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_snd @ A @ B @ X )
= Z2 ) ) ).
% sndI
thf(fact_4950_in__set__enumerate__eq,axiom,
! [A: $tType,P3: product_prod @ nat @ A,N2: nat,Xs: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P3 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N2 @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N2 @ ( product_fst @ nat @ A @ P3 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P3 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N2 ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P3 ) @ N2 ) )
= ( product_snd @ nat @ A @ P3 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_4951_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_4952_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Xy: product_prod @ A @ C] :
( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Yz: product_prod @ C @ B] :
( if @ ( list @ ( product_prod @ A @ B ) )
@ ( ( product_snd @ A @ C @ Xy )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys ) ) ) ) ).
% set_relcomp
thf(fact_4953_relcomp__empty2,axiom,
! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ A @ C )] :
( ( relcomp @ A @ C @ B @ R2 @ ( bot_bot @ ( set @ ( product_prod @ C @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty2
thf(fact_4954_relcomp__empty1,axiom,
! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) @ R2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty1
thf(fact_4955_relcomp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B ),T2: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ R2 @ ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ S @ T2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ R2 @ S ) @ ( relcomp @ A @ C @ B @ R2 @ T2 ) ) ) ).
% relcomp_distrib
thf(fact_4956_relcomp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S: set @ ( product_prod @ A @ C ),T2: set @ ( product_prod @ A @ C ),R2: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( sup_sup @ ( set @ ( product_prod @ A @ C ) ) @ S @ T2 ) @ R2 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( relcomp @ A @ C @ B @ S @ R2 ) @ ( relcomp @ A @ C @ B @ T2 @ R2 ) ) ) ).
% relcomp_distrib2
thf(fact_4957_relcomp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R7: set @ ( product_prod @ A @ B ),R3: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C ),S2: set @ ( product_prod @ B @ C )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R7 @ R3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S4 @ S2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R7 @ S4 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) ) ) ) ).
% relcomp_mono
thf(fact_4958_O__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R2: set @ ( product_prod @ A @ D ),S: set @ ( product_prod @ D @ C ),T2: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( relcomp @ A @ D @ C @ R2 @ S ) @ T2 )
= ( relcomp @ A @ D @ B @ R2 @ ( relcomp @ D @ C @ B @ S @ T2 ) ) ) ).
% O_assoc
thf(fact_4959_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) )
=> ~ ! [B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B5 ) @ R3 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_4960_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) )
= ( ? [A7: A,B6: B,C5: C] :
( ( A1 = A7 )
& ( A22 = C5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B6 ) @ R3 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B6 @ C5 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_4961_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B ),C2: C,S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C2 ) @ S2 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_4962_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R3: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R3 @ S2 ) )
=> ~ ! [X3: A,Y3: C,Z3: B] :
( ( Xz
= ( product_Pair @ A @ B @ X3 @ Z3 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_4963_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R3: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C2 ) @ ( relcomp @ A @ C @ B @ R3 @ S2 ) )
=> ~ ! [B5: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B5 ) @ R3 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B5 @ C2 ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_4964_union__comp__emptyL,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),C4: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A5 @ C4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ B4 @ C4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B4 ) @ C4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyL
thf(fact_4965_union__comp__emptyR,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A ),C4: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A5 @ B4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ A5 @ C4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ A5 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B4 @ C4 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyR
thf(fact_4966_relcomp__UNION__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S2: set @ ( product_prod @ A @ C ),R3: D > ( set @ ( product_prod @ C @ B ) ),I5: set @ D] :
( ( relcomp @ A @ C @ B @ S2 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ C @ B ) ) @ ( image2 @ D @ ( set @ ( product_prod @ C @ B ) ) @ R3 @ I5 ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I4: D] : ( relcomp @ A @ C @ B @ S2 @ ( R3 @ I4 ) )
@ I5 ) ) ) ).
% relcomp_UNION_distrib
thf(fact_4967_relcomp__UNION__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R3: D > ( set @ ( product_prod @ A @ C ) ),I5: set @ D,S2: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ C ) ) @ ( image2 @ D @ ( set @ ( product_prod @ A @ C ) ) @ R3 @ I5 ) ) @ S2 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I4: D] : ( relcomp @ A @ C @ B @ ( R3 @ I4 ) @ S2 )
@ I5 ) ) ) ).
% relcomp_UNION_distrib2
thf(fact_4968_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs12: list @ A,Xs23: list @ A,X1: A,X2: A,Xs: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs ) ) )
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs12 ) @ ( cons @ A @ X2 @ 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,X: A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs12 ) @ Xs23 ) ) ).
% mergesort_by_rel_split.simps(2)
thf(fact_4971_mergesort__by__rel__split_Oelims,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X @ Xa )
= Y )
=> ( ! [Xs1: list @ A,Xs22: list @ A] :
( ( X
= ( 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] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X3: A] :
( ( Xa
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs1 ) @ Xs22 ) ) ) )
=> ~ ! [Xs1: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X12: A,X22: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X12 @ ( cons @ A @ X22 @ Xs2 ) ) )
=> ( Y
!= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ Xs1 ) @ ( cons @ A @ X22 @ Xs22 ) ) @ Xs2 ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
thf(fact_4972_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S )
=> ( ( relcomp @ A @ B @ C @ R2 @ 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,Y4: B,A8: set @ ( product_prod @ A @ C )] :
( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [W3: B,Z7: C,A16: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y4 = W3 ) @ ( insert @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Z7 ) @ A16 ) @ A16 ) )
@ A8
@ S ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R2 ) ) ) ) ).
% relcomp_fold
thf(fact_4973_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R2: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X @ R2 ) @ 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,Z7: B,A16: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z7 ) @ A16 )
@ A16 ) )
@ ( relcomp @ C @ A @ B @ R2 @ S )
@ S ) ) ) ).
% insert_relcomp_fold
thf(fact_4974_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X6: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S ) @ X6 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z7: B,A16: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z7 ) @ A16 )
@ A16 ) )
@ X6
@ S ) ) ) ).
% insert_relcomp_union_fold
thf(fact_4975_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 ) )
=> ( ! [R10: 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 ) @ R10 @ Xs2 ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R10 @ ( 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 @ R10 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
=> ( P @ R10 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% mergesort_by_rel.pinduct
thf(fact_4976_mergesort__by__rel__split_Opelims,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ X @ Xa ) )
=> ( ! [Xs1: list @ A,Xs22: list @ A] :
( ( X
= ( 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] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X3: A] :
( ( Xa
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ 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 @ X3 @ ( nil @ A ) ) ) ) ) ) )
=> ~ ! [Xs1: list @ A,Xs22: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X12: A,X22: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X12 @ ( cons @ A @ X22 @ Xs2 ) ) )
=> ( ( Y
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X12 @ Xs1 ) @ ( cons @ A @ X22 @ 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 @ X22 @ Xs2 ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
thf(fact_4977_min__ext__compat,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S ) @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S ) @ ( insert @ ( 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 @ R2 ) ) ) ).
% min_ext_compat
thf(fact_4978_max__ext__compat,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S ) @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S ) @ ( insert @ ( 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 @ R2 ) ) ) ).
% max_ext_compat
thf(fact_4979_mergesort__by__rel_Opsimps,axiom,
! [A: $tType,R2: 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 ) @ R2 @ Xs ) )
=> ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R2 @ Xs )
= Xs ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R2 @ Xs )
= ( merges9089515139780605204_merge @ A @ R2 @ ( mergesort_by_rel @ A @ R2 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) @ ( mergesort_by_rel @ A @ R2 @ ( 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_4980_mergesort__by__rel_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ( ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y = Xa ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y
= ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) ) ) ) ) ).
% mergesort_by_rel.pelims
thf(fact_4981_mergesort__by__rel__simps_I1_J,axiom,
! [A: $tType,R2: A > A > $o] :
( ( mergesort_by_rel @ A @ R2 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% mergesort_by_rel_simps(1)
thf(fact_4982_set__mergesort__by__rel,axiom,
! [A: $tType,R2: A > A > $o,Xs: list @ A] :
( ( set2 @ A @ ( mergesort_by_rel @ A @ R2 @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_mergesort_by_rel
thf(fact_4983_mergesort__by__rel__merge__simps_I3_J,axiom,
! [A: $tType,R2: A > A > $o,Ys3: list @ A] :
( ( merges9089515139780605204_merge @ A @ R2 @ ( nil @ A ) @ Ys3 )
= Ys3 ) ).
% mergesort_by_rel_merge_simps(3)
thf(fact_4984_sorted__wrt__mergesort__by__rel__merge,axiom,
! [A: $tType,R2: A > A > $o,Xs: list @ A,Ys3: list @ A] :
( ! [X3: A,Y3: A] :
( ( R2 @ X3 @ Y3 )
| ( R2 @ Y3 @ X3 ) )
=> ( ! [X3: A,Y3: A,Z3: A] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ( sorted_wrt @ A @ R2 @ ( merges9089515139780605204_merge @ A @ R2 @ Xs @ Ys3 ) )
= ( ( sorted_wrt @ A @ R2 @ Xs )
& ( sorted_wrt @ A @ R2 @ Ys3 ) ) ) ) ) ).
% sorted_wrt_mergesort_by_rel_merge
thf(fact_4985_mergesort__by__rel__simps_I2_J,axiom,
! [A: $tType,R2: A > A > $o,X: A] :
( ( mergesort_by_rel @ A @ R2 @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% mergesort_by_rel_simps(2)
thf(fact_4986_set__mergesort__by__rel__merge,axiom,
! [A: $tType,R2: A > A > $o,Xs: list @ A,Ys3: list @ A] :
( ( set2 @ A @ ( merges9089515139780605204_merge @ A @ R2 @ Xs @ Ys3 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) ) ) ).
% set_mergesort_by_rel_merge
thf(fact_4987_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R2: A > A > $o,X1: A,X2: A,Xs: list @ A] :
( ( mergesort_by_rel @ A @ R2 @ ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs ) ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs13: list @ A,Xs24: list @ A] : ( merges9089515139780605204_merge @ A @ R2 @ ( mergesort_by_rel @ A @ R2 @ Xs13 ) @ ( mergesort_by_rel @ A @ R2 @ Xs24 ) )
@ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ ( nil @ A ) ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ Xs ) ) ) ).
% mergesort_by_rel_simps(3)
thf(fact_4988_mergesort__by__rel__merge__simps_I2_J,axiom,
! [A: $tType,R2: A > A > $o,Xs: list @ A] :
( ( merges9089515139780605204_merge @ A @ R2 @ Xs @ ( nil @ A ) )
= Xs ) ).
% mergesort_by_rel_merge_simps(2)
thf(fact_4989_mergesort__by__rel__merge__simps_I1_J,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys3: list @ A] :
( ( ( R2 @ X @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys3 ) )
= ( cons @ A @ X @ ( merges9089515139780605204_merge @ A @ R2 @ Xs @ ( cons @ A @ Y @ Ys3 ) ) ) ) )
& ( ~ ( R2 @ X @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys3 ) )
= ( cons @ A @ Y @ ( merges9089515139780605204_merge @ A @ R2 @ ( cons @ A @ X @ Xs ) @ Ys3 ) ) ) ) ) ).
% mergesort_by_rel_merge_simps(1)
thf(fact_4990_sorted__wrt__mergesort__by__rel,axiom,
! [X15: $tType,R2: X15 > X15 > $o,Xs: list @ X15] :
( ! [X3: X15,Y3: X15] :
( ( R2 @ X3 @ Y3 )
| ( R2 @ Y3 @ X3 ) )
=> ( ! [X3: X15,Y3: X15,Z3: X15] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( sorted_wrt @ X15 @ R2 @ ( mergesort_by_rel @ X15 @ R2 @ Xs ) ) ) ) ).
% sorted_wrt_mergesort_by_rel
thf(fact_4991_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_4992_mergesort__by__rel__merge_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X @ Xa @ Xb )
= Y )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xb
= ( cons @ A @ Y3 @ Ys4 ) )
=> ~ ( ( ( X @ X3 @ Y3 )
=> ( Y
= ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ X @ Xs2 @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) )
& ( ~ ( X @ X3 @ Y3 )
=> ( Y
= ( cons @ A @ Y3 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X3 @ Xs2 ) @ Ys4 ) ) ) ) ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V4: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V4 @ Va ) )
=> ( Y
!= ( cons @ A @ V4 @ Va ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.elims
thf(fact_4993_mergesort__by__rel__merge_Osimps_I3_J,axiom,
! [A: $tType,R2: A > A > $o,V2: A,Va2: list @ A] :
( ( merges9089515139780605204_merge @ A @ R2 @ ( nil @ A ) @ ( cons @ A @ V2 @ Va2 ) )
= ( cons @ A @ V2 @ Va2 ) ) ).
% mergesort_by_rel_merge.simps(3)
thf(fact_4994_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_4995_max__ext__additive,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,R2: set @ ( product_prod @ A @ A ),C4: set @ A,D4: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A5 @ B4 ) @ ( max_ext @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C4 @ D4 ) @ ( max_ext @ A @ R2 ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ C4 ) @ ( sup_sup @ ( set @ A ) @ B4 @ D4 ) ) @ ( max_ext @ A @ R2 ) ) ) ) ).
% max_ext_additive
thf(fact_4996_mergesort__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( mergesort @ A )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% mergesort_def
thf(fact_4997_max__ext_Ocases,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R2 ) )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X7: A] :
( ( member @ A @ X7 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa3 ) @ R2 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_4998_max__ext_Osimps,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R2 ) )
= ( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A1 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) ) ) ) ) ).
% max_ext.simps
thf(fact_4999_max__ext_Omax__extI,axiom,
! [A: $tType,X6: set @ A,Y7: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( Y7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa2 ) @ R2 ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X6 @ Y7 ) @ ( max_ext @ A @ R2 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_5000_mergesort__by__rel_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X @ Xa )
= Y )
=> ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y = Xa ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y
= ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) ) ) ).
% mergesort_by_rel.elims
thf(fact_5001_mergesort__by__rel_Osimps,axiom,
! [A: $tType] :
( ( mergesort_by_rel @ A )
= ( ^ [R6: A > A > $o,Xs3: list @ A] : ( if @ ( list @ A ) @ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Xs3 @ ( merges9089515139780605204_merge @ A @ R6 @ ( mergesort_by_rel @ A @ R6 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) @ ( mergesort_by_rel @ A @ R6 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) ) ) ) ) ).
% mergesort_by_rel.simps
thf(fact_5002_mergesort__by__rel__merge_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xb
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( ( ( X @ X3 @ Y3 )
=> ( Y
= ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ X @ Xs2 @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) )
& ( ~ ( X @ X3 @ Y3 )
=> ( Y
= ( cons @ A @ Y3 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X3 @ Xs2 ) @ Ys4 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V4: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V4 @ Va ) )
=> ( ( Y
= ( cons @ A @ V4 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
thf(fact_5003_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,Y4: A,A8: set @ ( product_prod @ C @ B )] :
( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z7: B,A16: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y4 = W3 ) @ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X4 @ Z7 ) @ A16 ) @ A16 ) )
@ A8
@ S ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_5004_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ( max_extp @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R6 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_5005_comp__fun__commute__const,axiom,
! [B: $tType,A: $tType,F3: B > B] :
( finite6289374366891150609ommute @ A @ B
@ ^ [Uu2: A] : F3 ) ).
% comp_fun_commute_const
thf(fact_5006_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 ) @ ( insert @ A @ X4 @ A16 ) @ A16 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_5007_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
! [B: $tType,A: $tType,F3: A > B > B,G3: A > nat] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( finite6289374366891150609ommute @ A @ B
@ ^ [X4: A] : ( compow @ ( B > B ) @ ( G3 @ X4 ) @ ( F3 @ X4 ) ) ) ) ).
% comp_fun_commute.comp_fun_commute_funpow
thf(fact_5008_comp__fun__commute_Ofoldl__f__commute,axiom,
! [B: $tType,A: $tType,F3: A > B > B,A3: A,B2: B,Xs: list @ A] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( ( F3 @ A3
@ ( foldl @ B @ A
@ ^ [A7: B,B6: A] : ( F3 @ B6 @ A7 )
@ B2
@ Xs ) )
= ( foldl @ B @ A
@ ^ [A7: B,B6: A] : ( F3 @ B6 @ A7 )
@ ( F3 @ A3 @ B2 )
@ Xs ) ) ) ).
% comp_fun_commute.foldl_f_commute
thf(fact_5009_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,Y4: B,A8: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X4 @ S ) @ ( insert @ B @ Y4 @ A8 ) @ A8 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_5010_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B4: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X4: B,Z7: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y4: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y4 ) )
@ Z7
@ B4 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_5011_max__extp__max__ext__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) )
= ( ^ [X4: set @ A,Y4: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y4 ) @ ( max_ext @ A @ R2 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_5012_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R4: A > A > $o,X4: set @ A,Y4: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y4 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_5013_map__of__distinct__upd,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_add @ A @ B
@ ( fun_upd @ A @ ( option @ B )
@ ^ [X4: A] : ( none @ B )
@ X
@ ( some @ B @ Y ) )
@ ( map_of @ A @ B @ Xs ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Xs ) @ X @ ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_upd
thf(fact_5014_relpow__finite__bounded1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R2 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 )
@ ( collect @ nat
@ ^ [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_5015_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 ) ) )
@ ^ [Uu2: A] : ( image2 @ C @ B @ ( comp @ ( product_prod @ A @ B ) @ B @ C @ ( product_snd @ A @ B ) @ F3 ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% range_prod
thf(fact_5016_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B2: B,B4: A > ( set @ B )] :
( ( member @ A @ A3 @ A5 )
=> ( ( member @ B @ B2 @ ( B4 @ A3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) ) ) ) ).
% SigmaI
thf(fact_5017_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
= ( ( member @ A @ A3 @ A5 )
& ( member @ B @ B2 @ ( B4 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_5018_map__add__find__right,axiom,
! [B: $tType,A: $tType,N2: B > ( option @ A ),K: B,Xx: A,M: B > ( option @ A )] :
( ( ( N2 @ K )
= ( some @ A @ Xx ) )
=> ( ( map_add @ B @ A @ M @ N2 @ K )
= ( some @ A @ Xx ) ) ) ).
% map_add_find_right
thf(fact_5019_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B6: B] :
( ( P @ A7 )
& ( Q @ B6 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu2: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_5020_empty__eq__map__add__iff,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),G3: A > ( option @ B )] :
( ( ( ^ [X4: A] : ( none @ B ) )
= ( map_add @ A @ B @ F3 @ G3 ) )
= ( ( F3
= ( ^ [X4: A] : ( none @ B ) ) )
& ( G3
= ( ^ [X4: A] : ( none @ B ) ) ) ) ) ).
% empty_eq_map_add_iff
thf(fact_5021_map__add__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( map_add @ A @ B @ M
@ ^ [X4: A] : ( none @ B ) )
= M ) ).
% map_add_empty
thf(fact_5022_empty__map__add,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( map_add @ A @ B
@ ^ [X4: A] : ( none @ B )
@ M )
= M ) ).
% empty_map_add
thf(fact_5023_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_5024_map__add__upd,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),G3: A > ( option @ B ),X: A,Y: B] :
( ( map_add @ A @ B @ F3 @ ( fun_upd @ A @ ( option @ B ) @ G3 @ X @ ( some @ B @ Y ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ F3 @ G3 ) @ X @ ( some @ B @ Y ) ) ) ).
% map_add_upd
thf(fact_5025_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_5026_Times__empty,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B4
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_5027_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,A5: set @ A,X6: set @ B] :
( ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : X6 )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : ( top_top @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_UNIV_cancel
thf(fact_5028_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : ( top_top @ ( set @ B ) ) ) )
= ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A5 )
@ ^ [Uu2: A] : ( top_top @ ( set @ B ) ) ) ) ).
% Compl_Times_UNIV2
thf(fact_5029_Compl__Times__UNIV1,axiom,
! [B: $tType,A: $tType,A5: set @ B] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : A5 ) )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : ( uminus_uminus @ ( set @ B ) @ A5 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_5030_finite__SigmaI,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A5 )
=> ( finite_finite2 @ B @ ( B4 @ A6 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_5031_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% UNIV_Times_UNIV
thf(fact_5032_pairself__image__cart,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B,B4: set @ B] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F3 )
@ ( product_Sigma @ B @ B @ A5
@ ^ [Uu2: B] : B4 ) )
= ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F3 @ A5 )
@ ^ [Uu2: A] : ( image2 @ B @ A @ F3 @ B4 ) ) ) ).
% pairself_image_cart
thf(fact_5033_finite__relpow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N2: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) ) ) ) ).
% finite_relpow
thf(fact_5034_fst__image__times,axiom,
! [B: $tType,A: $tType,B4: set @ B,A5: set @ A] :
( ( ( B4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) )
= A5 ) ) ) ).
% fst_image_times
thf(fact_5035_snd__image__times,axiom,
! [B: $tType,A: $tType,A5: set @ B,B4: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu2: B] : B4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu2: B] : B4 ) )
= B4 ) ) ) ).
% snd_image_times
thf(fact_5036_set__product,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys3: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys3 ) )
= ( product_Sigma @ A @ B @ ( set2 @ A @ Xs )
@ ^ [Uu2: A] : ( set2 @ B @ Ys3 ) ) ) ).
% set_product
thf(fact_5037_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B2: B,B4: set @ B] :
( ( product_Sigma @ A @ B @ ( insert @ A @ A3 @ A5 )
@ ^ [Uu2: A] : ( insert @ B @ B2 @ B4 ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : ( insert @ B @ B2 @ B4 ) )
@ ( product_Sigma @ A @ B @ ( insert @ A @ A3 @ A5 )
@ ^ [Uu2: A] : B4 ) ) ) ) ).
% insert_Times_insert
thf(fact_5038_card__SigmaI,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( finite_finite2 @ B @ ( B4 @ X3 ) ) )
=> ( ( finite_card @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A7: A] : ( finite_card @ B @ ( B4 @ A7 ) )
@ A5 ) ) ) ) ).
% card_SigmaI
thf(fact_5039_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y4: B] :
( ( P @ X4 )
& ( Q @ X4 @ Y4 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X4: A] : ( collect @ B @ ( Q @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_5040_times__eq__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B,C4: set @ A,D4: set @ B] :
( ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 )
= ( product_Sigma @ A @ B @ C4
@ ^ [Uu2: A] : D4 ) )
= ( ( ( A5 = C4 )
& ( B4 = D4 ) )
| ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B4
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C4
= ( bot_bot @ ( set @ A ) ) )
| ( D4
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_5041_map__add__first__le,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M: A > ( option @ B ),M8: A > ( option @ B ),N2: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M @ M8 )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M @ N2 ) @ ( map_add @ A @ B @ M8 @ N2 ) ) ) ) ).
% map_add_first_le
thf(fact_5042_map__add__left__None,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),K: B,G3: B > ( option @ A )] :
( ( ( F3 @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F3 @ G3 @ K )
= ( G3 @ K ) ) ) ).
% map_add_left_None
thf(fact_5043_map__add__find__left,axiom,
! [A: $tType,B: $tType,G3: B > ( option @ A ),K: B,F3: B > ( option @ A )] :
( ( ( G3 @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F3 @ G3 @ K )
= ( F3 @ K ) ) ) ).
% map_add_find_left
thf(fact_5044_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I4: A] : ( minus_minus @ ( set @ B ) @ ( A5 @ I4 ) @ ( B4 @ I4 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) @ ( product_Sigma @ A @ B @ I5 @ B4 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_5045_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,C4: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A5 @ B4 )
@ ^ [Uu2: A] : C4 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : C4 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu2: A] : C4 ) ) ) ).
% Times_Diff_distrib1
thf(fact_5046_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J5: set @ A,C4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I5 @ J5 ) @ C4 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C4 ) @ ( product_Sigma @ A @ B @ J5 @ C4 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_5047_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C4: set @ A,A5: set @ B,B4: set @ B] :
( ( member @ A @ X @ C4 )
=> ( ( ( product_Sigma @ B @ A @ A5
@ ^ [Uu2: B] : C4 )
= ( product_Sigma @ B @ A @ B4
@ ^ [Uu2: B] : C4 ) )
= ( A5 = B4 ) ) ) ).
% Times_eq_cancel2
thf(fact_5048_Sigma__cong,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,C4: A > ( set @ B ),D4: A > ( set @ B )] :
( ( A5 = B4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B4 )
=> ( ( C4 @ X3 )
= ( D4 @ X3 ) ) )
=> ( ( product_Sigma @ A @ B @ A5 @ C4 )
= ( product_Sigma @ A @ B @ B4 @ D4 ) ) ) ) ).
% Sigma_cong
thf(fact_5049_relpow__Suc__D2_H,axiom,
! [A: $tType,N2: nat,R2: set @ ( product_prod @ A @ A ),X7: A,Y6: A,Z6: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Y6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z6 ) @ R2 ) )
=> ? [W: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ W ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W @ Z6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_5050_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
=> ~ ( ( member @ A @ A3 @ A5 )
=> ~ ( member @ B @ B2 @ ( B4 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_5051_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
=> ( member @ B @ B2 @ ( B4 @ A3 ) ) ) ).
% SigmaD2
thf(fact_5052_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
=> ( member @ A @ A3 @ A5 ) ) ).
% SigmaD1
thf(fact_5053_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod @ A @ B,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ! [Y3: B] :
( ( member @ B @ Y3 @ ( B4 @ X3 ) )
=> ( C2
!= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_5054_Sigma__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,C4: set @ A,B4: A > ( set @ B ),D4: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B4 @ X3 ) @ ( D4 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) @ ( product_Sigma @ A @ B @ C4 @ D4 ) ) ) ) ).
% Sigma_mono
thf(fact_5055_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J5: set @ A,C4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I5 @ J5 ) @ C4 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C4 ) @ ( product_Sigma @ A @ B @ J5 @ C4 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_5056_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J5: set @ A,C4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I5 @ J5 ) @ C4 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C4 ) @ ( product_Sigma @ A @ B @ J5 @ C4 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_5057_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: set @ A,B4: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A5 @ B4 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) ) ) ).
% member_product
thf(fact_5058_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A8: set @ A,B7: set @ B] :
( product_Sigma @ A @ B @ A8
@ ^ [Uu2: A] : B7 ) ) ) ).
% Product_Type.product_def
thf(fact_5059_map__add__SomeD,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),N2: B > ( option @ A ),K: B,X: A] :
( ( ( map_add @ B @ A @ M @ N2 @ K )
= ( some @ A @ X ) )
=> ( ( ( N2 @ K )
= ( some @ A @ X ) )
| ( ( ( N2 @ K )
= ( none @ A ) )
& ( ( M @ K )
= ( some @ A @ X ) ) ) ) ) ).
% map_add_SomeD
thf(fact_5060_map__add__Some__iff,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),N2: B > ( option @ A ),K: B,X: A] :
( ( ( map_add @ B @ A @ M @ N2 @ K )
= ( some @ A @ X ) )
= ( ( ( N2 @ K )
= ( some @ A @ X ) )
| ( ( ( N2 @ K )
= ( none @ A ) )
& ( ( M @ K )
= ( some @ A @ X ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_5061_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_5062_relpow__0__E,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 ) )
=> ( X = Y ) ) ).
% relpow_0_E
thf(fact_5063_relpow__0__I,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 ) ) ).
% relpow_0_I
thf(fact_5064_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R2 ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 ) ) ) ).
% relpow_Suc_E
thf(fact_5065_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y: A,N2: nat,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R2 ) ) ) ) ).
% relpow_Suc_I
thf(fact_5066_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R2 ) )
=> ? [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_5067_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R2 ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_5068_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z2: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N2 ) @ R2 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_5069_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C4: set @ A,A5: set @ B,B4: set @ B] :
( ( member @ A @ X @ C4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu2: B] : C4 )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu2: B] : C4 ) )
= ( ord_less_eq @ ( set @ B ) @ A5 @ B4 ) ) ) ).
% Times_subset_cancel2
thf(fact_5070_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A5: set @ A,B4: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A5 )
& ( member @ B @ ( product_snd @ A @ B @ X ) @ B4 ) ) ) ).
% mem_Times_iff
thf(fact_5071_in__prod__fst__sndI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: set @ A,B4: set @ B] :
( ( member @ A @ ( product_fst @ A @ B @ X ) @ A5 )
=> ( ( member @ B @ ( product_snd @ A @ B @ X ) @ B4 )
=> ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) ) ) ) ).
% in_prod_fst_sndI
thf(fact_5072_card__cartesian__product,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) )
= ( times_times @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_5073_swap__product,axiom,
! [B: $tType,A: $tType,A5: set @ B,B4: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I4: B,J3: A] : ( product_Pair @ A @ B @ J3 @ I4 ) )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu2: B] : B4 ) )
= ( product_Sigma @ A @ B @ B4
@ ^ [Uu2: A] : A5 ) ) ).
% swap_product
thf(fact_5074_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I5: set @ A,X6: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I5 @ X6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ I5 )
=> ( ( X6 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_5075_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,C4: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A5 @ B4 )
@ ^ [Uu2: A] : C4 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : C4 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu2: A] : C4 ) ) ) ).
% Times_Int_distrib1
thf(fact_5076_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I4: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ I4 ) @ ( B4 @ I4 ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) @ ( product_Sigma @ A @ B @ I5 @ B4 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_5077_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B,C4: set @ A,D4: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 )
@ ( product_Sigma @ A @ B @ C4
@ ^ [Uu2: A] : D4 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A5 @ C4 )
@ ^ [Uu2: A] : ( inf_inf @ ( set @ B ) @ B4 @ D4 ) ) ) ).
% Times_Int_Times
thf(fact_5078_finite__cartesian__product,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_5079_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,C4: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A5 @ B4 )
@ ^ [Uu2: A] : C4 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : C4 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu2: A] : C4 ) ) ) ).
% Times_Un_distrib1
thf(fact_5080_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I4: A] : ( sup_sup @ ( set @ B ) @ ( A5 @ I4 ) @ ( B4 @ I4 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) @ ( product_Sigma @ A @ B @ I5 @ B4 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_5081_relcomp__subset__Sigma,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),A5: set @ A,B4: set @ B,S2: set @ ( product_prod @ B @ C ),C4: set @ C] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S2
@ ( product_Sigma @ B @ C @ B4
@ ^ [Uu2: B] : C4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R3 @ S2 )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu2: A] : C4 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_5082_Sigma__Union,axiom,
! [B: $tType,A: $tType,X6: set @ ( set @ A ),B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X6 ) @ B4 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A8: set @ A] : ( product_Sigma @ A @ B @ A8 @ B4 )
@ X6 ) ) ) ).
% Sigma_Union
thf(fact_5083_Id__on__subset__Times,axiom,
! [A: $tType,A5: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( id_on @ A @ A5 )
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ).
% Id_on_subset_Times
thf(fact_5084_map__add__def,axiom,
! [B: $tType,A: $tType] :
( ( map_add @ A @ B )
= ( ^ [M1: A > ( option @ B ),M22: A > ( option @ B ),X4: A] : ( case_option @ ( option @ B ) @ B @ ( M1 @ X4 ) @ ( some @ B ) @ ( M22 @ X4 ) ) ) ) ).
% map_add_def
thf(fact_5085_relpowp__relpow__eq,axiom,
! [A: $tType,N2: nat,R2: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N2
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R2 ) )
= ( ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_5086_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X: A,A5: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu2: A] : A5 ) )
= ( finite_card @ B @ A5 ) ) ).
% card_cartesian_product_singleton
thf(fact_5087_times__subset__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,C4: set @ B,B4: set @ A,D4: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : C4 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu2: A] : D4 ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( C4
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
& ( ord_less_eq @ ( set @ B ) @ C4 @ D4 ) ) ) ) ).
% times_subset_iff
thf(fact_5088_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F3: C > A,G3: D > B,A5: set @ C,B4: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y4: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ Y4 ) ) )
@ ( product_Sigma @ C @ D @ A5
@ ^ [Uu2: C] : B4 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F3 @ A5 )
@ ^ [Uu2: A] : ( image2 @ D @ B @ G3 @ B4 ) ) ) ).
% image_paired_Times
thf(fact_5089_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ( B4 @ X4 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A5 )
=> ( finite_finite2 @ B @ ( B4 @ A6 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) ) ) ) ).
% finite_SigmaI2
thf(fact_5090_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_cartesian_productD1
thf(fact_5091_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B4 ) ) ) ).
% finite_cartesian_productD2
thf(fact_5092_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B4
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A5 )
& ( finite_finite2 @ B @ B4 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_5093_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ( B4 @ X4 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_5094_relpow__E2,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X != Z2 ) )
=> ~ ! [Y3: A,M5: nat] :
( ( N2
= ( suc @ M5 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M5 @ R2 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_5095_relpow__E,axiom,
! [A: $tType,X: A,Z2: A,N2: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( X != Z2 ) )
=> ~ ! [Y3: A,M5: nat] :
( ( N2
= ( suc @ M5 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M5 @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 ) ) ) ) ) ).
% relpow_E
thf(fact_5096_UN__Times__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E6: C > ( set @ A ),F5: D > ( set @ B ),A5: set @ C,B4: set @ D] :
( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A7: C,B6: D] :
( product_Sigma @ A @ B @ ( E6 @ A7 )
@ ^ [Uu2: A] : ( F5 @ B6 ) ) )
@ ( product_Sigma @ C @ D @ A5
@ ^ [Uu2: C] : B4 ) ) )
= ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ E6 @ A5 ) )
@ ^ [Uu2: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ D @ ( set @ B ) @ F5 @ B4 ) ) ) ) ).
% UN_Times_distrib
thf(fact_5097_relpow__empty,axiom,
! [A: $tType,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% relpow_empty
thf(fact_5098_sum_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > C > A,B4: set @ C,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G3 @ X4 ) @ B4 )
@ A5 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G3 )
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu2: B] : B4 ) ) ) ) ).
% sum.cartesian_product
thf(fact_5099_prod_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > C > A,B4: set @ C,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G3 @ X4 ) @ B4 )
@ A5 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G3 )
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu2: B] : B4 ) ) ) ) ).
% prod.cartesian_product
thf(fact_5100_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A5: set @ B,B4: B > ( set @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A5 @ B4 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% snd_image_Sigma
thf(fact_5101_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5
@ ( product_Sigma @ A @ B @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 )
@ ^ [Uu2: A] : ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A5 ) ) ) ).
% subset_fst_snd
thf(fact_5102_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N2: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R2 ) )
= ( ? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= A3 )
& ( ( F4 @ N2 )
= B2 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R2 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_5103_sum_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: B > ( set @ C ),G3: B > C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( finite_finite2 @ C @ ( B4 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G3 @ X4 ) @ ( B4 @ X4 ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G3 ) @ ( product_Sigma @ B @ C @ A5 @ B4 ) ) ) ) ) ) ).
% sum.Sigma
thf(fact_5104_prod_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: B > ( set @ C ),G3: B > C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( finite_finite2 @ C @ ( B4 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G3 @ X4 ) @ ( B4 @ X4 ) )
@ A5 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G3 ) @ ( product_Sigma @ B @ C @ A5 @ B4 ) ) ) ) ) ) ).
% prod.Sigma
thf(fact_5105_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A8: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A8 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X4 ) @ A8 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_5106_relpow__finite__bounded,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R2 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 )
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_5107_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A8: set @ A,B7: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y4: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B7 @ X4 ) ) )
@ A8 ) ) ) ) ).
% Sigma_def
thf(fact_5108_product__fold,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: A,Z7: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y4: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) )
@ Z7
@ B4 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A5 ) ) ) ) ).
% product_fold
thf(fact_5109_lists__length__Suc__eq,axiom,
! [A: $tType,A5: set @ A,N2: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N2 ) ) ) )
= ( image2 @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs3: list @ A,N: A] : ( cons @ A @ N @ Xs3 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N2 ) ) )
@ ^ [Uu2: list @ A] : A5 ) ) ) ).
% lists_length_Suc_eq
thf(fact_5110_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N: nat,R6: 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 @ R6 )
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I4 )
& ( ord_less_eq @ nat @ I4 @ ( suc @ N ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_5111_infinite__cartesian__product,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ~ ( finite_finite2 @ B @ B4 )
=> ~ ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_5112_Restr__subset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu2: A] : B4 ) )
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ) ).
% Restr_subset
thf(fact_5113_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),As9: A > B] :
! [I4: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I4 @ J3 ) @ R4 )
=> ( ord_less_eq @ B @ ( As9 @ I4 ) @ ( As9 @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_5114_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_5115_trancl__finite__eq__relpow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( transitive_trancl @ A @ R2 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 )
@ ( collect @ nat
@ ^ [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_5116_trancl__empty,axiom,
! [A: $tType] :
( ( transitive_trancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trancl_empty
thf(fact_5117_same__fst__trancl,axiom,
! [B: $tType,A: $tType,P: A > $o,R2: A > ( set @ ( product_prod @ B @ B ) )] :
( ( transitive_trancl @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P @ R2 ) )
= ( same_fst @ A @ B @ P
@ ^ [X4: A] : ( transitive_trancl @ B @ ( R2 @ X4 ) ) ) ) ).
% same_fst_trancl
thf(fact_5118_trancl__single,axiom,
! [A: $tType,A3: A,B2: A] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) )
= ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trancl_single
thf(fact_5119_trancl__mono__mp,axiom,
! [A: $tType,U3: set @ ( product_prod @ A @ A ),V: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U3 @ V )
=> ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ U3 ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ V ) ) ) ) ).
% trancl_mono_mp
thf(fact_5120_trancl__sub,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( transitive_trancl @ A @ R2 ) ) ).
% trancl_sub
thf(fact_5121_trancl__mono,axiom,
! [A: $tType,P3: product_prod @ A @ A,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P3 @ ( transitive_trancl @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 )
=> ( member @ ( product_prod @ A @ A ) @ P3 @ ( transitive_trancl @ A @ S2 ) ) ) ) ).
% trancl_mono
thf(fact_5122_converse__trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ B2 ) @ R3 )
=> ( P @ Y3 ) )
=> ( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P @ Z3 )
=> ( P @ Y3 ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% converse_trancl_induct
thf(fact_5123_trancl__trans__induct,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),P: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( P @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P @ X3 @ Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( P @ Y3 @ Z3 )
=> ( P @ X3 @ Z3 ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_5124_trancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_5125_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_5126_irrefl__trancl__rD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_5127_converse__tranclE,axiom,
! [A: $tType,X: A,Z2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R3 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ) ).
% converse_tranclE
thf(fact_5128_r__r__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% r_r_into_trancl
thf(fact_5129_trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y3 ) @ R3 )
=> ( P @ Y3 ) )
=> ( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R3 )
=> ( ( P @ Y3 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_5130_trancl__trans,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_trans
thf(fact_5131_tranclE,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ~ ! [C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C3 @ B2 ) @ R3 ) ) ) ) ).
% tranclE
thf(fact_5132_trancl_Or__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ).
% trancl.r_into_trancl
thf(fact_5133_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R3 ) )
= ( ? [A7: A,B6: A] :
( ( A1 = A7 )
& ( A22 = B6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B6 ) @ R3 ) )
| ? [A7: A,B6: A,C5: A] :
( ( A1 = A7 )
& ( A22 = C5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B6 ) @ ( transitive_trancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ C5 ) @ R3 ) ) ) ) ).
% trancl.simps
thf(fact_5134_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ R3 )
=> ~ ! [B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B5 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A22 ) @ R3 ) ) ) ) ).
% trancl.cases
thf(fact_5135_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ! [A6: A,B5: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B5 ) ) @ R3 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: B,Aa3: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B5 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) ) @ R3 )
=> ( ( P @ A6 @ B5 )
=> ( P @ Aa3 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_5136_trancl__sub__insert__trancl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 ) @ ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ X @ R2 ) ) ) ).
% trancl_sub_insert_trancl
thf(fact_5137_trancl__unfold,axiom,
! [A: $tType] :
( ( transitive_trancl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( relcomp @ A @ A @ A @ ( transitive_trancl @ A @ R4 ) @ R4 ) ) ) ) ).
% trancl_unfold
thf(fact_5138_trancl__subset__Sigma,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 )
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ).
% trancl_subset_Sigma
thf(fact_5139_trancl__power,axiom,
! [A: $tType,P3: product_prod @ A @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P3 @ ( transitive_trancl @ A @ R2 ) )
= ( ? [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( member @ ( product_prod @ A @ A ) @ P3 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ) ).
% trancl_power
thf(fact_5140_trancl__Int__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 ) @ S2 ) @ R3 ) @ S2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 ) @ S2 ) ) ) ).
% trancl_Int_subset
thf(fact_5141_Restr__trancl__mono,axiom,
! [A: $tType,V2: A,W2: A,E6: set @ ( product_prod @ A @ A ),U3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E6
@ ( product_Sigma @ A @ A @ U3
@ ^ [Uu2: A] : U3 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ E6 ) ) ) ).
% Restr_trancl_mono
thf(fact_5142_less__eq,axiom,
! [M: nat,N2: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N2 ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% less_eq
thf(fact_5143_trancl__insert2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A3 ) @ ( transitive_trancl @ A @ R3 ) )
| ( X4 = A3 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y4 ) @ ( transitive_trancl @ A @ R3 ) )
| ( Y4 = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_5144_nth__step__trancl,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A ),N2: nat,M: 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 ) ) @ R2 ) )
=> ( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ M @ N2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N2 ) @ ( nth @ A @ Xs @ M ) ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ).
% nth_step_trancl
thf(fact_5145_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K: A,V2: B] :
( ( map_to_set @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V2 ) ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 )
@ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( map_to_set @ A @ B @ M )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu2: product_prod @ A @ B] :
? [V6: B] :
( Uu2
= ( product_Pair @ A @ B @ K @ V6 ) ) ) ) ) ) ).
% map_to_set_upd
thf(fact_5146_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( transitive_rtrancl @ A @ R2 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 )
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less_eq @ nat @ N @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_5147_rtrancl__last__visit__node,axiom,
! [A: $tType,S2: A,S4: A,R2: set @ ( product_prod @ A @ A ),Sh: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S2 @ S4 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( ( S2 != Sh )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S2 @ S4 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) )
| ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S2 @ Sh ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Sh @ S4 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit_node
thf(fact_5148_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
? [Y4: A] :
( ( P @ Y4 )
& ( Q @ X4 @ Y4 ) ) ) )
= ( ! [Y4: A] :
( ( P @ Y4 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] : ( Q @ X4 @ Y4 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_5149_eq__or__mem__image__simp,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: B,B4: set @ B] :
( ( collect @ A
@ ^ [Uu2: A] :
? [L2: B] :
( ( Uu2
= ( F3 @ L2 ) )
& ( ( L2 = A3 )
| ( member @ B @ L2 @ B4 ) ) ) )
= ( insert @ A @ ( F3 @ A3 )
@ ( collect @ A
@ ^ [Uu2: A] :
? [L2: B] :
( ( Uu2
= ( F3 @ L2 ) )
& ( member @ B @ L2 @ B4 ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_5150_Eps__Opt__eq__None,axiom,
! [A: $tType,P: A > $o] :
( ( ( eps_Opt @ A @ P )
= ( none @ A ) )
= ( ~ ? [X11: A] : ( P @ X11 ) ) ) ).
% Eps_Opt_eq_None
thf(fact_5151_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 )
@ ^ [Uu2: product_prod @ A @ A] :
? [A7: B,B6: B] :
( ( Uu2
= ( product_Pair @ A @ A @ ( F3 @ A7 ) @ ( F3 @ B6 ) ) )
& ( P @ A7 @ B6 ) ) ) ) ).
% pairself_image_eq
thf(fact_5152_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( U != V2 )
=> ~ ! [Vh: A] :
( ( U != Vh )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ).
% converse_rtranclE'
thf(fact_5153_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_5154_converse__rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P @ B2 )
=> ( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P @ Z3 )
=> ( P @ Y3 ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_5155_converse__rtranclE,axiom,
! [A: $tType,X: A,Z2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( X != Z2 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_5156_rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( P @ A3 )
=> ( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R3 )
=> ( ( P @ Y3 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_5157_rtrancl__trans,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% rtrancl_trans
thf(fact_5158_rtranclE,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( A3 != B2 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ B2 ) @ R3 ) ) ) ) ).
% rtranclE
thf(fact_5159_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_5160_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A3: A,R3: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_5161_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R3 ) )
= ( ? [A7: A] :
( ( A1 = A7 )
& ( A22 = A7 ) )
| ? [A7: A,B6: A,C5: A] :
( ( A1 = A7 )
& ( A22 = C5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B6 ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ C5 ) @ R3 ) ) ) ) ).
% rtrancl.simps
thf(fact_5162_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( A22 != A1 )
=> ~ ! [B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B5 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A22 ) @ R3 ) ) ) ) ).
% rtrancl.cases
thf(fact_5163_Union__SetCompr__eq,axiom,
! [B: $tType,A: $tType,F3: B > ( set @ A ),P: B > $o] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu2: set @ A] :
? [X4: B] :
( ( Uu2
= ( F3 @ X4 ) )
& ( P @ X4 ) ) ) )
= ( collect @ A
@ ^ [A7: A] :
? [X4: B] :
( ( P @ X4 )
& ( member @ A @ A7 @ ( F3 @ X4 ) ) ) ) ) ).
% Union_SetCompr_eq
thf(fact_5164_tranclp_Omono,axiom,
! [A: $tType,R3: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P6: A > A > $o,X13: A,X23: A] :
( ? [A7: A,B6: A] :
( ( X13 = A7 )
& ( X23 = B6 )
& ( R3 @ A7 @ B6 ) )
| ? [A7: A,B6: A,C5: A] :
( ( X13 = A7 )
& ( X23 = C5 )
& ( P6 @ A7 @ B6 )
& ( R3 @ B6 @ C5 ) ) ) ) ).
% tranclp.mono
thf(fact_5165_rtranclp_Omono,axiom,
! [A: $tType,R3: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P6: A > A > $o,X13: A,X23: A] :
( ? [A7: A] :
( ( X13 = A7 )
& ( X23 = A7 ) )
| ? [A7: A,B6: A,C5: A] :
( ( X13 = A7 )
& ( X23 = C5 )
& ( P6 @ A7 @ B6 )
& ( R3 @ B6 @ C5 ) ) ) ) ).
% rtranclp.mono
thf(fact_5166_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A8: set @ A,XS: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z7: list @ A] :
? [X4: A,Xs3: list @ A] :
( ( Z7
= ( cons @ A @ X4 @ Xs3 ) )
& ( member @ A @ X4 @ A8 )
& ( member @ ( list @ A ) @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_5167_ord_Olexordp_Omono,axiom,
! [A: $tType,Less: A > A > $o] :
( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X23: list @ A] :
( ? [Y4: A,Ys2: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) ) )
| ? [X4: A,Y4: A,Xs3: list @ A,Ys2: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) )
& ( Less @ X4 @ Y4 ) )
| ? [X4: A,Y4: A,Xs3: list @ A,Ys2: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) )
& ~ ( Less @ X4 @ Y4 )
& ~ ( Less @ Y4 @ X4 )
& ( P6 @ Xs3 @ Ys2 ) ) ) ) ).
% ord.lexordp.mono
thf(fact_5168_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X23: list @ A] :
( ? [Y4: A,Ys2: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) ) )
| ? [X4: A,Y4: A,Xs3: list @ A,Ys2: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) )
& ( ord_less @ A @ X4 @ Y4 ) )
| ? [X4: A,Y4: A,Xs3: list @ A,Ys2: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) )
& ~ ( ord_less @ A @ X4 @ Y4 )
& ~ ( ord_less @ A @ Y4 @ X4 )
& ( P6 @ Xs3 @ Ys2 ) ) ) ) ) ).
% lexordp.mono
thf(fact_5169_finite__image__set,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: A > B] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Uu2: B] :
? [X4: A] :
( ( Uu2
= ( F3 @ X4 ) )
& ( P @ X4 ) ) ) ) ) ).
% finite_image_set
thf(fact_5170_finite__image__set2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > $o,Q: B > $o,F3: A > B > C] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
=> ( finite_finite2 @ C
@ ( collect @ C
@ ^ [Uu2: C] :
? [X4: A,Y4: B] :
( ( Uu2
= ( F3 @ X4 @ Y4 ) )
& ( P @ X4 )
& ( Q @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_5171_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ ( transitive_rtrancl @ A @ R3 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel1_rtrancl_subset_rtrancl_listrel1
thf(fact_5172_rtrancl__mono__rightI,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ S3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ S3 ) ) ) ).
% rtrancl_mono_rightI
thf(fact_5173_rtrancl__mono__mp,axiom,
! [A: $tType,U3: set @ ( product_prod @ A @ A ),V: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U3 @ V )
=> ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ U3 ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ V ) ) ) ) ).
% rtrancl_mono_mp
thf(fact_5174_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_5175_rtrancl__subset__rtrancl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( transitive_rtrancl @ A @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ ( transitive_rtrancl @ A @ S2 ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_5176_rtrancl__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( transitive_rtrancl @ A @ S )
= ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% rtrancl_subset
thf(fact_5177_rtrancl__mono,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ ( transitive_rtrancl @ A @ S2 ) ) ) ).
% rtrancl_mono
thf(fact_5178_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P6: ( set @ A ) > $o,X4: set @ A] :
( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,A7: A] :
( ( X4
= ( insert @ A @ A7 @ A8 ) )
& ( P6 @ A8 ) ) ) ) ).
% finite.mono
thf(fact_5179_in__rtrancl__UnI,axiom,
! [A: $tType,X: product_prod @ A @ A,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ R2 ) )
| ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ S ) ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_5180_rtrancl__Un__rtrancl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 ) @ ( transitive_rtrancl @ A @ S ) ) )
= ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) ) ) ).
% rtrancl_Un_rtrancl
thf(fact_5181_in__rtrancl__insert,axiom,
! [A: $tType,X: product_prod @ A @ A,R2: set @ ( product_prod @ A @ A ),R3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ ( insert @ ( product_prod @ A @ A ) @ R3 @ R2 ) ) ) ) ).
% in_rtrancl_insert
thf(fact_5182_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 )
@ ^ [Uu2: product_prod @ A @ B] :
? [P6: product_prod @ A @ B] :
( ( Uu2 = P6 )
& ( member @ C @ ( F3 @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) @ S ) ) ) )
= ( collect @ A
@ ^ [A7: A] :
? [B6: B] : ( member @ C @ ( F3 @ A7 @ B6 ) @ S ) ) ) ).
% fs_contract
thf(fact_5183_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu2: A] :
? [X4: B] :
( ( Uu2
= ( F3 @ X4 ) )
& ( P @ X4 ) ) )
= ( image2 @ B @ A @ F3 @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_5184_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( collect @ A
@ ^ [Uu2: A] :
? [X4: B] :
( ( Uu2
= ( F3 @ X4 ) )
& ( member @ B @ X4 @ A5 ) ) )
= ( image2 @ B @ A @ F3 @ A5 ) ) ).
% Setcompr_eq_image
thf(fact_5185_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys3: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys3 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_5186_trancl__rtrancl__trancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_5187_rtrancl__trancl__trancl,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_5188_rtrancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_5189_rtrancl__into__trancl1,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_5190_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_5191_trancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ).
% trancl_into_rtrancl
thf(fact_5192_tranclD2,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ R2 ) ) ) ).
% tranclD2
thf(fact_5193_rtranclD,axiom,
! [A: $tType,A3: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( A3 = B2 )
| ( ( A3 != B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ).
% rtranclD
thf(fact_5194_tranclD,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% tranclD
thf(fact_5195_rtrancl__Un__separatorE,axiom,
! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X3 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ Q )
=> ( X3 = Y3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_5196_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ Q )
=> ( Y3 = X3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_5197_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( P @ Bx @ By )
=> ( ! [A6: A,B5: B,Aa3: 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 @ Aa3 @ Ba ) ) @ R3 )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( P @ Aa3 @ Ba )
=> ( P @ A6 @ B5 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_5198_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa: A,Xb: B,Za: A,Zb: B,R3: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za @ Zb ) )
=> ~ ! [A6: A,B5: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A6 @ B5 ) ) @ R3 )
=> ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_5199_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R3: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A6: A,B5: B,Aa3: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B5 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R3 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) ) @ R3 )
=> ( ( P @ A6 @ B5 )
=> ( P @ Aa3 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_5200_rtrancl__Un__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 ) @ ( transitive_rtrancl @ A @ S ) ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) ) ) ).
% rtrancl_Un_subset
thf(fact_5201_rtrancl__sub__insert__rtrancl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 ) @ ( transitive_rtrancl @ A @ ( insert @ ( product_prod @ A @ A ) @ X @ R2 ) ) ) ).
% rtrancl_sub_insert_rtrancl
thf(fact_5202_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( collect @ A
@ ^ [U2: A] :
? [X4: B] :
( U2
= ( F3 @ X4 ) ) )
= ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_5203_finite__inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A3: A,A5: set @ A] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ A5 ) )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [Uu2: A] :
? [B6: A] :
( ( Uu2
= ( inf_inf @ A @ A3 @ B6 ) )
& ( member @ A @ B6 @ A5 ) ) ) ) ) ) ).
% finite_inf_Sup
thf(fact_5204_ran__def,axiom,
! [B: $tType,A: $tType] :
( ( ran @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( collect @ B
@ ^ [B6: B] :
? [A7: A] :
( ( M3 @ A7 )
= ( some @ B @ B6 ) ) ) ) ) ).
% ran_def
thf(fact_5205_Pow__Compl,axiom,
! [A: $tType,A5: set @ A] :
( ( pow @ A @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( collect @ ( set @ A )
@ ^ [Uu2: set @ A] :
? [B7: set @ A] :
( ( Uu2
= ( uminus_uminus @ ( set @ A ) @ B7 ) )
& ( member @ ( set @ A ) @ A5 @ ( pow @ A @ B7 ) ) ) ) ) ).
% Pow_Compl
thf(fact_5206_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys2: list @ A] :
? [Us3: list @ A,Z7: A,Z9: A,Vs3: list @ A] :
( ( Xs3
= ( append @ A @ Us3 @ ( cons @ A @ Z7 @ Vs3 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z7 @ Z9 ) @ R4 )
& ( Ys2
= ( append @ A @ Us3 @ ( cons @ A @ Z9 @ Vs3 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_5207_trancl__union__outside,axiom,
! [A: $tType,V2: A,W2: A,E6: set @ ( product_prod @ A @ A ),U3: 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 ) ) @ E6 @ U3 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ E6 ) )
=> ? [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ X3 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E6 @ U3 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ U3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ W2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E6 @ U3 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_5208_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys3 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_5209_trancl__over__edgeE,axiom,
! [A: $tType,U: A,W2: A,V1: A,V22: A,E6: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W2 ) @ ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V1 @ V22 ) @ E6 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W2 ) @ ( transitive_trancl @ A @ E6 ) )
=> ~ ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V1 ) @ ( transitive_rtrancl @ A @ E6 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V22 @ W2 ) @ ( transitive_rtrancl @ A @ E6 ) ) ) ) ) ).
% trancl_over_edgeE
thf(fact_5210_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) )
=> ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_5211_Un__interval,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [B1: A,B22: A,B32: A,F3: A > B] :
( ( ord_less_eq @ A @ B1 @ B22 )
=> ( ( ord_less_eq @ A @ B22 @ B32 )
=> ( ( sup_sup @ ( set @ B )
@ ( collect @ B
@ ^ [Uu2: B] :
? [I4: A] :
( ( Uu2
= ( F3 @ I4 ) )
& ( ord_less_eq @ A @ B1 @ I4 )
& ( ord_less @ A @ I4 @ B22 ) ) )
@ ( collect @ B
@ ^ [Uu2: B] :
? [I4: A] :
( ( Uu2
= ( F3 @ I4 ) )
& ( ord_less_eq @ A @ B22 @ I4 )
& ( ord_less @ A @ I4 @ B32 ) ) ) )
= ( collect @ B
@ ^ [Uu2: B] :
? [I4: A] :
( ( Uu2
= ( F3 @ I4 ) )
& ( ord_less_eq @ A @ B1 @ I4 )
& ( ord_less @ A @ I4 @ B32 ) ) ) ) ) ) ) ).
% Un_interval
thf(fact_5212_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ? [Xys2: list @ A,X4: A,Y4: A,Xs5: list @ A,Ys6: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs5 ) ) )
& ( Ys2
= ( append @ A @ Xys2 @ ( cons @ A @ Y4 @ Ys6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R4 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_5213_Collect__ex__eq,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ( collect @ A
@ ^ [X4: A] :
? [X11: B] : ( P @ X4 @ X11 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] :
( collect @ A
@ ^ [X4: A] : ( P @ X4 @ Y4 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_ex_eq
thf(fact_5214_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
=> ( ( A3 = B2 )
| ( member @ A @ A3 @ A5 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_5215_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R4: set @ ( product_prod @ A @ C ),S6: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Z7: B] :
? [Y4: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y4 ) @ R4 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z7 ) @ S6 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_5216_Restr__rtrancl__mono,axiom,
! [A: $tType,V2: A,W2: A,E6: set @ ( product_prod @ A @ A ),U3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E6
@ ( product_Sigma @ A @ A @ U3
@ ^ [Uu2: A] : U3 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_rtrancl @ A @ E6 ) ) ) ).
% Restr_rtrancl_mono
thf(fact_5217_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu2: product_prod @ A @ B] :
? [A7: A,B6: B] :
( ( Uu2
= ( product_Pair @ A @ B @ A7 @ B6 ) )
& ( ( M3 @ A7 )
= ( some @ B @ B6 ) ) ) ) ) ) ).
% graph_def
thf(fact_5218_pred__nat__trancl__eq__le,axiom,
! [M: nat,N2: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N2 ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
= ( ord_less_eq @ nat @ M @ N2 ) ) ).
% pred_nat_trancl_eq_le
thf(fact_5219_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,E6: set @ ( product_prod @ A @ A ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ E6 ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ A3 ) @ ( F3 @ B2 ) ) @ ( transitive_rtrancl @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( pairself @ A @ B @ F3 ) @ E6 ) ) ) ) ).
% rtrancl_mapI
thf(fact_5220_set__map__filter,axiom,
! [B: $tType,A: $tType,G3: B > ( option @ A ),Xs: list @ B] :
( ( set2 @ A @ ( map_filter @ B @ A @ G3 @ Xs ) )
= ( collect @ A
@ ^ [Y4: A] :
? [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs ) )
& ( ( G3 @ X4 )
= ( some @ A @ Y4 ) ) ) ) ) ).
% set_map_filter
thf(fact_5221_ex__assn__def,axiom,
! [A: $tType] :
( ( ex_assn @ A )
= ( ^ [P4: A > assn] :
( abs_assn
@ ^ [H6: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
? [X4: A] : ( rep_assn @ ( P4 @ X4 ) @ H6 ) ) ) ) ).
% ex_assn_def
thf(fact_5222_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs3: list @ A] :
( collect @ A
@ ^ [Uu2: A] :
? [I4: nat] :
( ( Uu2
= ( nth @ A @ Xs3 @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_5223_rtrancl__last__visit_H,axiom,
! [A: $tType,Q4: A,Q8: A,R2: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q8 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q8 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : S ) ) ) )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Qt ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q8 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit'
thf(fact_5224_rtrancl__last__touch,axiom,
! [A: $tType,Q4: A,Q8: A,R2: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q8 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ A @ Q4 @ S )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Qt ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q8 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_touch
thf(fact_5225_rtrancl__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A3 ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y4 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_5226_rtrancl__is__UN__relpow,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R6 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtrancl_is_UN_relpow
thf(fact_5227_rtrancl__imp__UN__relpow,axiom,
! [A: $tType,P3: product_prod @ A @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P3 @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ P3
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtrancl_imp_UN_relpow
thf(fact_5228_rtrancl__last__visit,axiom,
! [A: $tType,Q4: A,Q8: A,R2: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q8 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Q8 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : S ) ) ) )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ Qt ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q8 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : S ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit
thf(fact_5229_trancl__insert,axiom,
! [A: $tType,Y: A,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R3 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A7: A,B6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B6 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_5230_set__drop__conv,axiom,
! [A: $tType,N2: nat,L: list @ A] :
( ( set2 @ A @ ( drop @ A @ N2 @ L ) )
= ( collect @ A
@ ^ [Uu2: A] :
? [I4: nat] :
( ( Uu2
= ( nth @ A @ L @ I4 ) )
& ( ord_less_eq @ nat @ N2 @ I4 )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ L ) ) ) ) ) ).
% set_drop_conv
thf(fact_5231_trancl__multi__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,M: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ X6
@ ^ [Uu2: A] : ( insert @ A @ M @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X3 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_5232_trancl__multi__insert2,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),M: A,X6: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( insert @ A @ M @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu2: A] : X6 ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ M ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_5233_brk__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( brk_rel @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B )] :
( sup_sup @ ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu2: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X4: A,Y4: B] :
( ( Uu2
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $false @ X4 ) @ ( product_Pair @ $o @ B @ $false @ Y4 ) ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R6 ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu2: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X4: A,Y4: B] :
( Uu2
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $true @ X4 ) @ ( product_Pair @ $o @ B @ $false @ Y4 ) ) ) ) ) ) ) ).
% brk_rel_def
thf(fact_5234_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),N: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N )
& ? [Xys2: list @ A,X4: A,Y4: A,Xs5: list @ A,Ys6: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs5 ) ) )
& ( Ys2
= ( append @ A @ Xys2 @ ( cons @ A @ Y4 @ Ys6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R4 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_5235_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A8: set @ C,F4: C > A,G4: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu2: product_prod @ A @ B] :
? [A7: C] :
( ( Uu2
= ( product_Pair @ A @ B @ ( F4 @ A7 ) @ ( G4 @ A7 ) ) )
& ( member @ C @ A7 @ A8 ) ) ) ) ) ).
% image2_def
thf(fact_5236_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_5237_lexn_Osimps_I1_J,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R3 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_5238_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F3: B > A,X: B,C2: C,G3: B > C,A5: set @ B] :
( ( B2
= ( F3 @ X ) )
=> ( ( C2
= ( G3 @ X ) )
=> ( ( member @ B @ X @ A5 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A5 @ F3 @ G3 ) ) ) ) ) ).
% image2_eqI
thf(fact_5239_lexn__length,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A ),N2: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lexn @ A @ R3 @ N2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= N2 )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 ) ) ) ).
% lexn_length
thf(fact_5240_lex__def,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( complete_Sup_Sup @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ nat @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( lexn @ A @ R4 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% lex_def
thf(fact_5241_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X4: list @ A,Y4: list @ A] :
? [A7: A,V3: list @ A] :
( ( Y4
= ( append @ A @ X4 @ ( cons @ A @ A7 @ V3 ) ) )
| ? [U2: list @ A,B6: A,C5: A,W3: list @ A,Z7: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ C5 ) @ R4 )
& ( X4
= ( append @ A @ U2 @ ( cons @ A @ B6 @ W3 ) ) )
& ( Y4
= ( append @ A @ U2 @ ( cons @ A @ C5 @ Z7 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_5242_list__collect__set__alt,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F4: B > ( set @ A ),L2: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu2: set @ A] :
? [I4: nat] :
( ( Uu2
= ( F4 @ ( nth @ B @ L2 @ I4 ) ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ B ) @ L2 ) ) ) ) ) ) ) ).
% list_collect_set_alt
thf(fact_5243_lexn_Osimps_I2_J,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),N2: nat] :
( ( lexn @ A @ R3 @ ( suc @ N2 ) )
= ( inf_inf @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ ( product_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_map_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) ) @ ( lex_prod @ A @ ( list @ A ) @ R3 @ ( lexn @ A @ R3 @ N2 ) ) )
@ ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N2 ) )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= ( suc @ N2 ) ) ) ) ) ) ) ).
% lexn.simps(2)
thf(fact_5244_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [Y4: B] : Y4 )
= ( ^ [Z7: product_prod @ A @ B] : Z7 ) ) ).
% map_prod_ident
thf(fact_5245_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > B,A3: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ ( product_Pair @ C @ D @ A3 @ B2 ) )
= ( product_Pair @ A @ B @ ( F3 @ A3 ) @ ( G3 @ B2 ) ) ) ).
% map_prod_simp
thf(fact_5246_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ X ) )
= ( F3 @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_5247_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > B,G3: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F3 @ G3 @ X ) )
= ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_5248_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F3: B > ( set @ A ),A3: B] :
( ( list_collect_set @ B @ A @ F3 @ ( cons @ B @ A3 @ ( nil @ B ) ) )
= ( F3 @ A3 ) ) ).
% list_collect_set_simps(2)
thf(fact_5249_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_5250_list__collect__set__simps_I3_J,axiom,
! [A: $tType,B: $tType,F3: B > ( set @ A ),A3: B,L: list @ B] :
( ( list_collect_set @ B @ A @ F3 @ ( cons @ B @ A3 @ L ) )
= ( sup_sup @ ( set @ A ) @ ( F3 @ A3 ) @ ( list_collect_set @ B @ A @ F3 @ L ) ) ) ).
% list_collect_set_simps(3)
thf(fact_5251_list__collect__set__simps_I4_J,axiom,
! [A: $tType,B: $tType,F3: B > ( set @ A ),L: list @ B,L3: list @ B] :
( ( list_collect_set @ B @ A @ F3 @ ( append @ B @ L @ L3 ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F3 @ L ) @ ( list_collect_set @ B @ A @ F3 @ L3 ) ) ) ).
% list_collect_set_simps(4)
thf(fact_5252_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R2: set @ ( product_prod @ A @ B ),F3: A > C,G3: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R2 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A3 ) @ ( G3 @ B2 ) ) @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_5253_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list @ A,B2: A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R3 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
| ( ( A3 = B2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_5254_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R3 ) )
= ( ? [A7: A,X4: list @ A] :
( Y
= ( cons @ A @ A7 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_5255_list__collect__set__map__simps_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > ( set @ A ),X: C > B,A3: C] :
( ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X @ ( cons @ C @ A3 @ ( nil @ C ) ) ) )
= ( F3 @ ( X @ A3 ) ) ) ).
% list_collect_set_map_simps(2)
thf(fact_5256_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > ( set @ A ),X: C > B] :
( ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X @ ( nil @ C ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_map_simps(1)
thf(fact_5257_list__collect__set__map__simps_I3_J,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > ( set @ A ),X: C > B,A3: C,L: list @ C] :
( ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X @ ( cons @ C @ A3 @ L ) ) )
= ( sup_sup @ ( set @ A ) @ ( F3 @ ( X @ A3 ) ) @ ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X @ L ) ) ) ) ).
% list_collect_set_map_simps(3)
thf(fact_5258_list__collect__set__map__simps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > ( set @ A ),X: C > B,L: list @ C,L3: list @ C] :
( ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X @ ( append @ C @ L @ L3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X @ L ) ) @ ( list_collect_set @ B @ A @ F3 @ ( map @ C @ B @ X @ L3 ) ) ) ) ).
% list_collect_set_map_simps(4)
thf(fact_5259_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F3: A > C,G3: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_map_prod
thf(fact_5260_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F3: A > D,G3: B > C] :
( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F3 @ G3 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ G3 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_map_prod
thf(fact_5261_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F: $tType,D: $tType,B: $tType,F3: C > E,G3: D > F,H: A > C,I2: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I2 ) )
= ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H ) @ ( comp @ D @ F @ B @ G3 @ I2 ) ) ) ).
% map_prod.comp
thf(fact_5262_map__prod_Ocompositionality,axiom,
! [D: $tType,F: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F3: C > E,G3: D > F,H: A > C,I2: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 @ ( product_map_prod @ A @ C @ B @ D @ H @ I2 @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H ) @ ( comp @ D @ F @ B @ G3 @ I2 ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_5263_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F: $tType,B: $tType,F1: E > C,F22: A > E,G1: F > D,G22: B > F] :
( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F @ D @ B @ G1 @ G22 ) )
= ( comp @ ( product_prod @ E @ F ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_5264_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F3: D > B,G3: E > C,X: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F3 @ G3 @ X ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L2: D,R4: E] : ( H @ ( F3 @ L2 ) @ ( G3 @ R4 ) )
@ X ) ) ).
% case_prod_map_prod
thf(fact_5265_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F3: C > A,G3: D > B,R2: 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 @ G3 ) @ R2 ) )
=> ~ ! [X3: C,Y3: D] :
( ( C2
= ( product_Pair @ A @ B @ ( F3 @ X3 ) @ ( G3 @ Y3 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_5266_lexord__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_irreflexive
thf(fact_5267_lexord__linear,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R3 )
| ( A6 = B5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
| ( X = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_linear
thf(fact_5268_lexord__Nil__right,axiom,
! [A: $tType,X: list @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R3 ) ) ).
% lexord_Nil_right
thf(fact_5269_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V2: list @ A,R3: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V2 ) ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_append_leftI
thf(fact_5270_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F4: A > C,G4: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X4: A,Y4: B] : ( product_Pair @ C @ D @ ( F4 @ X4 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% map_prod_def
thf(fact_5271_case__prod__o__map__prod,axiom,
! [B: $tType,D: $tType,C: $tType,E: $tType,A: $tType,F3: D > E > C,G1: A > D,G22: B > E] :
( ( comp @ ( product_prod @ D @ E ) @ C @ ( product_prod @ A @ B ) @ ( product_case_prod @ D @ E @ C @ F3 ) @ ( product_map_prod @ A @ D @ B @ E @ G1 @ G22 ) )
= ( product_case_prod @ A @ B @ C
@ ^ [L2: A,R4: B] : ( F3 @ ( G1 @ L2 ) @ ( G22 @ R4 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_5272_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A ),Ys3: list @ A,Zs: list @ A] :
( ! [X3: A,Y3: A,Z3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ R3 ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lexord @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys3 @ Zs ) @ ( lexord @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_5273_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V2: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V2 ) ) @ ( lexord @ A @ R3 ) )
=> ( ! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_append_leftD
thf(fact_5274_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ? [B12: A,Z6: list @ A] :
( Y
= ( cons @ A @ B12 @ Z6 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y ) ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_append_rightI
thf(fact_5275_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs: list @ A,Ys3: list @ A,Qs: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys3 @ Qs ) ) @ ( lexord @ A @ R3 ) )
=> ( ( Xs != Ys3 )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys3 ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lexord @ A @ R3 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_5276_list__collect__set__as__map,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F4: B > ( set @ A ),L2: list @ B] : ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( map @ B @ ( set @ A ) @ F4 @ L2 ) ) ) ) ) ).
% list_collect_set_as_map
thf(fact_5277_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F3: B > A,A5: set @ B,A17: set @ A,G3: D > C,B4: set @ D,B15: set @ C] :
( ( ( image2 @ B @ A @ F3 @ A5 )
= A17 )
=> ( ( ( image2 @ D @ C @ G3 @ B4 )
= B15 )
=> ( ( image2 @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F3 @ G3 )
@ ( product_Sigma @ B @ D @ A5
@ ^ [Uu2: B] : B4 ) )
= ( product_Sigma @ A @ C @ A17
@ ^ [Uu2: A] : B15 ) ) ) ) ).
% map_prod_surj_on
thf(fact_5278_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lex @ A @ R3 ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_5279_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F3: A > B,G3: C > D] :
( ( ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image2 @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_5280_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A3 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_append_left_rightI
thf(fact_5281_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,Zs: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys3 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R3 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R3 ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys3 @ Zs ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_5282_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W2: list @ A,R3: 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 @ R3 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V2 ) @ ( append @ A @ W2 @ Z2 ) ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_sufI
thf(fact_5283_list__collect__set__def,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F4: B > ( set @ A ),L2: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu2: set @ A] :
? [A7: B] :
( ( Uu2
= ( F4 @ A7 ) )
& ( member @ B @ A7 @ ( set2 @ B @ L2 ) ) ) ) ) ) ) ).
% list_collect_set_def
thf(fact_5284_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R4: A > A > $o,Xs3: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys2 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_5285_Gr__incl,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: A > B,B4: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A5 @ F3 )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ B4 ) ) ).
% Gr_incl
thf(fact_5286_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_apsnd @ D @ B @ C @ G3 @ X ) )
= ( product_Pair @ A @ B @ ( F3 @ ( product_fst @ C @ D @ X ) ) @ ( G3 @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_5287_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F3: C > A,X: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_Pair @ C @ B @ X @ Y ) )
= ( product_Pair @ A @ B @ ( F3 @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_5288_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,X: product_prod @ C @ B,G3: C > A] :
( ( ( product_apfst @ C @ A @ B @ F3 @ X )
= ( product_apfst @ C @ A @ B @ G3 @ X ) )
= ( ( F3 @ ( product_fst @ C @ B @ X ) )
= ( G3 @ ( product_fst @ C @ B @ X ) ) ) ) ).
% apfst_eq_conv
thf(fact_5289_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,X: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F3 @ X ) )
= ( F3 @ ( product_fst @ C @ B @ X ) ) ) ).
% fst_apfst
thf(fact_5290_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F3: C > B,X: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F3 @ X ) )
= ( product_snd @ C @ A @ X ) ) ).
% snd_apfst
thf(fact_5291_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_5292_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_5293_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G3: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G3 @ X ) )
= ( product_Pair @ A @ B @ ( G3 @ ( product_fst @ D @ C @ X ) ) @ ( F3 @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_5294_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G3: D > A,P3: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G3 @ P3 ) )
= ( product_apfst @ D @ A @ B @ G3 @ ( product_apsnd @ C @ B @ D @ F3 @ P3 ) ) ) ).
% apsnd_apfst_commute
thf(fact_5295_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > C,X: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_apfst @ D @ C @ B @ G3 @ X ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F3 @ G3 ) @ X ) ) ).
% apfst_compose
thf(fact_5296_GrD2,axiom,
! [A: $tType,B: $tType,X: A,Fx: B,A5: set @ A,F3: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F3 ) )
=> ( ( F3 @ X )
= Fx ) ) ).
% GrD2
thf(fact_5297_GrD1,axiom,
! [B: $tType,A: $tType,X: A,Fx: B,A5: set @ A,F3: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F3 ) )
=> ( member @ A @ X @ A5 ) ) ).
% GrD1
thf(fact_5298_Gr__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr @ A @ B )
= ( ^ [A8: set @ A,F4: A > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu2: product_prod @ A @ B] :
? [A7: A] :
( ( Uu2
= ( product_Pair @ A @ B @ A7 @ ( F4 @ A7 ) ) )
& ( member @ A @ A7 @ A8 ) ) ) ) ) ).
% Gr_def
thf(fact_5299_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q4: product_prod @ A @ B,F3: C > A,P3: product_prod @ C @ B] :
( ( Q4
= ( product_apfst @ C @ A @ B @ F3 @ P3 ) )
=> ~ ! [X3: C,Y3: B] :
( ( P3
= ( product_Pair @ C @ B @ X3 @ Y3 ) )
=> ( Q4
!= ( product_Pair @ A @ B @ ( F3 @ X3 ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_5300_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
= X ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I4 @ X )
= ( take @ A @ I4 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I4 ) @ ( nth @ A @ Y @ I4 ) ) @ R3 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_5301_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P3: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P3 ) )
= ( ? [A7: B] :
( P3
= ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_5302_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_min @ A @ A3 @ A3 )
= A3 ) ) ).
% min.idem
thf(fact_5303_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_min @ A @ A3 @ ( ord_min @ A @ A3 @ B2 ) )
= ( ord_min @ A @ A3 @ B2 ) ) ) ).
% min.left_idem
thf(fact_5304_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
= ( ord_min @ A @ A3 @ B2 ) ) ) ).
% min.right_idem
thf(fact_5305_min__eq__arg_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N2: A] :
( ( ( ord_min @ A @ M @ N2 )
= M )
= ( ord_less_eq @ A @ M @ N2 ) ) ) ).
% min_eq_arg(1)
thf(fact_5306_min__eq__arg_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N2: A] :
( ( ( ord_min @ A @ M @ N2 )
= N2 )
= ( ord_less_eq @ A @ N2 @ M ) ) ) ).
% min_eq_arg(2)
thf(fact_5307_min__arg__le_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N2: A,M: A] :
( ( ord_less_eq @ A @ N2 @ ( ord_min @ A @ M @ N2 ) )
= ( ( ord_min @ A @ M @ N2 )
= N2 ) ) ) ).
% min_arg_le(1)
thf(fact_5308_min__arg__le_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [M: A,N2: A] :
( ( ord_less_eq @ A @ M @ ( ord_min @ A @ M @ N2 ) )
= ( ( ord_min @ A @ M @ N2 )
= M ) ) ) ).
% min_arg_le(2)
thf(fact_5309_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% min.bounded_iff
thf(fact_5310_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_5311_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min.absorb1
thf(fact_5312_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N2: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N2 ) @ M ) )
= ( ( ord_min @ A @ M @ N2 )
= M ) ) ) ).
% min_arg_not_ge(1)
thf(fact_5313_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N2: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N2 ) @ N2 ) )
= ( ( ord_min @ A @ M @ N2 )
= N2 ) ) ) ).
% min_arg_not_ge(2)
thf(fact_5314_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% min_less_self_conv(1)
thf(fact_5315_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% min_less_self_conv(2)
thf(fact_5316_min__simps_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min_simps(1)
thf(fact_5317_min__simps_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min_simps(2)
thf(fact_5318_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min.absorb3
thf(fact_5319_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min.absorb4
thf(fact_5320_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ ( ord_min @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z2 @ X )
& ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_5321_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_5322_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_5323_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% min_top
thf(fact_5324_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% min_top2
thf(fact_5325_min__0R,axiom,
! [N2: nat] :
( ( ord_min @ nat @ N2 @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_5326_min__0L,axiom,
! [N2: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_5327_min__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( suc @ ( ord_min @ nat @ M @ N2 ) ) ) ).
% min_Suc_Suc
thf(fact_5328_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(1)
thf(fact_5329_Int__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atMost @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% Int_atMost
thf(fact_5330_min__Suc__gt_I2_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_min @ nat @ B2 @ ( suc @ A3 ) )
= ( suc @ A3 ) ) ) ).
% min_Suc_gt(2)
thf(fact_5331_min__Suc__gt_I1_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_min @ nat @ ( suc @ A3 ) @ B2 )
= ( suc @ A3 ) ) ) ).
% min_Suc_gt(1)
thf(fact_5332_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_5333_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(3)
thf(fact_5334_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_5335_Int__atLeastAtMostR1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ C2 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostR1
thf(fact_5336_Int__atLeastAtMostL1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atMost @ A @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostL1
thf(fact_5337_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI2
thf(fact_5338_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI1
thf(fact_5339_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A,B6: A] :
( ( A7
= ( ord_min @ A @ A7 @ B6 ) )
& ( A7 != B6 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_5340_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% min.strict_boundedE
thf(fact_5341_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
= ( ( ord_less @ A @ X @ Z2 )
| ( ord_less @ A @ Y @ Z2 ) ) ) ) ).
% min_less_iff_disj
thf(fact_5342_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X @ Y ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_min
thf(fact_5343_nat__mult__min__right,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ M @ ( ord_min @ nat @ N2 @ Q4 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% nat_mult_min_right
thf(fact_5344_nat__mult__min__left,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M @ N2 ) @ Q4 )
= ( ord_min @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N2 @ Q4 ) ) ) ).
% nat_mult_min_left
thf(fact_5345_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% min.assoc
thf(fact_5346_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_min @ A )
= ( ^ [A7: A,B6: A] : ( ord_min @ A @ B6 @ A7 ) ) ) ) ).
% min.commute
thf(fact_5347_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_min @ A @ B2 @ ( ord_min @ A @ A3 @ C2 ) )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% min.left_commute
thf(fact_5348_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A7: A,B6: A] : ( if @ A @ ( ord_less_eq @ A @ A7 @ B6 ) @ A7 @ B6 ) ) ) ) ).
% min_def_raw
thf(fact_5349_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X @ Z2 )
| ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% min_le_iff_disj
thf(fact_5350_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI2
thf(fact_5351_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI1
thf(fact_5352_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A7: A] :
( ( ord_min @ A @ A7 @ B6 )
= B6 ) ) ) ) ).
% min.absorb_iff2
thf(fact_5353_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B6: A] :
( ( ord_min @ A @ A7 @ B6 )
= A7 ) ) ) ) ).
% min.absorb_iff1
thf(fact_5354_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_5355_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 ) ) ).
% min.cobounded1
thf(fact_5356_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B6: A] :
( A7
= ( ord_min @ A @ A7 @ B6 ) ) ) ) ) ).
% min.order_iff
thf(fact_5357_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).
% min.boundedI
thf(fact_5358_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% min.boundedE
thf(fact_5359_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_min @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% min.orderI
thf(fact_5360_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_5361_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ C2 @ D3 ) ) ) ) ) ).
% min.mono
thf(fact_5362_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_min @ A @ X @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_5363_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_min @ A @ X @ Y )
= X ) ) ) ).
% min_absorb1
thf(fact_5364_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A7: A,B6: A] : ( if @ A @ ( ord_less_eq @ A @ A7 @ B6 ) @ A7 @ B6 ) ) ) ) ).
% min_def
thf(fact_5365_min__diff,axiom,
! [M: nat,I2: nat,N2: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I2 ) @ ( minus_minus @ nat @ N2 @ I2 ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M @ N2 ) @ I2 ) ) ).
% min_diff
thf(fact_5366_complete__linorder__inf__min,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% complete_linorder_inf_min
thf(fact_5367_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_5368_inf__min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% inf_min
thf(fact_5369_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,M: A,N2: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_min @ B @ ( F3 @ M ) @ ( F3 @ N2 ) )
= ( F3 @ ( ord_min @ A @ M @ N2 ) ) ) ) ) ).
% min_of_mono
thf(fact_5370_greaterThan__Int__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( set_ord_lessThan @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% greaterThan_Int_greaterThan
thf(fact_5371_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_5372_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( S
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ X @ S ) )
= X ) )
& ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ X @ S ) )
= ( ord_min @ A @ X @ ( complete_Inf_Inf @ A @ S ) ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_5373_min__Suc2,axiom,
! [M: nat,N2: nat] :
( ( ord_min @ nat @ M @ ( suc @ N2 ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M7: nat] : ( suc @ ( ord_min @ nat @ M7 @ N2 ) )
@ M ) ) ).
% min_Suc2
thf(fact_5374_min__Suc1,axiom,
! [N2: nat,M: nat] :
( ( ord_min @ nat @ ( suc @ N2 ) @ M )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M7: nat] : ( suc @ ( ord_min @ nat @ N2 @ M7 ) )
@ M ) ) ).
% min_Suc1
thf(fact_5375_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P3: product_prod @ A @ B] :
( ( A3
= ( product_fst @ A @ B @ P3 ) )
= ( ? [B6: B] :
( P3
= ( product_Pair @ A @ B @ A3 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_5376_min__list_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A] :
( ( min_list @ A @ ( cons @ A @ X @ Xs ) )
= ( case_list @ A @ A @ X
@ ^ [A7: A,List: list @ A] : ( ord_min @ A @ X @ ( min_list @ A @ Xs ) )
@ Xs ) ) ) ).
% min_list.simps
thf(fact_5377_set__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys3: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu2: product_prod @ A @ B] :
? [I4: nat] :
( ( Uu2
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I4 ) @ ( nth @ B @ Ys3 @ I4 ) ) )
& ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys3 ) ) ) ) ) ) ).
% set_zip
thf(fact_5378_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V2: A,E6: set @ ( product_prod @ A @ A ),R2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ E6
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : R2 ) ) ) )
=> ( ~ ( member @ A @ U @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E6 @ R2 ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_5379_zip__eq__zip__same__len,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,A4: list @ A,B3: list @ B] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ( ( zip @ A @ B @ A3 @ B2 )
= ( zip @ A @ B @ A4 @ B3 ) )
= ( ( A3 = A4 )
& ( B2 = B3 ) ) ) ) ) ).
% zip_eq_zip_same_len
thf(fact_5380_rel__restrict__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( rel_restrict @ A @ R2 @ ( bot_bot @ ( set @ A ) ) )
= R2 ) ).
% rel_restrict_empty
thf(fact_5381_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Y: B,Ys3: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys3 ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs @ Ys3 ) ) ) ).
% zip_Cons_Cons
thf(fact_5382_nth__zip,axiom,
! [A: $tType,B: $tType,I2: nat,Xs: list @ A,Ys3: list @ B] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) @ I2 )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I2 ) @ ( nth @ B @ Ys3 @ I2 ) ) ) ) ) ).
% nth_zip
thf(fact_5383_rel__restrict__sub,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R2 @ A5 ) @ R2 ) ).
% rel_restrict_sub
thf(fact_5384_rel__restrict__mono,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A ),R2: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A5 @ R2 ) @ ( rel_restrict @ A @ B4 @ R2 ) ) ) ).
% rel_restrict_mono
thf(fact_5385_finite__rel__restrict,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( rel_restrict @ A @ R2 @ A5 ) ) ) ).
% finite_rel_restrict
thf(fact_5386_zip__inj,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,A4: list @ A,B3: list @ B] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B3 ) )
=> ( ( ( zip @ A @ B @ A3 @ B2 )
= ( zip @ A @ B @ A4 @ B3 ) )
=> ( ( A3 = A4 )
& ( B2 = B3 ) ) ) ) ) ).
% zip_inj
thf(fact_5387_pair__list__split,axiom,
! [A: $tType,B: $tType,L: list @ ( product_prod @ A @ B )] :
~ ! [L12: list @ A,L23: list @ B] :
( ( L
= ( zip @ A @ B @ L12 @ L23 ) )
=> ( ( ( size_size @ ( list @ A ) @ L12 )
= ( size_size @ ( list @ B ) @ L23 ) )
=> ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ L )
!= ( size_size @ ( list @ B ) @ L23 ) ) ) ) ).
% pair_list_split
thf(fact_5388_rel__restrict__lift,axiom,
! [A: $tType,X: A,Y: A,E6: set @ ( product_prod @ A @ A ),R2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E6 @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E6 ) ) ).
% rel_restrict_lift
thf(fact_5389_rel__restrictI,axiom,
! [A: $tType,X: A,R2: set @ A,Y: A,E6: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X @ R2 )
=> ( ~ ( member @ A @ Y @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E6 @ R2 ) ) ) ) ) ).
% rel_restrictI
thf(fact_5390_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X: A,Y: A,A5: set @ ( product_prod @ A @ A ),R2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A5 @ R2 ) )
=> ~ ( member @ A @ X @ R2 ) ) ).
% rel_restrict_notR(1)
thf(fact_5391_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X: A,Y: A,A5: set @ ( product_prod @ A @ A ),R2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A5 @ R2 ) )
=> ~ ( member @ A @ Y @ R2 ) ) ).
% rel_restrict_notR(2)
thf(fact_5392_rel__restrict__union,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( rel_restrict @ A @ R2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( rel_restrict @ A @ ( rel_restrict @ A @ R2 @ A5 ) @ B4 ) ) ).
% rel_restrict_union
thf(fact_5393_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys3: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys3 @ 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,Y4: B,Z7: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y4 @ Z7 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys3 ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_5394_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I2: nat,X: A,Ys3: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs @ I2 @ X ) @ ( list_update @ B @ Ys3 @ I2 @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) @ I2 @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_update
thf(fact_5395_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys3: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys3 @ 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 ) )
@ ^ [Y4: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X4: A,Z7: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y4 @ Z7 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys3 @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_5396_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_5397_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V2: A,W2: A,E6: set @ ( product_prod @ A @ A ),R2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E6 @ R2 ) ) )
=> ~ ( member @ A @ W2 @ R2 ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_5398_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V2: A,W2: A,E6: set @ ( product_prod @ A @ A ),R2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E6 @ R2 ) ) )
=> ~ ( member @ A @ V2 @ R2 ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_5399_rel__restrict__trancl__mem,axiom,
! [A: $tType,A3: A,B2: A,A5: set @ ( product_prod @ A @ A ),R2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A5 @ R2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A5 ) @ R2 ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_5400_rel__restrict__mono2,axiom,
! [A: $tType,R2: set @ A,S: set @ A,A5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A5 @ S ) @ ( rel_restrict @ A @ A5 @ R2 ) ) ) ).
% rel_restrict_mono2
thf(fact_5401_rel__restrict__trancl__sub,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),R2: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A5 @ R2 ) ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A5 ) @ R2 ) ) ).
% rel_restrict_trancl_sub
thf(fact_5402_hd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys3
!= ( nil @ B ) )
=> ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) )
= ( product_Pair @ A @ B @ ( hd @ A @ Xs ) @ ( hd @ B @ Ys3 ) ) ) ) ) ).
% hd_zip
thf(fact_5403_zip__same,axiom,
! [A: $tType,A3: A,B2: A,Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
= ( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
& ( A3 = B2 ) ) ) ).
% zip_same
thf(fact_5404_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys3: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) ) )
=> ~ ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member @ B @ Y @ ( set2 @ B @ Ys3 ) ) ) ) ).
% in_set_zipE
thf(fact_5405_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,Xs: list @ A,Ys3: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_5406_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys3: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) ) )
=> ( member @ B @ Y @ ( set2 @ B @ Ys3 ) ) ) ).
% set_zip_rightD
thf(fact_5407_rel__restrict__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [V3: A,W3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V3 @ W3 ) @ R6 )
& ~ ( member @ A @ V3 @ A8 )
& ~ ( member @ A @ W3 @ A8 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_5408_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B,Xy2: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs @ Ys3 )
= ( cons @ ( product_prod @ A @ B ) @ Xy2 @ Xys ) )
=> ~ ! [X3: A,Xs6: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs6 ) )
=> ! [Y3: B,Ys5: list @ B] :
( ( Ys3
= ( cons @ B @ Y3 @ Ys5 ) )
=> ( ( Xy2
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs6 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_5409_map2__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,H: B > C > A,F3: D > B,Xs: list @ D,G3: D > C] :
( ( map @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ H ) @ ( zip @ B @ C @ ( map @ D @ B @ F3 @ Xs ) @ ( map @ D @ C @ G3 @ Xs ) ) )
= ( map @ D @ A
@ ^ [X4: D] : ( H @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_5410_set__zip__cart,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,L: list @ A,L3: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ L @ L3 ) ) )
=> ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ ( set2 @ A @ L )
@ ^ [Uu2: A] : ( set2 @ B @ L3 ) ) ) ) ).
% set_zip_cart
thf(fact_5411_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs3: list @ A,Ys2: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X4: B,Y4: A] : ( product_Pair @ A @ B @ Y4 @ X4 ) )
@ ( zip @ B @ A @ Ys2 @ Xs3 ) ) ) ) ).
% zip_commute
thf(fact_5412_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys3
!= ( nil @ B ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) )
= ( product_Pair @ A @ B @ ( last @ A @ Xs ) @ ( last @ B @ Ys3 ) ) ) ) ) ) ).
% last_zip
thf(fact_5413_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ Ys3 ) )
=> ~ ! [X3: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_5414_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ! [Y3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_5415_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Y4: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys3 ) @ X )
= ( some @ B @ Y4 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_5416_map__upds__fold__map__upd,axiom,
! [B: $tType,A: $tType] :
( ( map_upds @ A @ B )
= ( ^ [M3: A > ( option @ B ),Ks2: list @ A,Vs3: list @ B] :
( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
@ ^ [N: A > ( option @ B )] :
( product_case_prod @ A @ B @ ( A > ( option @ B ) )
@ ^ [K3: A,V3: B] : ( fun_upd @ A @ ( option @ B ) @ N @ K3 @ ( some @ B @ V3 ) ) )
@ M3
@ ( zip @ A @ B @ Ks2 @ Vs3 ) ) ) ) ).
% map_upds_fold_map_upd
thf(fact_5417_rel__restrict__compl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R2 @ A5 ) @ ( rel_restrict @ A @ R2 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% rel_restrict_compl
thf(fact_5418_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: ( product_prod @ B @ C ) > A,G3: D > B,Xs: list @ D,Ys3: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F3 @ ( zip @ B @ C @ ( map @ D @ B @ G3 @ Xs ) @ Ys3 ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X4: D,Y4: C] : ( F3 @ ( product_Pair @ B @ C @ ( G3 @ X4 ) @ Y4 ) ) )
@ ( zip @ D @ C @ Xs @ Ys3 ) ) ) ).
% map_zip_map
thf(fact_5419_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: ( product_prod @ B @ C ) > A,Xs: list @ B,G3: D > C,Ys3: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F3 @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G3 @ Ys3 ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X4: B,Y4: D] : ( F3 @ ( product_Pair @ B @ C @ X4 @ ( G3 @ Y4 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys3 ) ) ) ).
% map_zip_map2
thf(fact_5420_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys3: list @ A,Xs: list @ B,F3: C > A > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys3 ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldl @ C @ ( product_prod @ B @ A )
@ ^ [B6: C] :
( product_case_prod @ B @ A @ C
@ ^ [X4: B] : ( F3 @ B6 ) )
@ B2
@ ( zip @ B @ A @ Xs @ Ys3 ) )
= ( foldl @ C @ A @ F3 @ B2 @ Ys3 ) ) ) ).
% foldl_snd_zip
thf(fact_5421_homo__rel__restrict__mono,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),B4: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu2: A] : B4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R2 @ A5 )
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ B4 @ A5 )
@ ^ [Uu2: A] : ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% homo_rel_restrict_mono
thf(fact_5422_rel__restrict__alt__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A ),A8: set @ A] :
( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R6
@ ( product_Sigma @ A @ A @ ( uminus_uminus @ ( set @ A ) @ A8 )
@ ^ [Uu2: A] : ( uminus_uminus @ ( set @ A ) @ A8 ) ) ) ) ) ).
% rel_restrict_alt_def
thf(fact_5423_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,Xs: list @ C,Ys3: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F3 @ Xs ) @ Ys3 )
= ( 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 @ Ys3 ) ) ) ).
% zip_map1
thf(fact_5424_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F3: C > B,Ys3: list @ C] :
( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F3 @ Ys3 ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X4: A,Y4: C] : ( product_Pair @ A @ B @ X4 @ ( F3 @ Y4 ) ) )
@ ( zip @ A @ C @ Xs @ Ys3 ) ) ) ).
% zip_map2
thf(fact_5425_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > B,Xs: list @ C,Ys3: list @ D] :
( ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y4: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ Y4 ) ) )
@ ( zip @ C @ D @ Xs @ Ys3 ) )
= ( zip @ A @ B @ ( map @ C @ A @ F3 @ Xs ) @ ( map @ D @ B @ G3 @ Ys3 ) ) ) ).
% map_prod_fun_zip
thf(fact_5426_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys3: list @ B] :
( ( zip @ A @ B @ Xs @ ( cons @ B @ Y @ Ys3 ) )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Z7: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z7 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys3 ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_5427_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys3: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys3 )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y4: B,Ys2: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
@ Ys3 ) ) ).
% zip_Cons1
thf(fact_5428_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_5429_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys3: list @ B,Xs: list @ A,Zs: list @ B,X: A,Y: B,Z2: B] :
( ( ( size_size @ ( list @ B ) @ Ys3 )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( size_size @ ( list @ B ) @ Zs )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys3 ) ) @ X @ ( some @ B @ Y ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) @ X @ ( some @ B @ Z2 ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys3 ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_5430_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: nat > $o,Xs: list @ A,Is: list @ nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P6 ) ) )
@ ( zip @ A @ nat @ Xs @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P6 ) )
@ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_5431_rel__restrict__Sigma__sub,axiom,
! [A: $tType,A5: set @ A,R2: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) )
@ ( rel_restrict @ A
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
@ R2 )
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ A5 @ R2 )
@ ^ [Uu2: A] : ( minus_minus @ ( set @ A ) @ A5 @ R2 ) ) ) ) ).
% rel_restrict_Sigma_sub
thf(fact_5432_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B,I2: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys3 ) @ ( nth @ A @ Xs @ I2 ) )
= ( some @ B @ ( nth @ B @ Ys3 @ I2 ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_5433_in__set__zip,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,Xs: list @ A,Ys3: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) ) )
= ( ? [N: nat] :
( ( ( nth @ A @ Xs @ N )
= ( product_fst @ A @ B @ P3 ) )
& ( ( nth @ B @ Ys3 @ N )
= ( product_snd @ A @ B @ P3 ) )
& ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N @ ( size_size @ ( list @ B ) @ Ys3 ) ) ) ) ) ).
% in_set_zip
thf(fact_5434_nths__shift__lemma,axiom,
! [A: $tType,A5: set @ nat,Xs: list @ A,I2: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A5 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I2 @ ( plus_plus @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I2 ) @ A5 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_5435_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs3: list @ A,A8: set @ nat] :
( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A8 )
@ ( zip @ A @ nat @ Xs3 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ) ).
% nths_def
thf(fact_5436_mset__zip__take__Cons__drop__twice,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B,J: nat,X: A,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( append @ A @ ( take @ A @ J @ Xs ) @ ( cons @ A @ X @ ( drop @ A @ J @ Xs ) ) ) @ ( append @ B @ ( take @ B @ J @ Ys3 ) @ ( cons @ B @ Y @ ( drop @ B @ J @ Ys3 ) ) ) ) )
= ( add_mset @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) ) ) ) ) ) ).
% mset_zip_take_Cons_drop_twice
thf(fact_5437_min__list_Oelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs2 ) )
=> ( Y
!= ( case_list @ A @ A @ X3
@ ^ [A7: A,List: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ).
% min_list.elims
thf(fact_5438_mergesort__by__rel__permutes,axiom,
! [A: $tType,R2: A > A > $o,Xs: list @ A] :
( ( mset @ A @ ( mergesort_by_rel @ A @ R2 @ Xs ) )
= ( mset @ A @ Xs ) ) ).
% mergesort_by_rel_permutes
thf(fact_5439_nths__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_5440_quicksort__by__rel__permutes,axiom,
! [A: $tType,R2: A > A > $o,Sl2: list @ A,Xs: list @ A] :
( ( mset @ A @ ( quicksort_by_rel @ A @ R2 @ Sl2 @ Xs ) )
= ( mset @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% quicksort_by_rel_permutes
thf(fact_5441_mset__mergesort__by__rel__merge,axiom,
! [A: $tType,R2: A > A > $o,Xs: list @ A,Ys3: list @ A] :
( ( mset @ A @ ( merges9089515139780605204_merge @ A @ R2 @ Xs @ Ys3 ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys3 ) ) ) ).
% mset_mergesort_by_rel_merge
thf(fact_5442_Multiset_Omset__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( mset @ A
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) )
= ( add_mset @ A @ X @ ( mset @ A @ Xs ) ) ) ) ).
% Multiset.mset_insort
thf(fact_5443_sorted__list__of__multiset__mset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linord6283353356039996273ltiset @ A @ ( mset @ A @ Xs ) )
= ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs ) ) ) ).
% sorted_list_of_multiset_mset
thf(fact_5444_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_5445_mset__eq__finite,axiom,
! [A: $tType,Xs: list @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys2: list @ A] :
( ( mset @ A @ Ys2 )
= ( mset @ A @ Xs ) ) ) ) ).
% mset_eq_finite
thf(fact_5446_mset__eq__length__filter,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,Z2: A] :
( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys3 ) )
=> ( ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ Z2 )
@ Xs ) )
= ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ Z2 )
@ Ys3 ) ) ) ) ).
% mset_eq_length_filter
thf(fact_5447_set__nths__subset,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( nths @ A @ Xs @ I5 ) ) @ ( set2 @ A @ Xs ) ) ).
% set_nths_subset
thf(fact_5448_nths__all,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ nat @ I3 @ I5 ) )
=> ( ( nths @ A @ Xs @ I5 )
= Xs ) ) ).
% nths_all
thf(fact_5449_sorted__nths,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I5: set @ nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs @ I5 ) ) ) ) ).
% sorted_nths
thf(fact_5450_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X23: A] : X23 ) ) ).
% option.the_def
thf(fact_5451_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N: nat,Xs3: list @ A] : ( nths @ A @ Xs3 @ ( collect @ nat @ ( ord_less_eq @ nat @ N ) ) ) ) ) ).
% drop_eq_nths
thf(fact_5452_hd__def,axiom,
! [A: $tType] :
( ( hd @ A )
= ( case_list @ A @ A @ ( undefined @ A )
@ ^ [X21: A,X222: list @ A] : X21 ) ) ).
% hd_def
thf(fact_5453_properties__for__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Ys3: list @ A,Xs: list @ A] :
( ( ( mset @ A @ Ys3 )
= ( mset @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys3 )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X4: A] : X4
@ Xs )
= Ys3 ) ) ) ) ).
% properties_for_sort
thf(fact_5454_mset__swap,axiom,
! [A: $tType,I2: nat,Ls2: list @ A,J: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ls2 ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Ls2 ) )
=> ( ( mset @ A @ ( list_update @ A @ ( list_update @ A @ Ls2 @ J @ ( nth @ A @ Ls2 @ I2 ) ) @ I2 @ ( nth @ A @ Ls2 @ J ) ) )
= ( mset @ A @ Ls2 ) ) ) ) ).
% mset_swap
thf(fact_5455_nths__append,axiom,
! [A: $tType,L: list @ A,L3: list @ A,A5: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L3 ) @ A5 )
= ( append @ A @ ( nths @ A @ L @ A5 )
@ ( nths @ A @ L3
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( plus_plus @ nat @ J3 @ ( size_size @ ( list @ A ) @ L ) ) @ A5 ) ) ) ) ) ).
% nths_append
thf(fact_5456_filter__in__nths,axiom,
! [A: $tType,Xs: list @ A,S2: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ ( nths @ A @ Xs @ S2 ) ) )
@ Xs )
= ( nths @ A @ Xs @ S2 ) ) ) ).
% filter_in_nths
thf(fact_5457_length__nths,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I5 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I4 @ I5 ) ) ) ) ) ).
% length_nths
thf(fact_5458_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( nths @ A @ Xs3
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P4 @ ( nth @ A @ Xs3 @ I4 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_5459_properties__for__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Ys3: list @ B,Xs: list @ B,F3: B > A] :
( ( ( mset @ B @ Ys3 )
= ( mset @ B @ Xs ) )
=> ( ! [K2: B] :
( ( member @ B @ K2 @ ( set2 @ B @ Ys3 ) )
=> ( ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ K2 )
= ( F3 @ X4 ) )
@ Ys3 )
= ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ K2 )
= ( F3 @ X4 ) )
@ Xs ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Ys3 ) )
=> ( ( linorder_sort_key @ B @ A @ F3 @ Xs )
= Ys3 ) ) ) ) ) ).
% properties_for_sort_key
thf(fact_5460_nths__Cons,axiom,
! [A: $tType,X: A,L: list @ A,A5: set @ nat] :
( ( nths @ A @ ( cons @ A @ X @ L ) @ A5 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A5 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A5 ) ) ) ) ) ).
% nths_Cons
thf(fact_5461_mset__update,axiom,
! [A: $tType,I2: nat,Ls2: list @ A,V2: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ls2 ) )
=> ( ( mset @ A @ ( list_update @ A @ Ls2 @ I2 @ V2 ) )
= ( add_mset @ A @ V2 @ ( minus_minus @ ( multiset @ A ) @ ( mset @ A @ Ls2 ) @ ( add_mset @ A @ ( nth @ A @ Ls2 @ I2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% mset_update
thf(fact_5462_set__nths,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I5 ) )
= ( collect @ A
@ ^ [Uu2: A] :
? [I4: nat] :
( ( Uu2
= ( nth @ A @ Xs @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I4 @ I5 ) ) ) ) ).
% set_nths
thf(fact_5463_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X )
=> ( ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs2 ) )
=> ( ( Y
= ( case_list @ A @ A @ X3
@ ^ [A7: A,List: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X3 @ Xs2 ) ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_5464_mod__h__bot__normalize,axiom,
! [A: $tType,H: heap_ext @ product_unit,P: assn] :
( ( syntax7388354845996824322omatch @ A @ ( heap_ext @ product_unit ) @ ( undefined @ A ) @ H )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ ( undefined @ ( heap_ext @ product_unit ) ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_normalize
thf(fact_5465_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa )
= Y )
=> ( ! [X3: A] :
( ( Xa
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Y != X3 ) )
=> ( ! [X3: A,Y3: A,Zs2: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_5466_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,X: A,Y: A,Zs: list @ A] :
( ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) @ X @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_5467_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa ) )
=> ( ! [X3: A] :
( ( Xa
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ( Y = X3 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X3: A,Y3: A,Zs2: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_5468_mult_Osafe__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [X: A,Y: A,A3: A,B2: A] :
( ( syntax7388354845996824322omatch @ A @ A @ ( times_times @ A @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ B2 )
= ( times_times @ A @ B2 @ A3 ) ) ) ) ).
% mult.safe_commute
thf(fact_5469_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_5470_multiset_Omap__ident,axiom,
! [A: $tType,T6: multiset @ A] :
( ( image_mset @ A @ A
@ ^ [X4: A] : X4
@ T6 )
= T6 ) ).
% multiset.map_ident
thf(fact_5471_mset__map__split__orig,axiom,
! [B: $tType,A: $tType,F3: B > A,P: multiset @ B,M12: multiset @ A,M23: multiset @ A] :
( ( ( image_mset @ B @ A @ F3 @ P )
= ( plus_plus @ ( multiset @ A ) @ M12 @ M23 ) )
=> ~ ! [P12: multiset @ B,P22: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P12 @ P22 ) )
=> ( ( ( image_mset @ B @ A @ F3 @ P12 )
= M12 )
=> ( ( image_mset @ B @ A @ F3 @ P22 )
!= M23 ) ) ) ) ).
% mset_map_split_orig
thf(fact_5472_mset__map__id,axiom,
! [B: $tType,A: $tType,F3: B > A,G3: A > B,X6: multiset @ A] :
( ! [X3: A] :
( ( F3 @ ( G3 @ X3 ) )
= X3 )
=> ( ( image_mset @ B @ A @ F3 @ ( image_mset @ A @ B @ G3 @ X6 ) )
= X6 ) ) ).
% mset_map_id
thf(fact_5473_mset__map__split__orig__le,axiom,
! [B: $tType,A: $tType,F3: B > A,P: multiset @ B,M12: multiset @ A,M23: multiset @ A] :
( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F3 @ P ) @ ( plus_plus @ ( multiset @ A ) @ M12 @ M23 ) )
=> ~ ! [P12: multiset @ B,P22: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P12 @ P22 ) )
=> ( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F3 @ P12 ) @ M12 )
=> ~ ( subseteq_mset @ A @ ( image_mset @ B @ A @ F3 @ P22 ) @ M23 ) ) ) ) ).
% mset_map_split_orig_le
thf(fact_5474_mult_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.right_assoc
thf(fact_5475_mult_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% mult.right_commute
thf(fact_5476_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F3: C > A,G3: C > B,A5: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ A5 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F3 @ A5 )
@ ^ [X4: A] : ( image2 @ C @ B @ G3 @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F3 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_5477_enumerate__replicate__eq,axiom,
! [A: $tType,N2: nat,M: nat,A3: A] :
( ( enumerate @ A @ N2 @ ( replicate @ A @ M @ A3 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q7: nat] : ( product_Pair @ nat @ A @ Q7 @ A3 )
@ ( upt @ N2 @ ( plus_plus @ nat @ N2 @ M ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_5478_subset__eq__mset__impl,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A] :
( ( ( ( subset_eq_mset_impl @ A @ Xs @ Ys3 )
= ( none @ $o ) )
= ( ~ ( subseteq_mset @ A @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys3 ) ) ) )
& ( ( ( subset_eq_mset_impl @ A @ Xs @ Ys3 )
= ( some @ $o @ $true ) )
= ( subset_mset @ A @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys3 ) ) )
& ( ( ( subset_eq_mset_impl @ A @ Xs @ Ys3 )
= ( some @ $o @ $false ) )
=> ( ( mset @ A @ Xs )
= ( mset @ A @ Ys3 ) ) ) ) ).
% subset_eq_mset_impl
thf(fact_5479_vimage__eq,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,B4: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ B4 ) )
= ( member @ B @ ( F3 @ A3 ) @ B4 ) ) ).
% vimage_eq
thf(fact_5480_vimageI,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: B,B2: A,B4: set @ A] :
( ( ( F3 @ A3 )
= B2 )
=> ( ( member @ A @ B2 @ B4 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F3 @ B4 ) ) ) ) ).
% vimageI
thf(fact_5481_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,P: B > $o] :
( ( vimage @ A @ B @ F3 @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y4: A] : ( P @ ( F3 @ Y4 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_5482_vimage__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( vimage @ A @ A
@ ^ [X4: A] : X4
@ Y7 )
= Y7 ) ).
% vimage_ident
thf(fact_5483_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_5484_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_5485_vimage__Int,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ B,B4: set @ B] :
( ( vimage @ A @ B @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A5 ) @ ( vimage @ A @ B @ F3 @ B4 ) ) ) ).
% vimage_Int
thf(fact_5486_vimage__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ B,B4: set @ B] :
( ( vimage @ A @ B @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A5 ) @ ( vimage @ A @ B @ F3 @ B4 ) ) ) ).
% vimage_Un
thf(fact_5487_nth__replicate,axiom,
! [A: $tType,I2: nat,N2: nat,X: A] :
( ( ord_less @ nat @ I2 @ N2 )
=> ( ( nth @ A @ ( replicate @ A @ N2 @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_5488_vimage__const,axiom,
! [B: $tType,A: $tType,C2: B,A5: set @ B] :
( ( ( member @ B @ C2 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : C2
@ A5 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C2 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : C2
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_5489_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ A] :
( ( image2 @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A5 ) )
= ( inf_inf @ ( set @ A ) @ A5 @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_5490_zip__replicate,axiom,
! [A: $tType,B: $tType,I2: nat,X: A,J: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I2 @ X ) @ ( replicate @ B @ J @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I2 @ J ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_5491_vimage__if,axiom,
! [B: $tType,A: $tType,C2: B,A5: set @ B,D3: B,B4: set @ A] :
( ( ( member @ B @ C2 @ A5 )
=> ( ( ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B4 ) @ C2 @ D3 )
@ A5 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B4 ) @ C2 @ D3 )
@ A5 )
= B4 ) ) ) )
& ( ~ ( member @ B @ C2 @ A5 )
=> ( ( ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B4 ) @ C2 @ D3 )
@ A5 )
= ( uminus_uminus @ ( set @ A ) @ B4 ) ) )
& ( ~ ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ B4 ) @ C2 @ D3 )
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_5492_set__replicate,axiom,
! [A: $tType,N2: nat,X: A] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_5493_map__fst__mk__fst,axiom,
! [B: $tType,A: $tType,K: A,L: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_fst_mk_fst
thf(fact_5494_map__snd__mk__snd,axiom,
! [B: $tType,A: $tType,K: A,L: list @ B] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( map @ B @ ( product_prod @ B @ A )
@ ^ [X4: B] : ( product_Pair @ B @ A @ X4 @ K )
@ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_snd_mk_snd
thf(fact_5495_subset__mset_Olexordp_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o ) @ ( ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o )
@ ^ [P6: ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o,X13: list @ ( multiset @ A ),X23: list @ ( multiset @ A )] :
( ? [Y4: multiset @ A,Ys2: list @ ( multiset @ A )] :
( ( X13
= ( nil @ ( multiset @ A ) ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y4 @ Ys2 ) ) )
| ? [X4: multiset @ A,Y4: multiset @ A,Xs3: list @ ( multiset @ A ),Ys2: list @ ( multiset @ A )] :
( ( X13
= ( cons @ ( multiset @ A ) @ X4 @ Xs3 ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y4 @ Ys2 ) )
& ( subset_mset @ A @ X4 @ Y4 ) )
| ? [X4: multiset @ A,Y4: multiset @ A,Xs3: list @ ( multiset @ A ),Ys2: list @ ( multiset @ A )] :
( ( X13
= ( cons @ ( multiset @ A ) @ X4 @ Xs3 ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y4 @ Ys2 ) )
& ~ ( subset_mset @ A @ X4 @ Y4 )
& ~ ( subset_mset @ A @ Y4 @ X4 )
& ( P6 @ Xs3 @ Ys2 ) ) ) ) ).
% subset_mset.lexordp.mono
thf(fact_5496_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,B2: B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ ( insert @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F3 @ A3 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_5497_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ ( vimage @ B @ A @ F3 @ A5 ) ) @ A5 ) ).
% image_vimage_subset
thf(fact_5498_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A5 ) @ B4 )
= ( ord_less_eq @ ( set @ B ) @ A5 @ ( vimage @ B @ A @ F3 @ B4 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_5499_vimage__Compl,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ B] :
( ( vimage @ A @ B @ F3 @ ( uminus_uminus @ ( set @ B ) @ A5 ) )
= ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A5 ) ) ) ).
% vimage_Compl
thf(fact_5500_vimage__Diff,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ B,B4: set @ B] :
( ( vimage @ A @ B @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A5 ) @ ( vimage @ A @ B @ F3 @ B4 ) ) ) ).
% vimage_Diff
thf(fact_5501_vimage__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,F3: B > A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ A5 ) @ ( vimage @ B @ A @ F3 @ B4 ) ) ) ).
% vimage_mono
thf(fact_5502_subset__vimage__iff,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B,B4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( vimage @ A @ B @ F3 @ B4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F3 @ X4 ) @ B4 ) ) ) ) ).
% subset_vimage_iff
thf(fact_5503_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F3: A > B,Q: A > $o] :
( ! [X3: A] :
( ( P @ ( F3 @ X3 ) )
= ( Q @ X3 ) )
=> ( ( vimage @ A @ B @ F3 @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_5504_vimageI2,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: B,A5: set @ A] :
( ( member @ A @ ( F3 @ A3 ) @ A5 )
=> ( member @ B @ A3 @ ( vimage @ B @ A @ F3 @ A5 ) ) ) ).
% vimageI2
thf(fact_5505_vimageE,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,B4: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ B4 ) )
=> ( member @ B @ ( F3 @ A3 ) @ B4 ) ) ).
% vimageE
thf(fact_5506_vimageD,axiom,
! [A: $tType,B: $tType,A3: A,F3: A > B,A5: set @ B] :
( ( member @ A @ A3 @ ( vimage @ A @ B @ F3 @ A5 ) )
=> ( member @ B @ ( F3 @ A3 ) @ A5 ) ) ).
% vimageD
thf(fact_5507_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F4: A > B,B7: set @ B] :
( collect @ A
@ ^ [X4: A] : ( member @ B @ ( F4 @ X4 ) @ B7 ) ) ) ) ).
% vimage_def
thf(fact_5508_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B,G3: A > B,Y: set @ B] :
( ! [W: A] :
( ( member @ A @ W @ S )
=> ( ( F3 @ W )
= ( G3 @ W ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ Y ) @ S )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G3 @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_5509_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N2: nat,X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N2 @ X ) ) ) ).
% sorted_replicate
thf(fact_5510_subset__mset_Olift__Suc__mono__less,axiom,
! [A: $tType,F3: nat > ( multiset @ A ),N2: nat,N6: nat] :
( ! [N5: nat] : ( subset_mset @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
=> ( ( ord_less @ nat @ N2 @ N6 )
=> ( subset_mset @ A @ ( F3 @ N2 ) @ ( F3 @ N6 ) ) ) ) ).
% subset_mset.lift_Suc_mono_less
thf(fact_5511_subset__mset_Olift__Suc__mono__less__iff,axiom,
! [A: $tType,F3: nat > ( multiset @ A ),N2: nat,M: nat] :
( ! [N5: nat] : ( subset_mset @ A @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) )
=> ( ( subset_mset @ A @ ( F3 @ N2 ) @ ( F3 @ M ) )
= ( ord_less @ nat @ N2 @ M ) ) ) ).
% subset_mset.lift_Suc_mono_less_iff
thf(fact_5512_mset__subset__size,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A] :
( ( subset_mset @ A @ A5 @ B4 )
=> ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ A5 ) @ ( size_size @ ( multiset @ A ) @ B4 ) ) ) ).
% mset_subset_size
thf(fact_5513_vimage__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,B4: C > ( set @ B ),A5: set @ C] :
( ( vimage @ A @ B @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( vimage @ A @ B @ F3 @ ( B4 @ X4 ) )
@ A5 ) ) ) ).
% vimage_UN
thf(fact_5514_vimage__INT,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,B4: C > ( set @ B ),A5: set @ C] :
( ( vimage @ A @ B @ F3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( vimage @ A @ B @ F3 @ ( B4 @ X4 ) )
@ A5 ) ) ) ).
% vimage_INT
thf(fact_5515_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X4: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_5516_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > ( product_prod @ B @ C ),A5: set @ B,B4: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F3
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu2: B] : B4 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F3 ) @ A5 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F3 ) @ B4 ) ) ) ).
% vimage_Times
thf(fact_5517_vimage__Union,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ ( set @ B )] :
( ( vimage @ A @ B @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ A5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( vimage @ A @ B @ F3 ) @ A5 ) ) ) ).
% vimage_Union
thf(fact_5518_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X )
@ Xs ) )
@ X )
= ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_5519_subset__mset_Oordering__top__axioms,axiom,
! [A: $tType] :
( ordering_top @ ( multiset @ A )
@ ^ [A8: multiset @ A,B7: multiset @ A] : ( subseteq_mset @ A @ B7 @ A8 )
@ ^ [A8: multiset @ A,B7: multiset @ A] : ( subset_mset @ A @ B7 @ A8 )
@ ( zero_zero @ ( multiset @ A ) ) ) ).
% subset_mset.ordering_top_axioms
thf(fact_5520_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A5: set @ B,F3: B > ( set @ A )] :
( ( ( member @ B @ X @ A5 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A5 @ F3 ) )
= ( F3 @ X ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A5 @ F3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_5521_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ A] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F3 @ A5 )
= ( bot_bot @ ( set @ B ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_5522_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F3: B > A,B4: set @ A,A5: set @ B] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ B4 ) @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ).
% vimage_subsetD
thf(fact_5523_vimage__insert,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: B,B4: set @ B] :
( ( vimage @ A @ B @ F3 @ ( insert @ B @ A3 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F3 @ B4 ) ) ) ).
% vimage_insert
thf(fact_5524_sum__list__replicate,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat,C2: A] :
( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N2 @ C2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ C2 ) ) ) ).
% sum_list_replicate
thf(fact_5525_vimage__fst,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( vimage @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 )
= ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : ( top_top @ ( set @ B ) ) ) ) ).
% vimage_fst
thf(fact_5526_vimage__snd,axiom,
! [B: $tType,A: $tType,A5: set @ B] :
( ( vimage @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A5 )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : A5 ) ) ).
% vimage_snd
thf(fact_5527_size__psubset,axiom,
! [A: $tType,M6: multiset @ A,M9: multiset @ A] :
( ( subseteq_mset @ A @ M6 @ M9 )
=> ( ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ M6 ) @ ( size_size @ ( multiset @ A ) @ M9 ) )
=> ( subset_mset @ A @ M6 @ M9 ) ) ) ).
% size_psubset
thf(fact_5528_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ~ ! [Y3: A] :
( ( member @ A @ Y3 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_5529_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F3 @ A5 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_5530_finite__vimageD_H,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ F3 @ A5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
=> ( finite_finite2 @ B @ A5 ) ) ) ).
% finite_vimageD'
thf(fact_5531_map__replicate__trivial,axiom,
! [A: $tType,X: A,I2: nat] :
( ( map @ nat @ A
@ ^ [I4: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I2 ) )
= ( replicate @ A @ I2 @ X ) ) ).
% map_replicate_trivial
thf(fact_5532_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ F5 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ F5 ) @ A5 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_5533_set__replicate__conv__if,axiom,
! [A: $tType,N2: nat,X: A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N2 @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_5534_set__replicate__Suc,axiom,
! [A: $tType,N2: nat,X: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N2 ) @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_5535_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N2: nat,X: A] :
( ( replicate @ A @ ( suc @ N2 ) @ X )
= ( append @ A @ ( replicate @ A @ N2 @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% replicate_Suc_conv_snoc
thf(fact_5536_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F4: A > B,B7: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] : ( vimage @ A @ B @ F4 @ ( insert @ B @ Y4 @ ( bot_bot @ ( set @ B ) ) ) )
@ B7 ) ) ) ) ).
% vimage_eq_UN
thf(fact_5537_zip__replicate1,axiom,
! [A: $tType,B: $tType,N2: nat,X: A,Ys3: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N2 @ X ) @ Ys3 )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N2 @ Ys3 ) ) ) ).
% zip_replicate1
thf(fact_5538_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( image2 @ B @ A @ F3 @ A5 ) )
& ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_5539_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ~ ! [Y3: A] :
( ( member @ A @ Y3 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_5540_map__zip2,axiom,
! [A: $tType,B: $tType,K: A,L: list @ B] :
( ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L )
= ( zip @ A @ B @ ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) @ L ) ) ).
% map_zip2
thf(fact_5541_map__zip1,axiom,
! [A: $tType,B: $tType,K: B,L: list @ A] :
( ( map @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ K )
@ L )
= ( zip @ A @ B @ L @ ( replicate @ B @ ( size_size @ ( list @ A ) @ L ) @ K ) ) ) ).
% map_zip1
thf(fact_5542_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,N2: nat,Y: B] :
( ( zip @ A @ B @ Xs @ ( replicate @ B @ N2 @ Y ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ Y )
@ ( take @ A @ N2 @ Xs ) ) ) ).
% zip_replicate2
thf(fact_5543_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N2: nat,Y: A] :
( ( ( cons @ A @ X @ Xs )
= ( replicate @ A @ N2 @ Y ) )
= ( ( X = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_5544_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys3
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs @ Ys3 )
= ( append @ A @ Ys3 @ Xs ) )
=> ? [N5: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N5 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N5 @ Zs2 ) )
= ( append @ A @ Xs @ Ys3 ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_5545_subset__mset_Osum__pos,axiom,
! [A: $tType,B: $tType,I5: set @ B,F3: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F3 @ I3 ) ) )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ I5 ) ) ) ) ) ).
% subset_mset.sum_pos
thf(fact_5546_subset__mset_Osum__strict__mono,axiom,
! [A: $tType,B: $tType,A5: set @ B,F3: B > ( multiset @ A ),G3: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( subset_mset @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( subset_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F3 @ A5 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G3 @ A5 ) ) ) ) ) ).
% subset_mset.sum_strict_mono
thf(fact_5547_inv__image__partition,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys3: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Y3 ) )
=> ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( insert @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) )
= ( shuffles @ A @ Xs @ Ys3 ) ) ) ) ).
% inv_image_partition
thf(fact_5548_subset__mset_OgreaterThanLessThan__empty,axiom,
! [A: $tType,L: multiset @ A,K: multiset @ A] :
( ( subseteq_mset @ A @ L @ K )
=> ( ( set_gr287244882034783167ssThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ K @ L )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.greaterThanLessThan_empty
thf(fact_5549_subset__mset_OgreaterThanAtMost__empty__iff2,axiom,
! [A: $tType,K: multiset @ A,L: multiset @ A] :
( ( ( bot_bot @ ( set @ ( multiset @ A ) ) )
= ( set_gr3752724095348155675AtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ K @ L ) )
= ( ~ ( subset_mset @ A @ K @ L ) ) ) ).
% subset_mset.greaterThanAtMost_empty_iff2
thf(fact_5550_subset__mset_OgreaterThanAtMost__empty__iff,axiom,
! [A: $tType,K: multiset @ A,L: multiset @ A] :
( ( ( set_gr3752724095348155675AtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ K @ L )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
= ( ~ ( subset_mset @ A @ K @ L ) ) ) ).
% subset_mset.greaterThanAtMost_empty_iff
thf(fact_5551_subset__mset_OgreaterThanAtMost__empty,axiom,
! [A: $tType,L: multiset @ A,K: multiset @ A] :
( ( subseteq_mset @ A @ L @ K )
=> ( ( set_gr3752724095348155675AtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ K @ L )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.greaterThanAtMost_empty
thf(fact_5552_subset__mset_OgreaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( set_gr3752724095348155675AtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ A3 @ B2 )
= ( minus_minus @ ( set @ ( multiset @ A ) ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 ) @ ( insert @ ( multiset @ A ) @ A3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_5553_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [B: $tType,A: $tType,C: $tType,S: set @ A,F3: A > B > B,G3: C > A,R2: set @ C] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G3 @ ( top_top @ ( set @ C ) ) ) @ S )
=> ( finite673082921795544331dem_on @ C @ B @ R2 @ ( comp @ A @ ( B > B ) @ C @ F3 @ G3 ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_5554_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,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ) ).
% ring_1_class.of_int_def
thf(fact_5555_image__mset_Oidentity,axiom,
! [A: $tType] :
( ( image_mset @ A @ A
@ ^ [X4: A] : X4 )
= ( id @ ( multiset @ A ) ) ) ).
% image_mset.identity
thf(fact_5556_of__nat__eq__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( id @ nat ) ) ).
% of_nat_eq_id
thf(fact_5557_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_5558_id__funpow,axiom,
! [A: $tType,N2: nat] :
( ( compow @ ( A > A ) @ N2 @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_5559_subset__mset_OatLeastatMost__subset__iff,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ C2 @ D3 ) )
= ( ~ ( subseteq_mset @ A @ A3 @ B2 )
| ( ( subseteq_mset @ A @ C2 @ A3 )
& ( subseteq_mset @ A @ B2 @ D3 ) ) ) ) ).
% subset_mset.atLeastatMost_subset_iff
thf(fact_5560_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_5561_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_5562_subset__mset_OatLeastatMost__empty__iff2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( ( bot_bot @ ( set @ ( multiset @ A ) ) )
= ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 ) )
= ( ~ ( subseteq_mset @ A @ A3 @ B2 ) ) ) ).
% subset_mset.atLeastatMost_empty_iff2
thf(fact_5563_subset__mset_OatLeastatMost__empty__iff,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
= ( ~ ( subseteq_mset @ A @ A3 @ B2 ) ) ) ).
% subset_mset.atLeastatMost_empty_iff
thf(fact_5564_subset__mset_OatLeastatMost__empty,axiom,
! [A: $tType,B2: multiset @ A,A3: multiset @ A] :
( ( subset_mset @ A @ B2 @ A3 )
=> ( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.atLeastatMost_empty
thf(fact_5565_comp__the__Some,axiom,
! [A: $tType] :
( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
= ( id @ A ) ) ).
% comp_the_Some
thf(fact_5566_subset__mset_OatLeastAtMost__singleton,axiom,
! [A: $tType,A3: multiset @ A] :
( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ A3 )
= ( insert @ ( multiset @ A ) @ A3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.atLeastAtMost_singleton
thf(fact_5567_subset__mset_OatLeastAtMost__singleton__iff,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 )
= ( insert @ ( multiset @ A ) @ C2 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( ( A3 = B2 )
& ( B2 = C2 ) ) ) ).
% subset_mset.atLeastAtMost_singleton_iff
thf(fact_5568_set_Oidentity,axiom,
! [A: $tType] :
( ( vimage @ A @ A
@ ^ [X4: A] : X4 )
= ( id @ ( set @ A ) ) ) ).
% set.identity
thf(fact_5569_List_Omap_Oidentity,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X4: A] : X4 )
= ( id @ ( list @ A ) ) ) ).
% List.map.identity
thf(fact_5570_nat__def,axiom,
( nat2
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ nat @ nat @ rep_Integ @ ( id @ nat ) @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ) ).
% nat_def
thf(fact_5571_map__option_Oidentity,axiom,
! [A: $tType] :
( ( map_option @ A @ A
@ ^ [X4: A] : X4 )
= ( id @ ( option @ A ) ) ) ).
% map_option.identity
thf(fact_5572_map__fun_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( map_fun @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [X4: B] : X4 )
= ( id @ ( A > B ) ) ) ).
% map_fun.identity
thf(fact_5573_option_Omap__id0,axiom,
! [A: $tType] :
( ( map_option @ A @ A @ ( id @ A ) )
= ( id @ ( option @ A ) ) ) ).
% option.map_id0
thf(fact_5574_option_Omap__id,axiom,
! [A: $tType,T6: option @ A] :
( ( map_option @ A @ A @ ( id @ A ) @ T6 )
= T6 ) ).
% option.map_id
thf(fact_5575_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [X4: B] : X4 )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% map_prod.identity
thf(fact_5576_funpow__simps__right_I1_J,axiom,
! [A: $tType,F3: A > A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F3 )
= ( id @ A ) ) ).
% funpow_simps_right(1)
thf(fact_5577_apfst__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apfst @ A @ C @ B )
= ( ^ [F4: A > C] : ( product_map_prod @ A @ C @ B @ B @ F4 @ ( id @ B ) ) ) ) ).
% apfst_def
thf(fact_5578_apsnd__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apsnd @ B @ C @ A )
= ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% apsnd_def
thf(fact_5579_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,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ) ).
% less_int_def
thf(fact_5580_less__eq__int__def,axiom,
( ( ord_less_eq @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_5581_subset__mset_OatLeastatMost__psubset__iff,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
( ( ord_less @ ( set @ ( multiset @ A ) ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ C2 @ D3 ) )
= ( ( ~ ( subseteq_mset @ A @ A3 @ B2 )
| ( ( subseteq_mset @ A @ C2 @ A3 )
& ( subseteq_mset @ A @ B2 @ D3 )
& ( ( subset_mset @ A @ C2 @ A3 )
| ( subset_mset @ A @ B2 @ D3 ) ) ) )
& ( subseteq_mset @ A @ C2 @ D3 ) ) ) ).
% subset_mset.atLeastatMost_psubset_iff
thf(fact_5582_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_5583_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_5584_subset__mset_OatLeastAtMost__singleton_H,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( A3 = B2 )
=> ( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 )
= ( insert @ ( multiset @ A ) @ A3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.atLeastAtMost_singleton'
thf(fact_5585_comp__fun__idem__on_Ofold__insert__idem,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A5: set @ A,Z2: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z2 @ A5 ) ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem
thf(fact_5586_comp__fun__idem__on_Ofold__insert__idem2,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,X: A,A5: set @ A,Z2: B] :
( ( finite673082921795544331dem_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z2 ) @ A5 ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem2
thf(fact_5587_subset__mset_OatLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( set_atLeastLessThan @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ A3 @ B2 )
= ( minus_minus @ ( set @ ( multiset @ A ) ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 ) @ ( insert @ ( multiset @ A ) @ B2 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5588_finite__def,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P6: ( set @ A ) > $o,X4: set @ A] :
( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,A7: A] :
( ( X4
= ( insert @ A @ A7 @ A8 ) )
& ( P6 @ A8 ) ) ) ) ) ).
% finite_def
thf(fact_5589_revg__fun,axiom,
! [A: $tType] :
( ( revg @ A )
= ( ^ [A7: list @ A] : ( append @ A @ ( rev @ A @ A7 ) ) ) ) ).
% revg_fun
thf(fact_5590_subset__mset_OatLeastLessThan__empty,axiom,
! [A: $tType,B2: multiset @ A,A3: multiset @ A] :
( ( subseteq_mset @ A @ B2 @ A3 )
=> ( ( set_atLeastLessThan @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.atLeastLessThan_empty
thf(fact_5591_subset__mset_OatLeastLessThan__empty__iff,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( ( set_atLeastLessThan @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
= ( ~ ( subset_mset @ A @ A3 @ B2 ) ) ) ).
% subset_mset.atLeastLessThan_empty_iff
thf(fact_5592_subset__mset_OatLeastLessThan__empty__iff2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( ( bot_bot @ ( set @ ( multiset @ A ) ) )
= ( set_atLeastLessThan @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ A3 @ B2 ) )
= ( ~ ( subset_mset @ A @ A3 @ B2 ) ) ) ).
% subset_mset.atLeastLessThan_empty_iff2
thf(fact_5593_less__eq__integer__def,axiom,
( ( ord_less_eq @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > $o ) @ ( code_integer > $o ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ $o @ $o @ code_int_of_integer @ ( id @ $o ) ) @ ( ord_less_eq @ int ) ) ) ).
% less_eq_integer_def
thf(fact_5594_less__integer__def,axiom,
( ( ord_less @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > $o ) @ ( code_integer > $o ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ $o @ $o @ code_int_of_integer @ ( id @ $o ) ) @ ( ord_less @ int ) ) ) ).
% less_integer_def
thf(fact_5595_lfp__const,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [T6: A] :
( ( complete_lattice_lfp @ A
@ ^ [X4: A] : T6 )
= T6 ) ) ).
% lfp_const
thf(fact_5596_lfp__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,A5: A] :
( ! [U6: A] :
( ( ord_less_eq @ A @ ( F3 @ U6 ) @ U6 )
=> ( ord_less_eq @ A @ A5 @ U6 ) )
=> ( ord_less_eq @ A @ A5 @ ( complete_lattice_lfp @ A @ F3 ) ) ) ) ).
% lfp_greatest
thf(fact_5597_lfp__lowerbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,A5: A] :
( ( ord_less_eq @ A @ ( F3 @ A5 ) @ A5 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ A5 ) ) ) ).
% lfp_lowerbound
thf(fact_5598_lfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,G3: A > A] :
( ! [Z10: A] : ( ord_less_eq @ A @ ( F3 @ Z10 ) @ ( G3 @ Z10 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ ( complete_lattice_lfp @ A @ G3 ) ) ) ) ).
% lfp_mono
thf(fact_5599_lfp__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A > A] :
( ! [X3: A,Y3: A,W: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ W @ Z3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 @ W ) @ ( F3 @ Y3 @ Z3 ) ) ) )
=> ( ( complete_lattice_lfp @ A
@ ^ [X4: A] : ( complete_lattice_lfp @ A @ ( F3 @ X4 ) ) )
= ( complete_lattice_lfp @ A
@ ^ [X4: A] : ( F3 @ X4 @ X4 ) ) ) ) ) ).
% lfp_lfp
thf(fact_5600_lfp__rolling,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [G3: A > B,F3: B > A] :
( ( order_mono @ A @ B @ G3 )
=> ( ( order_mono @ B @ A @ F3 )
=> ( ( G3
@ ( complete_lattice_lfp @ A
@ ^ [X4: A] : ( F3 @ ( G3 @ X4 ) ) ) )
= ( complete_lattice_lfp @ B
@ ^ [X4: B] : ( G3 @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% lfp_rolling
thf(fact_5601_revg_Osimps_I2_J,axiom,
! [A: $tType,A3: A,As: list @ A,B2: list @ A] :
( ( revg @ A @ ( cons @ A @ A3 @ As ) @ B2 )
= ( revg @ A @ As @ ( cons @ A @ A3 @ B2 ) ) ) ).
% revg.simps(2)
thf(fact_5602_revg_Osimps_I1_J,axiom,
! [A: $tType,B2: list @ A] :
( ( revg @ A @ ( nil @ A ) @ B2 )
= B2 ) ).
% revg.simps(1)
thf(fact_5603_lfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F5: A > A,X: A] :
( ( order_mono @ A @ A @ F5 )
=> ( ( ( F5 @ X )
= X )
=> ( ! [Z3: A] :
( ( ( F5 @ Z3 )
= Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( complete_lattice_lfp @ A @ F5 )
= X ) ) ) ) ) ).
% lfp_eqI
thf(fact_5604_lfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_lfp @ A )
= ( ^ [F4: A > A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ ( F4 @ U2 ) @ U2 ) ) ) ) ) ) ).
% lfp_def
thf(fact_5605_def__lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: A,F3: A > A,P: A] :
( ( A5
= ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ ( inf_inf @ A @ A5 @ P ) ) @ P )
=> ( ord_less_eq @ A @ A5 @ P ) ) ) ) ) ).
% def_lfp_induct
thf(fact_5606_lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F3 ) @ P ) ) @ P )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ P ) ) ) ) ).
% lfp_induct
thf(fact_5607_lfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F3 )
=> ( ! [S9: A] :
( ( P @ S9 )
=> ( ( ord_less_eq @ A @ S9 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( P @ ( F3 @ S9 ) ) ) )
=> ( ! [M13: set @ A] :
( ! [X7: A] :
( ( member @ A @ X7 @ M13 )
=> ( P @ X7 ) )
=> ( P @ ( complete_Sup_Sup @ A @ M13 ) ) )
=> ( P @ ( complete_lattice_lfp @ A @ F3 ) ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_5608_lfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,N2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( complete_lattice_lfp @ A @ ( compow @ ( A > A ) @ ( suc @ N2 ) @ F3 ) )
= ( complete_lattice_lfp @ A @ F3 ) ) ) ) ).
% lfp_funpow
thf(fact_5609_revg_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [A6: A,As4: list @ A] :
( ( X
= ( cons @ A @ A6 @ As4 ) )
=> ( Y
!= ( revg @ A @ As4 @ ( cons @ A @ A6 @ Xa ) ) ) ) ) ) ).
% revg.elims
thf(fact_5610_subset__mset_OatLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 @ B2 ) @ ( set_atLeastLessThan @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ C2 @ D3 ) )
= ( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( subseteq_mset @ A @ C2 @ A3 )
& ( subset_mset @ A @ B2 @ D3 ) ) ) ) ).
% subset_mset.atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_5611_lfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,K: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F3 @ ( bot_bot @ A ) )
= ( compow @ ( A > A ) @ K @ F3 @ ( bot_bot @ A ) ) )
=> ( ( complete_lattice_lfp @ A @ F3 )
= ( compow @ ( A > A ) @ K @ F3 @ ( bot_bot @ A ) ) ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_5612_revg_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [A6: A,As4: list @ A] :
( ( X
= ( cons @ A @ A6 @ As4 ) )
=> ( ( Y
= ( revg @ A @ As4 @ ( cons @ A @ A6 @ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A6 @ As4 ) @ Xa ) ) ) ) ) ) ) ).
% revg.pelims
thf(fact_5613_subset__mset_OIcc__subset__Iic__iff,axiom,
! [A: $tType,L: multiset @ A,H: multiset @ A,H3: multiset @ A] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L @ H ) @ ( set_atMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ H3 ) )
= ( ~ ( subseteq_mset @ A @ L @ H )
| ( subseteq_mset @ A @ H @ H3 ) ) ) ).
% subset_mset.Icc_subset_Iic_iff
thf(fact_5614_swap__comp__swap,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% swap_comp_swap
thf(fact_5615_swap__swap,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P3 ) )
= P3 ) ).
% swap_swap
thf(fact_5616_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= ( product_Pair @ A @ B @ Y @ X ) ) ).
% swap_simp
thf(fact_5617_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B > A,P3: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y4: B,X4: C] : ( F3 @ X4 @ Y4 )
@ ( product_swap @ C @ B @ P3 ) )
= ( product_case_prod @ C @ B @ A @ F3 @ P3 ) ) ).
% case_swap
thf(fact_5618_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_5619_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_5620_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A5: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A5 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A5 ) ) ).
% pair_in_swap_image
thf(fact_5621_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_5622_ord_OatMost__def,axiom,
! [A: $tType] :
( ( set_atMost @ A )
= ( ^ [Less_eq2: A > A > $o,U2: A] :
( collect @ A
@ ^ [X4: A] : ( Less_eq2 @ X4 @ U2 ) ) ) ) ).
% ord.atMost_def
thf(fact_5623_subset__mset_OatMost__def,axiom,
! [A: $tType,U: multiset @ A] :
( ( set_atMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ U )
= ( collect @ ( multiset @ A )
@ ^ [X4: multiset @ A] : ( subseteq_mset @ A @ X4 @ U ) ) ) ).
% subset_mset.atMost_def
thf(fact_5624_def__lfp__induct__set,axiom,
! [A: $tType,A5: set @ A,F3: ( set @ A ) > ( set @ A ),A3: A,P: A > $o] :
( ( A5
= ( complete_lattice_lfp @ ( set @ A ) @ F3 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( F3 @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ) ) ).
% def_lfp_induct_set
thf(fact_5625_lfp__induct__set,axiom,
! [A: $tType,A3: A,F3: ( set @ A ) > ( set @ A ),P: A > $o] :
( ( member @ A @ A3 @ ( complete_lattice_lfp @ ( set @ A ) @ F3 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( F3 @ ( inf_inf @ ( set @ A ) @ ( complete_lattice_lfp @ ( set @ A ) @ F3 ) @ ( collect @ A @ P ) ) ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ) ).
% lfp_induct_set
thf(fact_5626_subset__mset_Onot__empty__eq__Iic__eq__empty,axiom,
! [A: $tType,H: multiset @ A] :
( ( bot_bot @ ( set @ ( multiset @ A ) ) )
!= ( set_atMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ H ) ) ).
% subset_mset.not_empty_eq_Iic_eq_empty
thf(fact_5627_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: 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 @ A3 @ B2 ) @ ( 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 @ A3 @ B2 ) ) ) ) ).
% lfp_induct2
thf(fact_5628_product__swap,axiom,
! [B: $tType,A: $tType,A5: set @ B,B4: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu2: B] : B4 ) )
= ( product_Sigma @ A @ B @ B4
@ ^ [Uu2: A] : A5 ) ) ).
% product_swap
thf(fact_5629_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P6: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P6 ) @ ( product_fst @ A @ B @ P6 ) ) ) ) ).
% prod.swap_def
thf(fact_5630_ord_OatLeastAtMost__def,axiom,
! [A: $tType] :
( ( set_atLeastAtMost @ A )
= ( ^ [Less_eq2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_atLeast @ A @ Less_eq2 @ L2 ) @ ( set_atMost @ A @ Less_eq2 @ U2 ) ) ) ) ).
% ord.atLeastAtMost_def
thf(fact_5631_ord_OgreaterThanAtMost__def,axiom,
! [A: $tType] :
( ( set_gr3752724095348155675AtMost @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_greaterThan @ A @ Less2 @ L2 ) @ ( set_atMost @ A @ Less_eq2 @ U2 ) ) ) ) ).
% ord.greaterThanAtMost_def
thf(fact_5632_subset__mset_OgreaterThanAtMost__def,axiom,
! [A: $tType,L: multiset @ A,U: multiset @ A] :
( ( set_gr3752724095348155675AtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ L @ U )
= ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_greaterThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ L ) @ ( set_atMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ U ) ) ) ).
% subset_mset.greaterThanAtMost_def
thf(fact_5633_subset__mset_OIcc__subset__Ici__iff,axiom,
! [A: $tType,L: multiset @ A,H: multiset @ A,L3: multiset @ A] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L @ H ) @ ( set_atLeast @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L3 ) )
= ( ~ ( subseteq_mset @ A @ L @ H )
| ( subseteq_mset @ A @ L3 @ L ) ) ) ).
% subset_mset.Icc_subset_Ici_iff
thf(fact_5634_ord_OgreaterThan__def,axiom,
! [A: $tType] :
( ( set_greaterThan @ A )
= ( ^ [Less2: A > A > $o,L2: A] : ( collect @ A @ ( Less2 @ L2 ) ) ) ) ).
% ord.greaterThan_def
thf(fact_5635_ord_OatLeast__def,axiom,
! [A: $tType] :
( ( set_atLeast @ A )
= ( ^ [Less_eq2: A > A > $o,L2: A] : ( collect @ A @ ( Less_eq2 @ L2 ) ) ) ) ).
% ord.atLeast_def
thf(fact_5636_subset__mset_OatLeast__def,axiom,
! [A: $tType,L: multiset @ A] :
( ( set_atLeast @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L )
= ( collect @ ( multiset @ A ) @ ( subseteq_mset @ A @ L ) ) ) ).
% subset_mset.atLeast_def
thf(fact_5637_subset__mset_OgreaterThan__def,axiom,
! [A: $tType,L: multiset @ A] :
( ( set_greaterThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ L )
= ( collect @ ( multiset @ A ) @ ( subset_mset @ A @ L ) ) ) ).
% subset_mset.greaterThan_def
thf(fact_5638_subset__mset_OIoi__le__Ico,axiom,
! [A: $tType,A3: multiset @ A] : ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ ( set_greaterThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ A3 ) @ ( set_atLeast @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A3 ) ) ).
% subset_mset.Ioi_le_Ico
thf(fact_5639_subset__mset_Onot__empty__eq__Ici__eq__empty,axiom,
! [A: $tType,L: multiset @ A] :
( ( bot_bot @ ( set @ ( multiset @ A ) ) )
!= ( set_atLeast @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L ) ) ).
% subset_mset.not_empty_eq_Ici_eq_empty
thf(fact_5640_subset__mset_OatLeastAtMost__def,axiom,
! [A: $tType,L: multiset @ A,U: multiset @ A] :
( ( set_atLeastAtMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L @ U )
= ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_atLeast @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L ) @ ( set_atMost @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ U ) ) ) ).
% subset_mset.atLeastAtMost_def
thf(fact_5641_subset__mset_OgreaterThanLessThan__def,axiom,
! [A: $tType,L: multiset @ A,U: multiset @ A] :
( ( set_gr287244882034783167ssThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ L @ U )
= ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_greaterThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ L ) @ ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ U ) ) ) ).
% subset_mset.greaterThanLessThan_def
thf(fact_5642_subset__mset_OgreaterThanLessThan__eq,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( set_gr287244882034783167ssThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ A3 @ B2 )
= ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_greaterThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ A3 ) @ ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ B2 ) ) ) ).
% subset_mset.greaterThanLessThan_eq
thf(fact_5643_ord_OgreaterThanLessThan__def,axiom,
! [A: $tType] :
( ( set_gr287244882034783167ssThan @ A )
= ( ^ [Less2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_greaterThan @ A @ Less2 @ L2 ) @ ( set_lessThan @ A @ Less2 @ U2 ) ) ) ) ).
% ord.greaterThanLessThan_def
thf(fact_5644_ord_OlessThan__def,axiom,
! [A: $tType] :
( ( set_lessThan @ A )
= ( ^ [Less2: A > A > $o,U2: A] :
( collect @ A
@ ^ [X4: A] : ( Less2 @ X4 @ U2 ) ) ) ) ).
% ord.lessThan_def
thf(fact_5645_subset__mset_OlessThan__def,axiom,
! [A: $tType,U: multiset @ A] :
( ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ U )
= ( collect @ ( multiset @ A )
@ ^ [X4: multiset @ A] : ( subset_mset @ A @ X4 @ U ) ) ) ).
% subset_mset.lessThan_def
thf(fact_5646_subset__mset_OIio__Int__singleton,axiom,
! [A: $tType,X: multiset @ A,K: multiset @ A] :
( ( ( subset_mset @ A @ X @ K )
=> ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ K ) @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) )
& ( ~ ( subset_mset @ A @ X @ K )
=> ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ K ) @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.Iio_Int_singleton
thf(fact_5647_subset__mset_OatLeastLessThan__def,axiom,
! [A: $tType,L: multiset @ A,U: multiset @ A] :
( ( set_atLeastLessThan @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( subset_mset @ A ) @ L @ U )
= ( inf_inf @ ( set @ ( multiset @ A ) ) @ ( set_atLeast @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ L ) @ ( set_lessThan @ ( multiset @ A ) @ ( subset_mset @ A ) @ U ) ) ) ).
% subset_mset.atLeastLessThan_def
thf(fact_5648_ord_OatLeastLessThan__def,axiom,
! [A: $tType] :
( ( set_atLeastLessThan @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_atLeast @ A @ Less_eq2 @ L2 ) @ ( set_lessThan @ A @ Less2 @ U2 ) ) ) ) ).
% ord.atLeastLessThan_def
thf(fact_5649_ord_OgreaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( set_gr287244882034783167ssThan @ A )
= ( ^ [Less2: A > A > $o,A7: A,B6: A] : ( inf_inf @ ( set @ A ) @ ( set_greaterThan @ A @ Less2 @ A7 ) @ ( set_lessThan @ A @ Less2 @ B6 ) ) ) ) ).
% ord.greaterThanLessThan_eq
thf(fact_5650_subset__mset_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp2 @ ( multiset @ A ) @ ( subset_mset @ A ) )
= ( complete_lattice_lfp @ ( ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o )
@ ^ [P6: ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o,X13: list @ ( multiset @ A ),X23: list @ ( multiset @ A )] :
( ? [Y4: multiset @ A,Ys2: list @ ( multiset @ A )] :
( ( X13
= ( nil @ ( multiset @ A ) ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y4 @ Ys2 ) ) )
| ? [X4: multiset @ A,Y4: multiset @ A,Xs3: list @ ( multiset @ A ),Ys2: list @ ( multiset @ A )] :
( ( X13
= ( cons @ ( multiset @ A ) @ X4 @ Xs3 ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y4 @ Ys2 ) )
& ( subset_mset @ A @ X4 @ Y4 ) )
| ? [X4: multiset @ A,Y4: multiset @ A,Xs3: list @ ( multiset @ A ),Ys2: list @ ( multiset @ A )] :
( ( X13
= ( cons @ ( multiset @ A ) @ X4 @ Xs3 ) )
& ( X23
= ( cons @ ( multiset @ A ) @ Y4 @ Ys2 ) )
& ~ ( subset_mset @ A @ X4 @ Y4 )
& ~ ( subset_mset @ A @ Y4 @ X4 )
& ( P6 @ Xs3 @ Ys2 ) ) ) ) ) ).
% subset_mset.lexordp_def
thf(fact_5651_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X23: list @ A] :
( ? [Y4: A,Ys2: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) ) )
| ? [X4: A,Y4: A,Xs3: list @ A,Ys2: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) )
& ( ord_less @ A @ X4 @ Y4 ) )
| ? [X4: A,Y4: A,Xs3: list @ A,Ys2: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) )
& ~ ( ord_less @ A @ X4 @ Y4 )
& ~ ( ord_less @ A @ Y4 @ X4 )
& ( P6 @ Xs3 @ Ys2 ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_5652_map__of__concat,axiom,
! [B: $tType,A: $tType,Xss: list @ ( list @ ( product_prod @ A @ B ) )] :
( ( map_of @ A @ B @ ( concat @ ( product_prod @ A @ B ) @ Xss ) )
= ( foldr @ ( list @ ( product_prod @ A @ B ) ) @ ( A > ( option @ B ) )
@ ^ [Xs3: list @ ( product_prod @ A @ B ),F4: A > ( option @ B )] : ( map_add @ A @ B @ F4 @ ( map_of @ A @ B @ Xs3 ) )
@ Xss
@ ^ [X4: A] : ( none @ B ) ) ) ).
% map_of_concat
thf(fact_5653_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A,Y: A,Ys3: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys3 ) )
= ( ( ord_less @ A @ X @ Y )
| ( ~ ( ord_less @ A @ Y @ X )
& ( ord_lexordp @ A @ Xs @ Ys3 ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_5654_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_5655_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys3: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys3 ) ) ) ) ).
% lexordp.Cons
thf(fact_5656_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys3: list @ A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ~ ( ord_less @ A @ Y @ X )
=> ( ( ord_lexordp @ A @ Xs @ Ys3 )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys3 ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_5657_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_5658_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys3: list @ A,Xs: list @ B,F3: A > C > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys3 ) @ ( 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 @ Ys3 )
@ B2 )
= ( foldr @ A @ C @ F3 @ Ys3 @ B2 ) ) ) ).
% foldr_snd_zip
thf(fact_5659_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_5660_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F4: B > A > A,Xs3: list @ B,A7: A] :
( foldl @ A @ B
@ ^ [X4: A,Y4: B] : ( F4 @ Y4 @ X4 )
@ A7
@ ( rev @ B @ Xs3 ) ) ) ) ).
% foldr_conv_foldl
thf(fact_5661_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F4: A > B > A,A7: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X4: B,Y4: A] : ( F4 @ Y4 @ X4 )
@ ( rev @ B @ Xs3 )
@ A7 ) ) ) ).
% foldl_conv_foldr
thf(fact_5662_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp @ A @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y3: A,Ys4: list @ A] :
( A22
!= ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys4: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys4 ) )
=> ~ ( ord_less @ A @ X3 @ Y3 ) ) )
=> ~ ! [X3: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys4: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X3 )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_5663_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Y4: A,Ys2: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y4 @ Ys2 ) ) )
| ? [X4: A,Y4: A,Xs3: list @ A,Ys2: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y4 @ Ys2 ) )
& ( ord_less @ A @ X4 @ Y4 ) )
| ? [X4: A,Y4: A,Xs3: list @ A,Ys2: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y4 @ Ys2 ) )
& ~ ( ord_less @ A @ X4 @ Y4 )
& ~ ( ord_less @ A @ Y4 @ X4 )
& ( ord_lexordp @ A @ Xs3 @ Ys2 ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_5664_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys3: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys3 )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y3: A,Ys5: list @ A] :
( Ys3
!= ( cons @ A @ Y3 @ Ys5 ) ) )
=> ( ! [X3: A] :
( ? [Xs6: list @ A] :
( Xs
= ( cons @ A @ X3 @ Xs6 ) )
=> ! [Y3: A] :
( ? [Ys5: list @ A] :
( Ys3
= ( cons @ A @ Y3 @ Ys5 ) )
=> ~ ( ord_less @ A @ X3 @ Y3 ) ) )
=> ~ ! [X3: A,Xs6: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs6 ) )
=> ! [Ys5: list @ A] :
( ( Ys3
= ( cons @ A @ X3 @ Ys5 ) )
=> ~ ( ord_lexordp @ A @ Xs6 @ Ys5 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_5665_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys3: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys3 )
=> ( ! [Y3: A,Ys4: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys4 ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: A,Ys4: list @ A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ! [X3: A,Xs2: list @ A,Ys4: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys4 )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ X3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys3 ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_5666_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Us: list @ A,Xs: list @ A,Ys3: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys3 ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_5667_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys2: list @ A] :
( ? [X4: A,Vs3: list @ A] :
( Ys2
= ( append @ A @ Xs3 @ ( cons @ A @ X4 @ Vs3 ) ) )
| ? [Us3: list @ A,A7: A,B6: A,Vs3: list @ A,Ws: list @ A] :
( ( ord_less @ A @ A7 @ B6 )
& ( Xs3
= ( append @ A @ Us3 @ ( cons @ A @ A7 @ Vs3 ) ) )
& ( Ys2
= ( append @ A @ Us3 @ ( cons @ A @ B6 @ Ws ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_5668_comp__fun__commute_Ofoldr__conv__foldl,axiom,
! [B: $tType,A: $tType,F3: A > B > B,Xs: list @ A,A3: B] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( ( foldr @ A @ B @ F3 @ Xs @ A3 )
= ( foldl @ B @ A
@ ^ [A7: B,B6: A] : ( F3 @ B6 @ A7 )
@ A3
@ Xs ) ) ) ).
% comp_fun_commute.foldr_conv_foldl
thf(fact_5669_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys2 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_5670_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A7: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X4: B,B6: A] : ( plus_plus @ A @ ( F4 @ X4 ) @ ( times_times @ A @ A7 @ B6 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_5671_ord_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o] :
( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X13: list @ A,X23: list @ A] :
( ? [Y4: A,Ys2: list @ A] :
( ( X13
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) ) )
| ? [X4: A,Y4: A,Xs3: list @ A,Ys2: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) )
& ( Less2 @ X4 @ Y4 ) )
| ? [X4: A,Y4: A,Xs3: list @ A,Ys2: list @ A] :
( ( X13
= ( cons @ A @ X4 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y4 @ Ys2 ) )
& ~ ( Less2 @ X4 @ Y4 )
& ~ ( Less2 @ Y4 @ X4 )
& ( P6 @ Xs3 @ Ys2 ) ) ) ) ) ) ).
% ord.lexordp_def
thf(fact_5672_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_5673_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X6: set @ A,F3: A > ( set @ B )] :
( ( finite_finite2 @ A @ X6 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [X4: A] : ( principal @ B @ ( F3 @ X4 ) )
@ X6 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ X6 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_5674_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_5675_max__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_max @ nat @ A3 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_5676_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_5677_max__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A3 @ B2 ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_5678_max__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% max_nat.right_neutral
thf(fact_5679_max__0L,axiom,
! [N2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N2 )
= N2 ) ).
% max_0L
thf(fact_5680_max__0R,axiom,
! [N2: nat] :
( ( ord_max @ nat @ N2 @ ( zero_zero @ nat ) )
= N2 ) ).
% max_0R
thf(fact_5681_max__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( suc @ ( ord_max @ nat @ M @ N2 ) ) ) ).
% max_Suc_Suc
thf(fact_5682_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ B2 )
= ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.right_idem
thf(fact_5683_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_max @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) )
= ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.left_idem
thf(fact_5684_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_max @ A @ A3 @ A3 )
= A3 ) ) ).
% max.idem
thf(fact_5685_principal__inject,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( principal @ A @ A5 )
= ( principal @ A @ B4 ) )
= ( A5 = B4 ) ) ).
% principal_inject
thf(fact_5686_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% max.bounded_iff
thf(fact_5687_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_max @ A @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_5688_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_max @ A @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb1
thf(fact_5689_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_max @ A @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb3
thf(fact_5690_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_max @ A @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb4
thf(fact_5691_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y ) @ Z2 )
= ( ( ord_less @ A @ X @ Z2 )
& ( ord_less @ A @ Y @ Z2 ) ) ) ) ).
% max_less_iff_conj
thf(fact_5692_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_5693_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_5694_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_5695_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_5696_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ X @ ( ord_min @ A @ X @ Y ) )
= X ) ) ).
% max_min_same(1)
thf(fact_5697_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ X )
= X ) ) ).
% max_min_same(2)
thf(fact_5698_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ Y )
= Y ) ) ).
% max_min_same(3)
thf(fact_5699_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ( ord_max @ A @ Y @ ( ord_min @ A @ X @ Y ) )
= Y ) ) ).
% max_min_same(4)
thf(fact_5700_sup__principal,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( filter @ A ) @ ( principal @ A @ A5 ) @ ( principal @ A @ B4 ) )
= ( principal @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% sup_principal
thf(fact_5701_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(1)
thf(fact_5702_principal__le__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( principal @ A @ A5 ) @ ( principal @ A @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% principal_le_iff
thf(fact_5703_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_5704_inf__principal,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( inf_inf @ ( filter @ A ) @ ( principal @ A @ A5 ) @ ( principal @ A @ B4 ) )
= ( principal @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% inf_principal
thf(fact_5705_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(2)
thf(fact_5706_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_5707_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_5708_Int__atLeastAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMost
thf(fact_5709_Int__atLeastLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastLessThan
thf(fact_5710_SUP__principal,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B] :
( ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [I4: B] : ( principal @ A @ ( A5 @ I4 ) )
@ I5 ) )
= ( principal @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) ) ) ).
% SUP_principal
thf(fact_5711_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ).
% Max_insert
thf(fact_5712_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,M: A,N2: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_max @ B @ ( F3 @ M ) @ ( F3 @ N2 ) )
= ( F3 @ ( ord_max @ A @ M @ N2 ) ) ) ) ) ).
% max_of_mono
thf(fact_5713_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_5714_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ A3 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_5715_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B6: A,A7: A] :
( ( A7
= ( ord_max @ A @ A7 @ B6 ) )
& ( A7 != B6 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_5716_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% max.strict_boundedE
thf(fact_5717_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z2 @ X )
| ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_5718_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A7: A,B6: A] : ( if @ A @ ( ord_less_eq @ A @ A7 @ B6 ) @ B6 @ A7 ) ) ) ) ).
% max_def_raw
thf(fact_5719_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A7: A,B6: A] : ( if @ A @ ( ord_less_eq @ A @ A7 @ B6 ) @ B6 @ A7 ) ) ) ) ).
% max_def
thf(fact_5720_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_max @ A @ X @ Y )
= X ) ) ) ).
% max_absorb1
thf(fact_5721_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_max @ A @ X @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_5722_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D3 ) @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_5723_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_5724_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_max @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% max.orderI
thf(fact_5725_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% max.boundedE
thf(fact_5726_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% max.boundedI
thf(fact_5727_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A7: A] :
( A7
= ( ord_max @ A @ A7 @ B6 ) ) ) ) ) ).
% max.order_iff
thf(fact_5728_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_5729_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_5730_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less_eq @ A @ Z2 @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less_eq @ A @ Z2 @ X )
| ( ord_less_eq @ A @ Z2 @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_5731_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A7: A] :
( ( ord_max @ A @ A7 @ B6 )
= A7 ) ) ) ) ).
% max.absorb_iff1
thf(fact_5732_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B6: A] :
( ( ord_max @ A @ A7 @ B6 )
= B6 ) ) ) ) ).
% max.absorb_iff2
thf(fact_5733_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_5734_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_5735_Max_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max.in_idem
thf(fact_5736_top__eq__principal__UNIV,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( principal @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% top_eq_principal_UNIV
thf(fact_5737_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_max @ A @ B2 @ ( ord_max @ A @ A3 @ C2 ) )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% max.left_commute
thf(fact_5738_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_max @ A )
= ( ^ [A7: A,B6: A] : ( ord_max @ A @ B6 @ A7 ) ) ) ) ).
% max.commute
thf(fact_5739_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ C2 )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% max.assoc
thf(fact_5740_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_5741_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq2: A > A > $o,A7: A,B6: A] : ( if @ A @ ( Less_eq2 @ A7 @ B6 ) @ B6 @ A7 ) ) ) ).
% ord.max_def
thf(fact_5742_nat__mult__max__left,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N2 ) @ Q4 )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N2 @ Q4 ) ) ) ).
% nat_mult_max_left
thf(fact_5743_nat__mult__max__right,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N2 @ Q4 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N2 ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% nat_mult_max_right
thf(fact_5744_nat__add__max__right,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N2 @ Q4 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( plus_plus @ nat @ M @ Q4 ) ) ) ).
% nat_add_max_right
thf(fact_5745_nat__add__max__left,axiom,
! [M: nat,N2: nat,Q4: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N2 ) @ Q4 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q4 ) @ ( plus_plus @ nat @ N2 @ Q4 ) ) ) ).
% nat_add_max_left
thf(fact_5746_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X @ Y ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_max
thf(fact_5747_complete__linorder__sup__max,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% complete_linorder_sup_max
thf(fact_5748_sup__max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% sup_max
thf(fact_5749_min__max__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_min @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) )
= ( ord_max @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ A3 @ C2 ) ) ) ) ).
% min_max_distrib2
thf(fact_5750_min__max__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_min @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
= ( ord_max @ A @ ( ord_min @ A @ B2 @ A3 ) @ ( ord_min @ A @ C2 @ A3 ) ) ) ) ).
% min_max_distrib1
thf(fact_5751_max__min__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_max @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ord_min @ A @ ( ord_max @ A @ A3 @ B2 ) @ ( ord_max @ A @ A3 @ C2 ) ) ) ) ).
% max_min_distrib2
thf(fact_5752_max__min__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_max @ A @ ( ord_min @ A @ B2 @ C2 ) @ A3 )
= ( ord_min @ A @ ( ord_max @ A @ B2 @ A3 ) @ ( ord_max @ A @ C2 @ A3 ) ) ) ) ).
% max_min_distrib1
thf(fact_5753_nat__minus__add__max,axiom,
! [N2: nat,M: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ M ) @ M )
= ( ord_max @ nat @ N2 @ M ) ) ).
% nat_minus_add_max
thf(fact_5754_lessThan__Int__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( set_ord_greaterThan @ A @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% lessThan_Int_lessThan
thf(fact_5755_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_5756_Misc_Ofoldr__Cons,axiom,
! [A: $tType,Xs: list @ A] :
( ( foldr @ A @ ( list @ A ) @ ( cons @ A ) @ Xs @ ( nil @ A ) )
= Xs ) ).
% Misc.foldr_Cons
thf(fact_5757_bot__eq__principal__empty,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_eq_principal_empty
thf(fact_5758_principal__eq__bot__iff,axiom,
! [A: $tType,X6: set @ A] :
( ( ( principal @ A @ X6 )
= ( bot_bot @ ( filter @ A ) ) )
= ( X6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% principal_eq_bot_iff
thf(fact_5759_max__Suc1,axiom,
! [N2: nat,M: nat] :
( ( ord_max @ nat @ ( suc @ N2 ) @ M )
= ( case_nat @ nat @ ( suc @ N2 )
@ ^ [M7: nat] : ( suc @ ( ord_max @ nat @ N2 @ M7 ) )
@ M ) ) ).
% max_Suc1
thf(fact_5760_max__Suc2,axiom,
! [M: nat,N2: nat] :
( ( ord_max @ nat @ M @ ( suc @ N2 ) )
= ( case_nat @ nat @ ( suc @ N2 )
@ ^ [M7: nat] : ( suc @ ( ord_max @ nat @ M7 @ N2 ) )
@ M ) ) ).
% max_Suc2
thf(fact_5761_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X4: nat,Y4: nat] : ( ord_less_eq @ nat @ Y4 @ X4 )
@ ^ [X4: nat,Y4: nat] : ( ord_less @ nat @ Y4 @ X4 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_5762_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs3: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) )
@ Xss
@ ( 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 )
@ ^ [Ys2: list @ B] :
( Ys2
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_5763_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_5764_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_5765_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_5766_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P3 @ X ) @ ( times_times @ A @ P3 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_5767_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P3 )
= ( ord_max @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P3 )
= ( ord_min @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_5768_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P3 )
= ( ord_min @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P3 )
= ( ord_max @ A @ ( times_times @ A @ X @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_5769_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A,X: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( S
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ S ) )
= X ) )
& ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ S ) )
= ( ord_max @ A @ X @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_5770_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P3 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P3 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_5771_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P3: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P3 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P3 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_5772_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N7: set @ A] :
( ! [X3: A,Y3: A] :
( ( H @ ( ord_max @ A @ X3 @ Y3 ) )
= ( ord_max @ A @ ( H @ X3 ) @ ( H @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798349783984er_Max @ A @ N7 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H @ N7 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_5773_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B4 ) @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ).
% Max.subset
thf(fact_5774_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_5775_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( ord_max @ A @ X3 @ Y3 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Max.closed
thf(fact_5776_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ ( lattic643756798349783984er_Max @ A @ B4 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_5777_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X @ A5 ) ) ) ) ).
% Max.eq_fold
thf(fact_5778_filter__conv__foldr,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( foldr @ A @ ( list @ A )
@ ^ [X4: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P4 @ X4 ) @ ( cons @ A @ X4 @ Xt ) @ Xt )
@ Xs3
@ ( nil @ A ) ) ) ) ).
% filter_conv_foldr
thf(fact_5779_foldr__length__aux,axiom,
! [A: $tType,L: list @ A,A3: nat] :
( ( foldr @ A @ nat
@ ^ [X4: A] : suc
@ L
@ A3 )
= ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldr_length_aux
thf(fact_5780_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_5781_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_5782_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( ord_max @ A @ X4 ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_5783_map__add__map__of__foldr,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),Ps: list @ ( product_prod @ A @ B )] :
( ( map_add @ A @ B @ M @ ( map_of @ A @ B @ Ps ) )
= ( foldr @ ( product_prod @ A @ B ) @ ( A > ( option @ B ) )
@ ( product_case_prod @ A @ B @ ( ( A > ( option @ B ) ) > A > ( option @ B ) )
@ ^ [K3: A,V3: B,M3: A > ( option @ B )] : ( fun_upd @ A @ ( option @ B ) @ M3 @ K3 @ ( some @ B @ V3 ) ) )
@ Ps
@ M ) ) ).
% map_add_map_of_foldr
thf(fact_5784_at__bot__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_bot_def
thf(fact_5785_at__bot__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( set_ord_atMost @ A @ C2 ) ) ) ) ) ).
% at_bot_sub
thf(fact_5786_finite__subsets__at__top__finite,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite5375528669736107172at_top @ A @ A5 )
= ( principal @ ( set @ A ) @ ( insert @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ).
% finite_subsets_at_top_finite
thf(fact_5787_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_5788_finite__subsets__at__top__neq__bot,axiom,
! [A: $tType,A5: set @ A] :
( ( finite5375528669736107172at_top @ A @ A5 )
!= ( bot_bot @ ( filter @ ( set @ A ) ) ) ) ).
% finite_subsets_at_top_neq_bot
thf(fact_5789_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_bot @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_bot_linorder
thf(fact_5790_finite__subsets__at__top__def,axiom,
! [A: $tType] :
( ( finite5375528669736107172at_top @ A )
= ( ^ [A8: set @ A] :
( complete_Inf_Inf @ ( filter @ ( set @ A ) )
@ ( image2 @ ( set @ A ) @ ( filter @ ( set @ A ) )
@ ^ [X11: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y10: set @ A] :
( ( finite_finite2 @ A @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ X11 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A8 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X11: set @ A] :
( ( finite_finite2 @ A @ X11 )
& ( ord_less_eq @ ( set @ A ) @ X11 @ A8 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_5791_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( sup_sup @ A @ X4 ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_5792_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( inf_inf @ A @ X4 ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_5793_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X4: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X4 @ ( ord_min @ A @ X4 ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_5794_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Min_singleton
thf(fact_5795_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Sup_fin.singleton
thf(fact_5796_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Inf_fin.singleton
thf(fact_5797_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ( inf_inf @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= A3 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_5798_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A3 )
= A3 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_5799_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_5800_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_5801_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ B,C2: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [Uu2: B] : C2
@ A5 ) )
= C2 ) ) ) ) ).
% Min_const
thf(fact_5802_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_5803_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_5804_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min_insert
thf(fact_5805_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_5806_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_5807_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A3 ) ) ) ) ).
% Min.coboundedI
thf(fact_5808_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( member @ A @ X @ A5 )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= X ) ) ) ) ) ).
% Min_eqI
thf(fact_5809_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X ) ) ) ) ).
% Min_le
thf(fact_5810_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_5811_Inf__fin__Min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% Inf_fin_Min
thf(fact_5812_Sup__fin__Max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% Sup_fin_Max
thf(fact_5813_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A5 ) ) ) ) ).
% Min_in
thf(fact_5814_Min_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ).
% Min.in_idem
thf(fact_5815_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A3 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_5816_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ord_less_eq @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_5817_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_5818_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_5819_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A5 )
= M )
= ( ( member @ A @ M @ A5 )
& ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ M @ X4 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_5820_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_5821_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ( member @ A @ M @ A5 )
& ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ M @ X4 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_5822_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
=> ! [A15: A] :
( ( member @ A @ A15 @ A5 )
=> ( ord_less_eq @ A @ X @ A15 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_5823_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A5 )
=> ( ord_less_eq @ A @ X @ A6 ) )
=> ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_5824_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_5825_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A5 )
=> ( ord_less_eq @ A @ A3 @ B5 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ A3 @ A5 ) )
= A3 ) ) ) ) ).
% Min_insert2
thf(fact_5826_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ).
% Min_Inf
thf(fact_5827_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= ( lattic643756798350308766er_Min @ A @ X6 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_5828_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_5829_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
=> ! [A15: A] :
( ( member @ A @ A15 @ A5 )
=> ( ord_less_eq @ A @ X @ A15 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_5830_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A5 )
=> ( ord_less_eq @ A @ X @ A6 ) )
=> ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_5831_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X )
=> ! [A15: A] :
( ( member @ A @ A15 @ A5 )
=> ( ord_less_eq @ A @ A15 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_5832_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A5 )
=> ( ord_less_eq @ A @ A6 @ X ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_5833_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X @ X4 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_5834_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_5835_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X6 )
= ( lattic5882676163264333800up_fin @ A @ X6 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_5836_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_5837_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_5838_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X6: set @ A] :
( ( finite_finite2 @ A @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X6 )
= ( lattic7752659483105999362nf_fin @ A @ X6 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_5839_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_5840_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_5841_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M6: set @ A,N7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M6 @ N7 )
=> ( ( M6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N7 ) @ ( lattic643756798350308766er_Min @ A @ M6 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_5842_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B4 ) @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_5843_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N7: set @ A] :
( ! [X3: A,Y3: A] :
( ( H @ ( ord_min @ A @ X3 @ Y3 ) )
= ( ord_min @ A @ ( H @ X3 ) @ ( H @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798350308766er_Min @ A @ N7 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H @ N7 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_5844_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B4 ) @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.subset
thf(fact_5845_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_5846_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( ord_min @ A @ X3 @ Y3 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Min.closed
thf(fact_5847_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F3 @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_5848_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( lattic643756798350308766er_Min @ A @ B4 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_5849_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X @ A5 ) ) ) ) ).
% Min.eq_fold
thf(fact_5850_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S: set @ B,F3: B > A,K: A] :
( ( finite_finite2 @ B @ S )
=> ( ( S
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ K )
@ S ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F3 @ S ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_5851_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B4 ) @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_5852_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B4 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_5853_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H: A > A,N7: set @ A] :
( ! [X3: A,Y3: A] :
( ( H @ ( inf_inf @ A @ X3 @ Y3 ) )
= ( inf_inf @ A @ ( H @ X3 ) @ ( H @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic7752659483105999362nf_fin @ A @ N7 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H @ N7 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_5854_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H: A > A,N7: set @ A] :
( ! [X3: A,Y3: A] :
( ( H @ ( sup_sup @ A @ X3 @ Y3 ) )
= ( sup_sup @ A @ ( H @ X3 ) @ ( H @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic5882676163264333800up_fin @ A @ N7 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H @ N7 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_5855_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B4 ) @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_5856_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B4 ) @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_5857_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_5858_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_5859_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y3: A] : ( member @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_5860_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_5861_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic7752659483105999362nf_fin @ A @ B4 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_5862_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B4 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_5863_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A5 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_5864_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A5 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_5865_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B4 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu2: A] :
? [A7: A,B6: A] :
( ( Uu2
= ( inf_inf @ A @ A7 @ B6 ) )
& ( member @ A @ A7 @ A5 )
& ( member @ A @ B6 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_5866_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu2: A] :
? [A7: A] :
( ( Uu2
= ( inf_inf @ A @ X @ A7 ) )
& ( member @ A @ A7 @ A5 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_5867_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic7752659483105999362nf_fin @ A @ B4 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu2: A] :
? [A7: A,B6: A] :
( ( Uu2
= ( sup_sup @ A @ A7 @ B6 ) )
& ( member @ A @ A7 @ A5 )
& ( member @ A @ B6 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_5868_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu2: A] :
? [A7: A] :
( ( Uu2
= ( sup_sup @ A @ X @ A7 ) )
& ( member @ A @ A7 @ A5 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_5869_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_5870_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_5871_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_5872_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_5873_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_5874_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_5875_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ Xs ) )
& ( P @ X7 ) )
=> ( ( 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_5876_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_5877_card__Min__le__sum,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A5 ) @ ( lattic643756798350308766er_Min @ nat @ ( image2 @ A @ nat @ F3 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) ) ) ).
% card_Min_le_sum
thf(fact_5878_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_5879_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_5880_image__o__collect,axiom,
! [B: $tType,C: $tType,A: $tType,G3: C > B,F5: set @ ( A > ( set @ C ) )] :
( ( bNF_collect @ A @ B @ ( image2 @ ( A > ( set @ C ) ) @ ( A > ( set @ B ) ) @ ( comp @ ( set @ C ) @ ( set @ B ) @ A @ ( image2 @ C @ B @ G3 ) ) @ F5 ) )
= ( comp @ ( set @ C ) @ ( set @ B ) @ A @ ( image2 @ C @ B @ G3 ) @ ( bNF_collect @ A @ C @ F5 ) ) ) ).
% image_o_collect
thf(fact_5881_subset__mset_Omin__arg__le_I2_J,axiom,
! [A: $tType,M: multiset @ A,N2: multiset @ A] :
( ( subseteq_mset @ A @ M @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N2 ) )
= ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N2 )
= M ) ) ).
% subset_mset.min_arg_le(2)
thf(fact_5882_subset__mset_Omin__arg__le_I1_J,axiom,
! [A: $tType,N2: multiset @ A,M: multiset @ A] :
( ( subseteq_mset @ A @ N2 @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N2 ) )
= ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N2 )
= N2 ) ) ).
% subset_mset.min_arg_le(1)
thf(fact_5883_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq2: A > A > $o,A7: A,B6: A] : ( if @ A @ ( Less_eq2 @ A7 @ B6 ) @ A7 @ B6 ) ) ) ).
% ord.min_def
thf(fact_5884_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_5885_linorder_OMax_Ocong,axiom,
! [A: $tType] :
( ( lattices_Max @ A )
= ( lattices_Max @ A ) ) ).
% linorder.Max.cong
thf(fact_5886_collect__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_collect @ B @ A )
= ( ^ [F7: set @ ( B > ( set @ A ) ),X4: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( B > ( set @ A ) ) @ ( set @ A )
@ ^ [F4: B > ( set @ A )] : ( F4 @ X4 )
@ F7 ) ) ) ) ).
% collect_def
thf(fact_5887_collect__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F5: set @ ( C > ( set @ B ) ),G3: A > C] :
( ( comp @ C @ ( set @ B ) @ A @ ( bNF_collect @ C @ B @ F5 ) @ G3 )
= ( bNF_collect @ A @ B
@ ( image2 @ ( C > ( set @ B ) ) @ ( A > ( set @ B ) )
@ ^ [F4: C > ( set @ B )] : ( comp @ C @ ( set @ B ) @ A @ F4 @ G3 )
@ F5 ) ) ) ).
% collect_comp
thf(fact_5888_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,T6: list @ ( product_prod @ A @ C ),K: A,X: C] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T6 @ K )
= ( some @ C @ X ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K3: A] : ( product_Pair @ B @ C @ ( F3 @ K3 ) ) )
@ T6 )
@ ( F3 @ K ) )
= ( some @ C @ X ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_5889_total__on__singleton,axiom,
! [A: $tType,X: A] : ( total_on @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_5890_nth__rotate,axiom,
! [A: $tType,N2: nat,Xs: list @ A,M: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate @ A @ M @ Xs ) @ N2 )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ N2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_5891_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_5892_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_5893_rotate__length01,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N2 @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_5894_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A
@ ^ [B6: A] : ( divide_divide @ A @ B6 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_5895_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ A @ C @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_apfst
thf(fact_5896_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_5897_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > C,A5: set @ A] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : ( top_top @ ( set @ B ) ) ) )
= ( inj_on @ A @ C @ F3 @ A5 ) ) ).
% inj_on_apfst
thf(fact_5898_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: B > C,A5: set @ B] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu2: A] : A5 ) )
= ( inj_on @ B @ C @ F3 @ A5 ) ) ).
% inj_on_apsnd
thf(fact_5899_inj__on__insert,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: A,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert @ A @ A3 @ A5 ) )
= ( ( inj_on @ A @ B @ F3 @ A5 )
& ~ ( member @ B @ ( F3 @ A3 ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_5900_inj__on__Un__image__eq__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( ( ( image2 @ A @ B @ F3 @ A5 )
= ( image2 @ A @ B @ F3 @ B4 ) )
= ( A5 = B4 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_5901_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,B4: set @ A,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ ( image2 @ A @ B @ F3 @ B4 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_5902_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ( ( image2 @ A @ B @ F3 @ A5 )
= ( image2 @ A @ B @ F3 @ B4 ) )
= ( A5 = B4 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_5903_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,B4: set @ A,A3: A,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ B4 )
=> ( ( member @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ B @ ( F3 @ A3 ) @ ( image2 @ A @ B @ F3 @ A5 ) )
= ( member @ A @ A3 @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_5904_subset__image__inj,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: B > A,T2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( image2 @ B @ A @ F3 @ T2 ) )
= ( ? [U7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U7 @ T2 )
& ( inj_on @ B @ A @ F3 @ U7 )
& ( S
= ( image2 @ B @ A @ F3 @ U7 ) ) ) ) ) ).
% subset_image_inj
thf(fact_5905_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_5906_inj__fun,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ ( C > B )
@ ^ [X4: A,Y4: C] : ( F3 @ X4 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fun
thf(fact_5907_option_Oinj__map,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( option @ A ) @ ( option @ B ) @ ( map_option @ A @ B @ F3 ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) ) ).
% option.inj_map
thf(fact_5908_inj__fn,axiom,
! [A: $tType,F3: A > A,N2: nat] :
( ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N2 @ F3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fn
thf(fact_5909_inj__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ).
% inj_of_nat
thf(fact_5910_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F3: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( F3 @ X3 )
!= ( F3 @ Y3 ) ) )
=> ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_5911_inj__on__Int,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B4: set @ A] :
( ( ( inj_on @ A @ B @ F3 @ A5 )
| ( inj_on @ A @ B @ F3 @ B4 ) )
=> ( inj_on @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% inj_on_Int
thf(fact_5912_subset__inj__on,axiom,
! [B: $tType,A: $tType,F3: A > B,B4: set @ A,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( inj_on @ A @ B @ F3 @ A5 ) ) ) ).
% subset_inj_on
thf(fact_5913_inj__on__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( inj_on @ A @ B @ F3 @ B4 ) ) ) ).
% inj_on_subset
thf(fact_5914_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,F3: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( F3 @ X3 )
!= ( F3 @ Y3 ) ) ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) )
=> ( inj_on @ A @ B @ F3 @ A5 ) ) ) ) ).
% linorder_inj_onI
thf(fact_5915_total__on__lex__prod,axiom,
! [A: $tType,B: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),B4: set @ B,S2: set @ ( product_prod @ B @ B )] :
( ( total_on @ A @ A5 @ R3 )
=> ( ( total_on @ B @ B4 @ S2 )
=> ( total_on @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 )
@ ( lex_prod @ A @ B @ R3 @ S2 ) ) ) ) ).
% total_on_lex_prod
thf(fact_5916_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,A5: set @ A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A5 ) ) ) ).
% inj_on_mult
thf(fact_5917_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: B > A,D4: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ B @ A @ F3 @ D4 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ D4 )
& ( member @ A @ ( F3 @ J3 ) @ A5 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_5918_inj__Some,axiom,
! [A: $tType,A5: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) ).
% inj_Some
thf(fact_5919_inj__Suc,axiom,
! [N7: set @ nat] : ( inj_on @ nat @ nat @ suc @ N7 ) ).
% inj_Suc
thf(fact_5920_inj__on__id2,axiom,
! [A: $tType,A5: set @ A] :
( inj_on @ A @ A
@ ^ [X4: A] : X4
@ A5 ) ).
% inj_on_id2
thf(fact_5921_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A5: set @ A] :
( inj_on @ A @ A
@ ^ [B6: A] : ( plus_plus @ A @ B6 @ A3 )
@ A5 ) ) ).
% inj_on_add'
thf(fact_5922_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ A8 )
=> ( ( X4 != Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R4 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_5923_total__onI,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) )
=> ( total_on @ A @ A5 @ R3 ) ) ).
% total_onI
thf(fact_5924_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F3: A > B,X6: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( F3 @ X4 ) )
@ X6 ) ).
% inj_on_convol_ident
thf(fact_5925_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_5926_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_5927_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F3: A > B,A5: set @ A,G3: C > D,B4: set @ C] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( inj_on @ C @ D @ G3 @ B4 )
=> ( inj_on @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu2: A] : B4 ) ) ) ) ).
% map_prod_inj_on
thf(fact_5928_inj__swap,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A5 ) ).
% inj_swap
thf(fact_5929_total__on__empty,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R3 ) ).
% total_on_empty
thf(fact_5930_inj__on__Inter,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ A ),F3: A > B] :
( ( S
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [A13: set @ A] :
( ( member @ ( set @ A ) @ A13 @ S )
=> ( inj_on @ A @ B @ F3 @ A13 ) )
=> ( inj_on @ A @ B @ F3 @ ( complete_Inf_Inf @ ( set @ A ) @ S ) ) ) ) ).
% inj_on_Inter
thf(fact_5931_inj__on__fst__map__to__set,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M ) ) ).
% inj_on_fst_map_to_set
thf(fact_5932_inj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( inj_on @ A @ A
@ ^ [B6: A] : ( minus_minus @ A @ B6 @ A3 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_diff_right
thf(fact_5933_inj__singleton,axiom,
! [A: $tType,A5: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X4: A] : ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) ).
% inj_singleton
thf(fact_5934_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
@ ^ [A7: B] : ( member @ A @ ( F3 @ A7 ) @ S ) ) ) ) ) ).
% finite_Collect
thf(fact_5935_finite__inverse__image,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: B > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] : ( member @ A @ ( F3 @ J3 ) @ A5 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_5936_sum_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,A5: set @ B] :
( ( inj_on @ B @ A @ G3 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : X4
@ ( image2 @ B @ A @ G3 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ).
% sum.image_eq
thf(fact_5937_prod_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A,A5: set @ B] :
( ( inj_on @ B @ A @ G3 @ A5 )
=> ( ( groups7121269368397514597t_prod @ A @ A
@ ^ [X4: A] : X4
@ ( image2 @ B @ A @ G3 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ).
% prod.image_eq
thf(fact_5938_inj__on__diff__nat,axiom,
! [N7: set @ nat,K: nat] :
( ! [N5: nat] :
( ( member @ nat @ N5 @ N7 )
=> ( ord_less_eq @ nat @ K @ N5 ) )
=> ( inj_on @ nat @ nat
@ ^ [N: nat] : ( minus_minus @ nat @ N @ K )
@ N7 ) ) ).
% inj_on_diff_nat
thf(fact_5939_swap__inj__on,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I4: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I4 ) )
@ A5 ) ).
% swap_inj_on
thf(fact_5940_inj__split__Cons,axiom,
! [A: $tType,X6: 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,N: A] : ( cons @ A @ N @ Xs3 ) )
@ X6 ) ).
% inj_split_Cons
thf(fact_5941_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A5: set @ A,A17: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A5 ) @ A17 ) ) )
= ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ A17 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_5942_finite__surj__inj,axiom,
! [A: $tType,A5: set @ A,F3: A > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( image2 @ A @ A @ F3 @ A5 ) )
=> ( inj_on @ A @ A @ F3 @ A5 ) ) ) ).
% finite_surj_inj
thf(fact_5943_inj__on__finite,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B4: set @ B] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ B4 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( finite_finite2 @ A @ A5 ) ) ) ) ).
% inj_on_finite
thf(fact_5944_endo__inj__surj,axiom,
! [A: $tType,A5: set @ A,F3: A > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ A @ A @ F3 @ A5 ) @ A5 )
=> ( ( inj_on @ A @ A @ F3 @ A5 )
=> ( ( image2 @ A @ A @ F3 @ A5 )
= A5 ) ) ) ) ).
% endo_inj_surj
thf(fact_5945_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ ( image2 @ A @ B @ F3 @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% inj_image_subset_iff
thf(fact_5946_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F3: A > B,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ( image2 @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ ( image2 @ A @ B @ F3 @ B4 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_5947_image__Int,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ ( image2 @ A @ B @ F3 @ B4 ) ) ) ) ).
% image_Int
thf(fact_5948_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F3: A > B,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ ( image2 @ A @ B @ F3 @ B4 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_5949_pigeonhole,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( finite_card @ B @ A5 ) )
=> ~ ( inj_on @ B @ A @ F3 @ A5 ) ) ).
% pigeonhole
thf(fact_5950_map__inj__on,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys3: list @ B] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ B @ A @ F3 @ Ys3 ) )
=> ( ( inj_on @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys3 ) ) )
=> ( Xs = Ys3 ) ) ) ).
% map_inj_on
thf(fact_5951_inj__on__map__eq__map,axiom,
! [B: $tType,A: $tType,F3: A > B,Xs: list @ A,Ys3: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys3 ) ) )
=> ( ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ F3 @ Ys3 ) )
= ( Xs = Ys3 ) ) ) ).
% inj_on_map_eq_map
thf(fact_5952_finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ( inj_on @ B @ A @ H @ A5 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ F5 ) @ A5 ) ) ) ) ).
% finite_vimage_IntI
thf(fact_5953_inj__graph,axiom,
! [B: $tType,A: $tType] :
( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [F4: A > B] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y4: B] :
( Y4
= ( F4 @ X4 ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_5954_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),F3: B > C] :
( ! [I3: A,J2: A] :
( ( member @ A @ I3 @ I5 )
=> ( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A5 @ I3 ) @ ( A5 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A5 @ J2 ) @ ( A5 @ I3 ) ) ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( inj_on @ B @ C @ F3 @ ( A5 @ I3 ) ) )
=> ( inj_on @ B @ C @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_5955_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,Y: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert @ A @ Y @ ( set2 @ A @ Xs ) ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
( ( F3 @ Y )
= ( F3 @ X4 ) )
@ Xs )
= ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ Y )
@ Xs ) ) ) ).
% inj_on_filter_key_eq
thf(fact_5956_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,F3: B > C,A5: A > ( set @ B )] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( inj_on @ B @ C @ F3 @ ( A5 @ I3 ) ) )
=> ( inj_on @ B @ C @ F3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_5957_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [N5: nat,F2: nat > A] :
( ( A5
= ( image2 @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N5 ) ) ) )
& ( inj_on @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N5 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_5958_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ~ ! [F2: A > nat] :
( ? [N5: nat] :
( ( image2 @ A @ nat @ F2 @ A5 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N5 ) ) )
=> ~ ( inj_on @ A @ nat @ F2 @ A5 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_5959_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [F2: A > nat,N5: nat] :
( ( ( image2 @ A @ nat @ F2 @ A5 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N5 ) ) )
& ( inj_on @ A @ nat @ F2 @ A5 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_5960_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_5961_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S: set @ A,T2: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ( ( finite_card @ A @ S )
= ( finite_card @ B @ T2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ S ) @ T2 )
=> ( ( ! [X4: B] :
( ( member @ B @ X4 @ T2 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ S )
& ( ( F3 @ Y4 )
= X4 ) ) ) )
= ( inj_on @ A @ B @ F3 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_5962_card__bij__eq,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B4: set @ B,G3: B > A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ B4 )
=> ( ( inj_on @ B @ A @ G3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G3 @ B4 ) @ A5 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( finite_card @ A @ A5 )
= ( finite_card @ B @ B4 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_5963_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F3: A > B,B4: set @ B,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ ( image2 @ A @ B @ F3 @ A5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ B4 ) @ A5 ) ) ) ).
% vimage_subsetI
thf(fact_5964_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_5965_inj__on__nth,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ! [X3: nat] :
( ( member @ nat @ X3 @ I5 )
=> ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_5966_infinite__iff__countable__subset,axiom,
! [A: $tType,S: set @ A] :
( ( ~ ( finite_finite2 @ A @ S ) )
= ( ? [F4: nat > A] :
( ( inj_on @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_5967_infinite__countable__subset,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ? [F2: nat > A] :
( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_5968_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,G3: A > B,B4: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( inj_on @ A @ B @ G3 @ B4 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ ( image2 @ A @ B @ G3 @ B4 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ A5 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_5969_set__to__map__simp,axiom,
! [B: $tType,A: $tType,S: set @ ( product_prod @ A @ B ),K: A,V2: B] :
( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S )
=> ( ( ( set_to_map @ A @ B @ S @ K )
= ( some @ B @ V2 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ S ) ) ) ).
% set_to_map_simp
thf(fact_5970_image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B,C4: set @ A,A5: set @ C,B4: C > ( set @ A ),J: C] :
( ( inj_on @ A @ B @ F3 @ C4 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( B4 @ X3 ) @ C4 ) )
=> ( ( member @ C @ J @ A5 )
=> ( ( image2 @ A @ B @ F3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ A @ B @ F3 @ ( B4 @ X4 ) )
@ A5 ) ) ) ) ) ) ).
% image_INT
thf(fact_5971_inj__on__funpow__least,axiom,
! [A: $tType,N2: nat,F3: A > A,S2: A] :
( ( ( compow @ ( A > A ) @ N2 @ F3 @ S2 )
= S2 )
=> ( ! [M5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( ord_less @ nat @ M5 @ N2 )
=> ( ( compow @ ( A > A ) @ M5 @ F3 @ S2 )
!= S2 ) ) )
=> ( inj_on @ nat @ A
@ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F3 @ S2 )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% inj_on_funpow_least
thf(fact_5972_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A5 ) @ B4 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B4 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_5973_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B4: set @ B] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ B4 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B4 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_5974_card__le__inj,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B4 ) )
=> ? [F2: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A5 ) @ B4 )
& ( inj_on @ A @ B @ F2 @ A5 ) ) ) ) ) ).
% card_le_inj
thf(fact_5975_inj__on__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ( inj_on @ A @ B @ F3 @ A5 )
& ( inj_on @ A @ B @ F3 @ B4 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_5976_card__vimage__inj,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( finite_card @ A @ ( vimage @ A @ B @ F3 @ A5 ) )
= ( finite_card @ B @ A5 ) ) ) ) ).
% card_vimage_inj
thf(fact_5977_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F3: A > B,D4: set @ A,A5: set @ B] :
( ( inj_on @ A @ B @ F3 @ D4 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A5 ) @ D4 ) ) @ ( finite_card @ B @ A5 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_5978_Ex__inj__on__UNION__Sigma,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B] :
? [F2: A > ( product_prod @ B @ A )] :
( ( inj_on @ A @ ( product_prod @ B @ A ) @ F2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ A @ ( product_prod @ B @ A ) @ F2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) ) @ ( product_Sigma @ B @ A @ I5 @ A5 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_5979_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,Ys3: list @ B] :
( ( inj_on @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys3 ) ) )
=> ( ( 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 @ Ys3 ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Ys3 ) )
=> ( ( ( set2 @ B @ Xs )
= ( set2 @ B @ Ys3 ) )
=> ( Xs = Ys3 ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_5980_sort__key__inj__key__eq,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,Ys3: list @ B,F3: B > A] :
( ( ( mset @ B @ Xs )
= ( mset @ B @ Ys3 ) )
=> ( ( inj_on @ B @ A @ F3 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Ys3 ) )
=> ( ( linorder_sort_key @ B @ A @ F3 @ Xs )
= Ys3 ) ) ) ) ) ).
% sort_key_inj_key_eq
thf(fact_5981_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F3: A > B,G3: C > B,A5: set @ A,B4: set @ C] :
( ( inj_on @ ( product_prod @ A @ C ) @ B
@ ( product_case_prod @ A @ C @ B
@ ^ [A7: A,B6: C] : ( times_times @ B @ ( F3 @ A7 ) @ ( G3 @ B6 ) ) )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu2: A] : B4 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G3 @ B4 ) )
= ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
@ ( collect @ B
@ ^ [Uu2: B] :
? [A7: A,B6: C] :
( ( Uu2
= ( times_times @ B @ ( F3 @ A7 ) @ ( G3 @ B6 ) ) )
& ( member @ A @ A7 @ A5 )
& ( member @ C @ B6 @ B4 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_5982_funpow__inj__finite,axiom,
! [A: $tType,P3: A > A,X: A] :
( ( inj_on @ A @ A @ P3 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y4: A] :
? [N: nat] :
( Y4
= ( compow @ ( A > A ) @ N @ P3 @ X ) ) ) )
=> ~ ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( compow @ ( A > A ) @ N5 @ P3 @ X )
!= X ) ) ) ) ).
% funpow_inj_finite
thf(fact_5983_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L: list @ A,A5: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L ) @ A5 )
=> ( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( map @ B @ A @ ( inv_on @ A @ B @ F3 @ A5 ) @ ( map @ A @ B @ F3 @ L ) )
= L ) ) ) ).
% inj_on_map_inv_f
thf(fact_5984_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,A3: B] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ A5 )
@ ( insert @ A
@ ( the @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ( F3 @ X4 )
= A3 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_5985_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F3: A > B,A3: B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert @ A
@ ( the @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= A3 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_5986_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X8: A,Y8: B] :
( ( X = X8 )
& ( Y = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% The_split_eq
thf(fact_5987_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,X: A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( inv_on @ A @ B @ F3 @ A5 @ ( F3 @ X ) )
= X ) ) ) ).
% inv_on_f_f
thf(fact_5988_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y: A,F3: B > A,A5: set @ B] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( F3 @ ( inv_on @ B @ A @ F3 @ A5 @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_5989_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y: A,F3: B > A,A5: set @ B] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( member @ B @ ( inv_on @ B @ A @ F3 @ A5 @ Y ) @ A5 ) ) ).
% inv_on_f_range
thf(fact_5990_The__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_5991_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X11: set @ A] :
( the @ A
@ ^ [X4: A] :
( X11
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_5992_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F13: A > B > T,X4: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F13 @ X4 ) ) ) ) ).
% old.rec_prod_def
thf(fact_5993_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F13: T,F23: nat > T > T,X4: nat] : ( the @ T @ ( rec_set_nat @ T @ F13 @ F23 @ X4 ) ) ) ) ).
% old.rec_nat_def
thf(fact_5994_the__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_5995_the__eq__trivial,axiom,
! [A: $tType,A3: A] :
( ( the @ A
@ ^ [X4: A] : X4 = A3 )
= A3 ) ).
% the_eq_trivial
thf(fact_5996_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_5997_the1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X7: A] :
( ( P @ X7 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X7 ) ) )
=> ( ( P @ A3 )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_5998_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X7: A] :
( ( P @ X7 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X7 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_5999_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P4: $o,X4: A,Y4: A] :
( the @ A
@ ^ [Z7: A] :
( ( P4
=> ( Z7 = X4 ) )
& ( ~ P4
=> ( Z7 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_6000_theI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_6001_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X7: A] :
( ( P @ X7 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X7 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_6002_theI,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_6003_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X4: rat] :
( the @ int
@ ^ [Z7: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z7 ) @ X4 )
& ( ord_less @ rat @ X4 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z7 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_6004_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),X: B,Y: A,Z2: A] :
( ( ( M @ X )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M @ ( dom @ B @ A @ M ) )
=> ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ X @ ( some @ A @ Z2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_6005_restrict__map__self,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M @ ( dom @ A @ B @ M ) )
= M ) ).
% restrict_map_self
thf(fact_6006_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_6007_dom__map__add,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),N2: A > ( option @ B )] :
( ( dom @ A @ B @ ( map_add @ A @ B @ M @ N2 ) )
= ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ N2 ) @ ( dom @ A @ B @ M ) ) ) ).
% dom_map_add
thf(fact_6008_dom__restrict,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),A5: set @ A] :
( ( dom @ A @ B @ ( restrict_map @ A @ B @ M @ A5 ) )
= ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M ) @ A5 ) ) ).
% dom_restrict
thf(fact_6009_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X4: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_6010_map__update__eta__repair_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V2: B,M: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X4: A] : ( if @ ( option @ B ) @ ( X4 = K ) @ ( some @ B @ V2 ) @ ( M @ X4 ) ) )
= ( insert @ A @ K @ ( dom @ A @ B @ M ) ) ) ).
% map_update_eta_repair(1)
thf(fact_6011_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_6012_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_6013_map__add__upd__left,axiom,
! [A: $tType,B: $tType,M: A,E22: A > ( option @ B ),E1: A > ( option @ B ),U1: B] :
( ~ ( member @ A @ M @ ( dom @ A @ B @ E22 ) )
=> ( ( map_add @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ E1 @ M @ ( some @ B @ U1 ) ) @ E22 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ E1 @ E22 ) @ M @ ( some @ B @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_6014_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option @ B,F3: A > ( option @ B ),X: A] :
( ( ( Y
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ Y ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ Y ) )
= ( insert @ A @ X @ ( dom @ A @ B @ F3 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_6015_dom__map__upds,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),Xs: list @ A,Ys3: list @ B] :
( ( dom @ A @ B @ ( map_upds @ A @ B @ M @ Xs @ Ys3 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys3 ) @ Xs ) ) @ ( dom @ A @ B @ M ) ) ) ).
% dom_map_upds
thf(fact_6016_ran__add,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),G3: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F3 ) @ ( dom @ A @ B @ G3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ F3 @ G3 ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ F3 ) @ ( ran @ A @ B @ G3 ) ) ) ) ).
% ran_add
thf(fact_6017_ran__map__add,axiom,
! [B: $tType,A: $tType,M14: A > ( option @ B ),M24: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M14 ) @ ( dom @ A @ B @ M24 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ M14 @ M24 ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ M14 ) @ ( ran @ A @ B @ M24 ) ) ) ) ).
% ran_map_add
thf(fact_6018_le__map__dom__mono,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M: A > ( option @ B ),M8: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M @ M8 )
=> ( ord_less_eq @ ( set @ A ) @ ( dom @ A @ B @ M ) @ ( dom @ A @ B @ M8 ) ) ) ) ).
% le_map_dom_mono
thf(fact_6019_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X4: rat,Y4: rat] :
( ( ord_less @ rat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% less_eq_rat_def
thf(fact_6020_abs__rat__def,axiom,
( ( abs_abs @ rat )
= ( ^ [A7: rat] : ( if @ rat @ ( ord_less @ rat @ A7 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A7 ) @ A7 ) ) ) ).
% abs_rat_def
thf(fact_6021_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A7: rat] :
( if @ rat
@ ( A7
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A7 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_6022_insert__dom,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X: B,Y: A] :
( ( ( F3 @ X )
= ( some @ A @ Y ) )
=> ( ( insert @ B @ X @ ( dom @ B @ A @ F3 ) )
= ( dom @ B @ A @ F3 ) ) ) ).
% insert_dom
thf(fact_6023_domI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A3: B,B2: A] :
( ( ( M @ A3 )
= ( some @ A @ B2 ) )
=> ( member @ B @ A3 @ ( dom @ B @ A @ M ) ) ) ).
% domI
thf(fact_6024_domD,axiom,
! [A: $tType,B: $tType,A3: A,M: A > ( option @ B )] :
( ( member @ A @ A3 @ ( dom @ A @ B @ M ) )
=> ? [B5: B] :
( ( M @ A3 )
= ( some @ B @ B5 ) ) ) ).
% domD
thf(fact_6025_nempty__dom,axiom,
! [B: $tType,A: $tType,E3: A > ( option @ B )] :
( ( E3
!= ( ^ [X4: A] : ( none @ B ) ) )
=> ~ ! [M5: A] :
~ ( member @ A @ M5 @ ( dom @ A @ B @ E3 ) ) ) ).
% nempty_dom
thf(fact_6026_dom__def,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( collect @ A
@ ^ [A7: A] :
( ( M3 @ A7 )
!= ( none @ B ) ) ) ) ) ).
% dom_def
thf(fact_6027_dom__map__option,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > C > B,M: A > ( option @ C )] :
( ( dom @ A @ B
@ ^ [K3: A] : ( map_option @ C @ B @ ( F3 @ K3 ) @ ( M @ K3 ) ) )
= ( dom @ A @ C @ M ) ) ).
% dom_map_option
thf(fact_6028_map__dom__ran__finite,axiom,
! [B: $tType,A: $tType,M6: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M6 ) )
=> ( finite_finite2 @ B @ ( ran @ A @ B @ M6 ) ) ) ).
% map_dom_ran_finite
thf(fact_6029_dom__if,axiom,
! [B: $tType,A: $tType,P: A > $o,F3: A > ( option @ B ),G3: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X4: A] : ( if @ ( option @ B ) @ ( P @ X4 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F3 ) @ ( collect @ A @ P ) )
@ ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ G3 )
@ ( collect @ A
@ ^ [X4: A] :
~ ( P @ X4 ) ) ) ) ) ).
% dom_if
thf(fact_6030_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A5: A > ( option @ B ),B4: A > ( option @ B ),C4: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ A5 ) @ ( dom @ A @ B @ B4 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ A5 @ ( map_add @ A @ B @ B4 @ C4 ) )
= ( map_add @ A @ B @ B4 @ ( map_add @ A @ B @ A5 @ C4 ) ) ) ) ).
% map_add_left_comm
thf(fact_6031_map__add__comm,axiom,
! [B: $tType,A: $tType,M14: A > ( option @ B ),M24: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M14 ) @ ( dom @ A @ B @ M24 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ M14 @ M24 )
= ( map_add @ A @ B @ M24 @ M14 ) ) ) ).
% map_add_comm
thf(fact_6032_restrict__map__eq_I1_J,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A5: set @ B,K: B] :
( ( ( restrict_map @ B @ A @ M @ A5 @ K )
= ( none @ A ) )
= ( ~ ( member @ B @ K @ ( inf_inf @ ( set @ B ) @ ( dom @ B @ A @ M ) @ A5 ) ) ) ) ).
% restrict_map_eq(1)
thf(fact_6033_map__card__eq__iff,axiom,
! [B: $tType,A: $tType,M6: A > ( option @ B ),X: A,Y: A] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M6 ) )
=> ( ( ( finite_card @ A @ ( dom @ A @ B @ M6 ) )
= ( finite_card @ B @ ( ran @ A @ B @ M6 ) ) )
=> ( ( member @ A @ X @ ( dom @ A @ B @ M6 ) )
=> ( ( ( M6 @ X )
= ( M6 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% map_card_eq_iff
thf(fact_6034_finite__map__to__set,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M ) )
= ( finite_finite2 @ A @ ( dom @ A @ B @ M ) ) ) ).
% finite_map_to_set
thf(fact_6035_map__to__set__dom,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M3: A > ( option @ B )] : ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M3 ) ) ) ) ).
% map_to_set_dom
thf(fact_6036_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_6037_finite__set__of__finite__maps,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( finite_finite2 @ ( A > ( option @ B ) )
@ ( collect @ ( A > ( option @ B ) )
@ ^ [M3: A > ( option @ B )] :
( ( ( dom @ A @ B @ M3 )
= A5 )
& ( ord_less_eq @ ( set @ B ) @ ( ran @ A @ B @ M3 ) @ B4 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_6038_card__map__to__set,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( finite_card @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M ) )
= ( finite_card @ A @ ( dom @ A @ B @ M ) ) ) ).
% card_map_to_set
thf(fact_6039_finite__Map__induct,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),P: ( A > ( option @ B ) ) > $o] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M ) )
=> ( ( P
@ ^ [X4: A] : ( none @ B ) )
=> ( ! [K2: A,V4: B,M5: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M5 ) )
=> ( ~ ( member @ A @ K2 @ ( dom @ A @ B @ M5 ) )
=> ( ( P @ M5 )
=> ( P @ ( fun_upd @ A @ ( option @ B ) @ M5 @ K2 @ ( some @ B @ V4 ) ) ) ) ) )
=> ( P @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_6040_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_6041_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M: A > ( option @ B ),M8: A > ( option @ B ),N2: A > ( option @ B ),N6: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M @ M8 )
=> ( ( ord_less_eq @ ( A > ( option @ B ) ) @ N2 @ N6 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M8 ) @ ( dom @ A @ B @ N6 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M @ N2 ) @ ( map_add @ A @ B @ M8 @ N6 ) ) ) ) ) ) ).
% map_add_distinct_le
thf(fact_6042_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( the2 @ B @ ( M3 @ X4 ) ) )
@ ( dom @ A @ B @ M3 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_6043_graph__map__add,axiom,
! [B: $tType,A: $tType,M14: A > ( option @ B ),M24: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M14 ) @ ( dom @ A @ B @ M24 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( graph @ A @ B @ ( map_add @ A @ B @ M14 @ M24 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( graph @ A @ B @ M14 ) @ ( graph @ A @ B @ M24 ) ) ) ) ).
% graph_map_add
thf(fact_6044_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X: A] :
( ( ( dom @ A @ B @ F3 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V3: B] :
( F3
= ( fun_upd @ A @ ( option @ B )
@ ^ [X4: A] : ( none @ B )
@ X
@ ( some @ B @ V3 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_6045_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list @ A,M: A > ( option @ B )] :
( ( ( set2 @ A @ Xs )
= ( dom @ A @ B @ M ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( the2 @ B @ ( M @ K3 ) ) )
@ Xs ) )
= M ) ) ).
% map_of_map_keys
thf(fact_6046_inj__on__map__the,axiom,
! [B: $tType,A: $tType,D4: set @ A,M: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ D4 @ ( dom @ A @ B @ M ) )
=> ( ( inj_on @ A @ ( option @ B ) @ M @ D4 )
=> ( inj_on @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ M ) @ D4 ) ) ) ).
% inj_on_map_the
thf(fact_6047_dom__override__on,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),G3: A > ( option @ B ),A5: set @ A] :
( ( dom @ A @ B @ ( override_on @ A @ ( option @ B ) @ F3 @ G3 @ A5 ) )
= ( sup_sup @ ( set @ A )
@ ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F3 )
@ ( collect @ A
@ ^ [A7: A] : ( member @ A @ A7 @ ( minus_minus @ ( set @ A ) @ A5 @ ( dom @ A @ B @ G3 ) ) ) ) )
@ ( collect @ A
@ ^ [A7: A] : ( member @ A @ A7 @ ( inf_inf @ ( set @ A ) @ A5 @ ( dom @ A @ B @ G3 ) ) ) ) ) ) ).
% dom_override_on
thf(fact_6048_normalize__negative,axiom,
! [Q4: int,P3: int] :
( ( ord_less @ int @ Q4 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P3 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P3 ) @ ( uminus_uminus @ int @ Q4 ) ) ) ) ) ).
% normalize_negative
thf(fact_6049_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ( override_on @ A @ B @ F3 @ G3 @ ( bot_bot @ ( set @ A ) ) )
= F3 ) ).
% override_on_emptyset
thf(fact_6050_normalize__denom__zero,axiom,
! [P3: int] :
( ( normalize @ ( product_Pair @ int @ int @ P3 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_6051_normalize__denom__pos,axiom,
! [R3: product_prod @ int @ int,P3: int,Q4: int] :
( ( ( normalize @ R3 )
= ( product_Pair @ int @ int @ P3 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% normalize_denom_pos
thf(fact_6052_normalize__crossproduct,axiom,
! [Q4: int,S2: int,P3: int,R3: int] :
( ( Q4
!= ( zero_zero @ int ) )
=> ( ( S2
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P3 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ R3 @ S2 ) ) )
=> ( ( times_times @ int @ P3 @ S2 )
= ( times_times @ int @ R3 @ Q4 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_6053_normalize__def,axiom,
( normalize
= ( ^ [P6: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P6 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P6 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_6054_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_6055_gcd__pos__int,axiom,
! [M: int,N2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ M @ N2 ) )
= ( ( M
!= ( zero_zero @ int ) )
| ( N2
!= ( zero_zero @ int ) ) ) ) ).
% gcd_pos_int
thf(fact_6056_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_Gcd @ A @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% Gcd_2
thf(fact_6057_quotient__of__number_I3_J,axiom,
! [K: num] :
( ( quotient_of @ ( numeral_numeral @ rat @ K ) )
= ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(3)
thf(fact_6058_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_6059_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_6060_quotient__of__number_I5_J,axiom,
! [K: num] :
( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(5)
thf(fact_6061_gcd__ge__0__int,axiom,
! [X: int,Y: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X @ Y ) ) ).
% gcd_ge_0_int
thf(fact_6062_gcd__dvd__prod,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ).
% gcd_dvd_prod
thf(fact_6063_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,K: A,N2: A] :
( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K @ M ) @ N2 ) )
= ( gcd_gcd @ A @ M @ N2 ) ) ) ).
% gcd_add_mult
thf(fact_6064_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_6065_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_6066_quotient__of__div,axiom,
! [R3: rat,N2: int,D3: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair @ int @ int @ N2 @ D3 ) )
=> ( R3
= ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N2 ) @ ( ring_1_of_int @ rat @ D3 ) ) ) ) ).
% quotient_of_div
thf(fact_6067_gcd__le1__int,axiom,
! [A3: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A3 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B2 ) @ A3 ) ) ).
% gcd_le1_int
thf(fact_6068_gcd__le2__int,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B2 ) @ B2 ) ) ).
% gcd_le2_int
thf(fact_6069_gcd__cases__int,axiom,
! [X: int,Y: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( P @ ( gcd_gcd @ int @ X @ Y ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_6070_gcd__unique__int,axiom,
! [D3: int,A3: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D3 )
& ( dvd_dvd @ int @ D3 @ A3 )
& ( dvd_dvd @ int @ D3 @ B2 )
& ! [E5: int] :
( ( ( dvd_dvd @ int @ E5 @ A3 )
& ( dvd_dvd @ int @ E5 @ B2 ) )
=> ( dvd_dvd @ int @ E5 @ D3 ) ) )
= ( D3
= ( gcd_gcd @ int @ A3 @ B2 ) ) ) ).
% gcd_unique_int
thf(fact_6071_gcd__non__0__int,axiom,
! [Y: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Y )
=> ( ( gcd_gcd @ int @ X @ Y )
= ( gcd_gcd @ int @ Y @ ( modulo_modulo @ int @ X @ Y ) ) ) ) ).
% gcd_non_0_int
thf(fact_6072_quotient__of__denom__pos,axiom,
! [R3: rat,P3: int,Q4: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair @ int @ int @ P3 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% quotient_of_denom_pos
thf(fact_6073_quotient__of__denom__pos_H,axiom,
! [R3: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R3 ) ) ) ).
% quotient_of_denom_pos'
thf(fact_6074_gcd__is__Max__divisors__int,axiom,
! [N2: int,M: int] :
( ( N2
!= ( zero_zero @ int ) )
=> ( ( gcd_gcd @ int @ M @ N2 )
= ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D6: int] :
( ( dvd_dvd @ int @ D6 @ M )
& ( dvd_dvd @ int @ D6 @ N2 ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_6075_rat__uminus__code,axiom,
! [P3: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A7 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_uminus_code
thf(fact_6076_rat__times__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( times_times @ rat @ P3 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B6: int,D6: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A7 @ B6 ) @ ( times_times @ int @ C5 @ D6 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_times_code
thf(fact_6077_rat__divide__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P3 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B6: int,D6: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A7 @ D6 ) @ ( times_times @ int @ C5 @ B6 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_divide_code
thf(fact_6078_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P6: rat,Q7: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A7: int,C5: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B6: int,D6: int] : ( ord_less @ int @ ( times_times @ int @ A7 @ D6 ) @ ( times_times @ int @ C5 @ B6 ) )
@ ( quotient_of @ Q7 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_code
thf(fact_6079_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P6: rat,Q7: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A7: int,C5: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B6: int,D6: int] : ( ord_less_eq @ int @ ( times_times @ int @ A7 @ D6 ) @ ( times_times @ int @ C5 @ B6 ) )
@ ( quotient_of @ Q7 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_eq_code
thf(fact_6080_rat__floor__code,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [P6: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P6 ) ) ) ) ).
% rat_floor_code
thf(fact_6081_rat__abs__code,axiom,
! [P3: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A7 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_abs_code
thf(fact_6082_rat__plus__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P3 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B6: int,D6: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A7 @ D6 ) @ ( times_times @ int @ B6 @ C5 ) ) @ ( times_times @ int @ C5 @ D6 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_plus_code
thf(fact_6083_rat__minus__code,axiom,
! [P3: rat,Q4: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P3 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B6: int,D6: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A7 @ D6 ) @ ( times_times @ int @ B6 @ C5 ) ) @ ( times_times @ int @ C5 @ D6 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_minus_code
thf(fact_6084_rat__sgn__code,axiom,
! [P3: rat] :
( ( quotient_of @ ( sgn_sgn @ rat @ P3 ) )
= ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P3 ) ) ) @ ( one_one @ int ) ) ) ).
% rat_sgn_code
thf(fact_6085_quotient__of__int,axiom,
! [A3: int] :
( ( quotient_of @ ( of_int @ A3 ) )
= ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_6086_rat__inverse__code,axiom,
! [P3: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,B6: int] :
( if @ ( product_prod @ int @ int )
@ ( A7
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A7 ) @ B6 ) @ ( abs_abs @ int @ A7 ) ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_inverse_code
thf(fact_6087_gcd__pos__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M @ N2 ) )
= ( ( M
!= ( zero_zero @ nat ) )
| ( N2
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_pos_nat
thf(fact_6088_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_6089_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_6090_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_6091_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_6092_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_6093_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_6094_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_6095_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_6096_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_6097_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_6098_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_6099_gcd__le1__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ A3 ) ) ).
% gcd_le1_nat
thf(fact_6100_gcd__le2__nat,axiom,
! [B2: nat,A3: nat] :
( ( B2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ B2 ) ) ).
% gcd_le2_nat
thf(fact_6101_gcd__diff1__nat,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M @ N2 ) @ N2 )
= ( gcd_gcd @ nat @ M @ N2 ) ) ) ).
% gcd_diff1_nat
thf(fact_6102_gcd__diff2__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N2 @ M ) @ N2 )
= ( gcd_gcd @ nat @ M @ N2 ) ) ) ).
% gcd_diff2_nat
thf(fact_6103_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_6104_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A7: A,B6: A] : ( times_times @ A @ ( inverse_inverse @ A @ B6 ) @ A7 ) ) ) ) ).
% divide_inverse_commute
thf(fact_6105_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A7: A,B6: A] : ( times_times @ A @ A7 @ ( inverse_inverse @ A @ B6 ) ) ) ) ) ).
% divide_inverse
thf(fact_6106_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A7: A,B6: A] : ( times_times @ A @ A7 @ ( inverse_inverse @ A @ B6 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_6107_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_6108_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N2: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N2 ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N2 ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_6109_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: nat,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_6110_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y: A,X: A] :
( ( ( times_times @ A @ Y @ X )
= ( times_times @ A @ X @ Y ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ Y ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_6111_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_6112_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_6113_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_6114_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_6115_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_6116_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_6117_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_6118_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_6119_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_6120_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_6121_Gcd__in,axiom,
! [A5: set @ nat] :
( ! [A6: nat,B5: nat] :
( ( member @ nat @ A6 @ A5 )
=> ( ( member @ nat @ B5 @ A5 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A6 @ B5 ) @ A5 ) ) )
=> ( ( A5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A5 ) @ A5 ) ) ) ).
% Gcd_in
thf(fact_6122_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_6123_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_6124_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_6125_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_6126_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_le_1_iff
thf(fact_6127_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_6128_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_less_inverse
thf(fact_6129_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_6130_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_6131_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_6132_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ B2 @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_6133_bezout__gcd__nat_H,axiom,
! [B2: nat,A3: nat] :
? [X3: nat,Y3: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ ( times_times @ nat @ A3 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B2 @ Y3 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ ( times_times @ nat @ B2 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A3 @ Y3 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_6134_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_6135_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_6136_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_6137_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_less_1_iff
thf(fact_6138_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_le_inverse
thf(fact_6139_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N5: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N5 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_6140_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M3: nat,N: nat] :
( ( dvd_dvd @ nat @ M3 @ N )
& ( M3 != N ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_6141_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N5 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_6142_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N2: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ M ) )
= ( times_times @ A @ ( power_power @ A @ X @ N2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_6143_gcd__is__Max__divisors__nat,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( gcd_gcd @ nat @ M @ N2 )
= ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D6: nat] :
( ( dvd_dvd @ nat @ D6 @ M )
& ( dvd_dvd @ nat @ D6 @ N2 ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_6144_gcd__nat_Opelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_6145_Frct__code__post_I5_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) )
= ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ).
% Frct_code_post(5)
thf(fact_6146_Frct__code__post_I2_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_6147_Frct__code__post_I1_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A3 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_6148_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_6149_Frct__code__post_I8_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( uminus_uminus @ int @ B2 ) ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(8)
thf(fact_6150_Frct__code__post_I7_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A3 ) @ B2 ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(7)
thf(fact_6151_Frct__code__post_I4_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) )
= ( numeral_numeral @ rat @ K ) ) ).
% Frct_code_post(4)
thf(fact_6152_Frct__code__post_I6_J,axiom,
! [K: num,L: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_6153_rat__floor__lemma,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( fract @ A3 @ B2 ) )
& ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% rat_floor_lemma
thf(fact_6154_If__the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,I2: A,C4: set @ A,G3: A > B,X: A] :
( ( member @ A @ I2 @ C4 )
=> ( ( inj_on @ A @ B @ G3 @ C4 )
=> ( ( comp @ B @ A @ A
@ ^ [I4: B] : ( if @ A @ ( member @ B @ I4 @ ( image2 @ A @ B @ G3 @ C4 ) ) @ ( the_inv_into @ A @ B @ C4 @ G3 @ I4 ) @ X )
@ G3
@ I2 )
= ( id @ A @ I2 ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_6155_less__rat,axiom,
! [B2: int,D3: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D3 ) )
= ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D3 ) @ ( times_times @ int @ B2 @ D3 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ) ).
% less_rat
thf(fact_6156_le__rat,axiom,
! [B2: int,D3: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D3 ) )
= ( ord_less_eq @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D3 ) @ ( times_times @ int @ B2 @ D3 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ) ).
% le_rat
thf(fact_6157_quotient__of__eq,axiom,
! [A3: int,B2: int,P3: int,Q4: int] :
( ( ( quotient_of @ ( fract @ A3 @ B2 ) )
= ( product_Pair @ int @ int @ P3 @ Q4 ) )
=> ( ( fract @ P3 @ Q4 )
= ( fract @ A3 @ B2 ) ) ) ).
% quotient_of_eq
thf(fact_6158_Rat__induct__pos,axiom,
! [P: rat > $o,Q4: rat] :
( ! [A6: int,B5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( P @ ( fract @ A6 @ B5 ) ) )
=> ( P @ Q4 ) ) ).
% Rat_induct_pos
thf(fact_6159_normalize__eq,axiom,
! [A3: int,B2: int,P3: int,Q4: int] :
( ( ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) )
= ( product_Pair @ int @ int @ P3 @ Q4 ) )
=> ( ( fract @ P3 @ Q4 )
= ( fract @ A3 @ B2 ) ) ) ).
% normalize_eq
thf(fact_6160_the__inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( the_inv_into @ A @ B )
= ( ^ [A8: set @ A,F4: A > B,X4: B] :
( the @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ A8 )
& ( ( F4 @ Y4 )
= X4 ) ) ) ) ) ).
% the_inv_into_def
thf(fact_6161_quotient__of__Fract,axiom,
! [A3: int,B2: int] :
( ( quotient_of @ ( fract @ A3 @ B2 ) )
= ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ).
% quotient_of_Fract
thf(fact_6162_Fract__less__zero__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% Fract_less_zero_iff
thf(fact_6163_zero__less__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% zero_less_Fract_iff
thf(fact_6164_Fract__less__one__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less @ int @ A3 @ B2 ) ) ) ).
% Fract_less_one_iff
thf(fact_6165_one__less__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less @ int @ B2 @ A3 ) ) ) ).
% one_less_Fract_iff
thf(fact_6166_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,X: B,B4: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( member @ B @ X @ ( image2 @ A @ B @ F3 @ A5 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A5 @ F3 @ X ) @ B4 ) ) ) ) ).
% the_inv_into_into
thf(fact_6167_zero__le__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% zero_le_Fract_iff
thf(fact_6168_Fract__le__zero__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% Fract_le_zero_iff
thf(fact_6169_one__le__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less_eq @ int @ B2 @ A3 ) ) ) ).
% one_le_Fract_iff
thf(fact_6170_Fract__le__one__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less_eq @ int @ A3 @ B2 ) ) ) ).
% Fract_le_one_iff
thf(fact_6171_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G3: A > B,C4: set @ A,B4: set @ A,X: A] :
( ( inj_on @ A @ B @ G3 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ ( sup_sup @ ( set @ A ) @ B4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I4: B] : ( if @ A @ ( member @ B @ I4 @ ( image2 @ A @ B @ G3 @ C4 ) ) @ ( the_inv_into @ A @ B @ C4 @ G3 @ I4 ) @ X )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_6172_positive__rat,axiom,
! [A3: int,B2: int] :
( ( positive @ ( fract @ A3 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A3 @ B2 ) ) ) ).
% positive_rat
thf(fact_6173_Func__non__emp,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: set @ B] :
( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A5 @ B4 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_6174_Func__is__emp,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A5 @ B4 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_6175_Func__empty,axiom,
! [B: $tType,A: $tType,B4: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B4 )
= ( insert @ ( A > B )
@ ^ [X4: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_6176_less__rat__def,axiom,
( ( ord_less @ rat )
= ( ^ [X4: rat,Y4: rat] : ( positive @ ( minus_minus @ rat @ Y4 @ X4 ) ) ) ) ).
% less_rat_def
thf(fact_6177_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A18: set @ B,B16: set @ A,F22: C > D,B23: set @ C,A25: set @ D] :
( ( ( image2 @ B @ A @ F1 @ A18 )
= B16 )
=> ( ( inj_on @ C @ D @ F22 @ B23 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image2 @ C @ D @ F22 @ B23 ) @ A25 )
=> ( ( ( B23
= ( bot_bot @ ( set @ C ) ) )
=> ( A25
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B23 @ B16 )
= ( image2 @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B23 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A25 @ A18 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_6178_positive__def,axiom,
( positive
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
@ ^ [X4: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ X4 ) ) ) ) ) ).
% positive_def
thf(fact_6179_Func__map,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,G3: A > B,A25: set @ A,A18: set @ B,F1: B > C,B16: set @ C,F22: D > A,B23: set @ D] :
( ( member @ ( A > B ) @ G3 @ ( bNF_Wellorder_Func @ A @ B @ A25 @ A18 ) )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ F1 @ A18 ) @ B16 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ F22 @ B23 ) @ A25 )
=> ( member @ ( D > C ) @ ( bNF_We4925052301507509544nc_map @ D @ B @ C @ A @ B23 @ F1 @ F22 @ G3 ) @ ( bNF_Wellorder_Func @ D @ C @ B23 @ B16 ) ) ) ) ) ).
% Func_map
thf(fact_6180_positive_Orep__eq,axiom,
( positive
= ( ^ [X4: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X4 ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X4 ) ) ) ) ) ) ).
% positive.rep_eq
thf(fact_6181_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ) ).
% plus_rat_def
thf(fact_6182_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_6183_one__rat__def,axiom,
( ( one_one @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% one_rat_def
thf(fact_6184_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: 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 @ Xa4 @ X4 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_6185_zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% zero_rat_def
thf(fact_6186_Fract__def,axiom,
( fract
= ( map_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat ) @ ( id @ int ) @ ( map_fun @ int @ int @ ( product_prod @ int @ int ) @ rat @ ( id @ int ) @ abs_Rat )
@ ^ [A7: int,B6: int] :
( if @ ( product_prod @ int @ int )
@ ( B6
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A7 @ B6 ) ) ) ) ).
% Fract_def
thf(fact_6187_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_6188_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ) ).
% times_rat_def
thf(fact_6189_of__rat__def,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ A @ A @ rep_Rat @ ( id @ A )
@ ^ [X4: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X4 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X4 ) ) ) ) ) ) ).
% of_rat_def
thf(fact_6190_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_6191_zero__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R3 ) )
= ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 ) ) ) ).
% zero_less_of_rat_iff
thf(fact_6192_of__rat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( zero_zero @ A ) )
= ( ord_less @ rat @ R3 @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_less_0_iff
thf(fact_6193_zero__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R3 ) )
= ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ R3 ) ) ) ).
% zero_le_of_rat_iff
thf(fact_6194_of__rat__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ rat @ R3 @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_le_0_iff
thf(fact_6195_one__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R3 ) )
= ( ord_less @ rat @ ( one_one @ rat ) @ R3 ) ) ) ).
% one_less_of_rat_iff
thf(fact_6196_of__rat__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( one_one @ A ) )
= ( ord_less @ rat @ R3 @ ( one_one @ rat ) ) ) ) ).
% of_rat_less_1_iff
thf(fact_6197_one__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R3 ) )
= ( ord_less_eq @ rat @ ( one_one @ rat ) @ R3 ) ) ) ).
% one_le_of_rat_iff
thf(fact_6198_of__rat__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ rat @ R3 @ ( one_one @ rat ) ) ) ) ).
% of_rat_le_1_iff
thf(fact_6199_of__rat__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat,S2: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( field_char_0_of_rat @ A @ S2 ) )
= ( ord_less @ rat @ R3 @ S2 ) ) ) ).
% of_rat_less
thf(fact_6200_of__rat__prod,axiom,
! [A: $tType,B: $tType] :
( ( field_char_0 @ A )
=> ! [F3: B > rat,A5: set @ B] :
( ( field_char_0_of_rat @ A @ ( groups7121269368397514597t_prod @ B @ rat @ F3 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A7: B] : ( field_char_0_of_rat @ A @ ( F3 @ A7 ) )
@ A5 ) ) ) ).
% of_rat_prod
thf(fact_6201_of__rat__sum,axiom,
! [A: $tType,B: $tType] :
( ( field_char_0 @ A )
=> ! [F3: B > rat,A5: set @ B] :
( ( field_char_0_of_rat @ A @ ( groups7311177749621191930dd_sum @ B @ rat @ F3 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A7: B] : ( field_char_0_of_rat @ A @ ( F3 @ A7 ) )
@ A5 ) ) ) ).
% of_rat_sum
thf(fact_6202_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_6203_of__rat__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R3: rat,S2: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R3 ) @ ( field_char_0_of_rat @ A @ S2 ) )
= ( ord_less_eq @ rat @ R3 @ S2 ) ) ) ).
% of_rat_less_eq
thf(fact_6204_of__rat__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat,B2: rat] :
( ( field_char_0_of_rat @ A @ ( times_times @ rat @ A3 @ B2 ) )
= ( times_times @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% of_rat_mult
thf(fact_6205_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_6206_ratrel__def,axiom,
( ratrel
= ( ^ [X4: product_prod @ int @ int,Y4: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X4 )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y4 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X4 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_6207_uminus__rat_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( uminus_uminus @ rat @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% uminus_rat.abs_eq
thf(fact_6208_times__rat_Oabs__eq,axiom,
! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( times_times @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_fst @ int @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_6209_positive_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( positive @ ( abs_Rat @ X ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% positive.abs_eq
thf(fact_6210_inverse__rat_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( inverse_inverse @ rat @ ( abs_Rat @ X ) )
= ( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_6211_cr__rat__def,axiom,
( cr_rat
= ( ^ [X4: product_prod @ int @ int,Y4: rat] :
( ( ratrel @ X4 @ X4 )
& ( ( abs_Rat @ X4 )
= Y4 ) ) ) ) ).
% cr_rat_def
thf(fact_6212_Code__Numeral_Odup__def,axiom,
( code_dup
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int
@ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ) ).
% Code_Numeral.dup_def
thf(fact_6213_quotient__of__def,axiom,
( quotient_of
= ( ^ [X4: rat] :
( the @ ( product_prod @ int @ int )
@ ^ [Pair: product_prod @ int @ int] :
( ( X4
= ( fract @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Pair ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) ) ) ) ) ).
% quotient_of_def
thf(fact_6214_ntrancl__Suc,axiom,
! [A: $tType,N2: nat,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( suc @ N2 ) @ R2 )
= ( relcomp @ A @ A @ A @ ( transitive_ntrancl @ A @ N2 @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ R2 ) ) ) ).
% ntrancl_Suc
thf(fact_6215_coprime__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ ( times_times @ A @ A3 @ B2 ) )
= ( ( algebr8660921524188924756oprime @ A @ C2 @ A3 )
& ( algebr8660921524188924756oprime @ A @ C2 @ B2 ) ) ) ) ).
% coprime_mult_right_iff
thf(fact_6216_coprime__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
& ( algebr8660921524188924756oprime @ A @ B2 @ C2 ) ) ) ) ).
% coprime_mult_left_iff
thf(fact_6217_pair__in__Id__conv,axiom,
! [A: $tType,A3: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
= ( A3 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_6218_IdI,axiom,
! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_6219_Id__O__R,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( relcomp @ A @ A @ B @ ( id2 @ A ) @ R2 )
= R2 ) ).
% Id_O_R
thf(fact_6220_R__O__Id,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( relcomp @ A @ B @ B @ R2 @ ( id2 @ B ) )
= R2 ) ).
% R_O_Id
thf(fact_6221_bijective__Id,axiom,
! [A: $tType] : ( bijective @ A @ A @ ( id2 @ A ) ) ).
% bijective_Id
thf(fact_6222_rtrancl__empty,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( id2 @ A ) ) ).
% rtrancl_empty
thf(fact_6223_rtrancl__reflcl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
= ( transitive_rtrancl @ A @ R2 ) ) ).
% rtrancl_reflcl
thf(fact_6224_rtrancl__reflcl__absorb,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 ) @ ( id2 @ A ) )
= ( transitive_rtrancl @ A @ R2 ) ) ).
% rtrancl_reflcl_absorb
thf(fact_6225_total__on__diff__Id,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A5 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
= ( total_on @ A @ A5 @ R3 ) ) ).
% total_on_diff_Id
thf(fact_6226_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_6227_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_6228_trancl__reflcl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
= ( transitive_rtrancl @ A @ R3 ) ) ).
% trancl_reflcl
thf(fact_6229_normalize__stable,axiom,
! [Q4: int,P3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q4 )
=> ( ( algebr8660921524188924756oprime @ int @ P3 @ Q4 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P3 @ Q4 ) )
= ( product_Pair @ int @ int @ P3 @ Q4 ) ) ) ) ).
% normalize_stable
thf(fact_6230_prod__coprime__left,axiom,
! [B: $tType,A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ B,F3: B > A,A3: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( algebr8660921524188924756oprime @ A @ ( F3 @ I3 ) @ A3 ) )
=> ( algebr8660921524188924756oprime @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ A3 ) ) ) ).
% prod_coprime_left
thf(fact_6231_prod__coprime__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ B,A3: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( algebr8660921524188924756oprime @ A @ A3 @ ( F3 @ I3 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ).
% prod_coprime_right
thf(fact_6232_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
=> ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_right_right_cancel
thf(fact_6233_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
=> ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_right_left_cancel
thf(fact_6234_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_left_right_cancel
thf(fact_6235_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_left_left_cancel
thf(fact_6236_quotient__of__coprime,axiom,
! [R3: rat,P3: int,Q4: int] :
( ( ( quotient_of @ R3 )
= ( product_Pair @ int @ int @ P3 @ Q4 ) )
=> ( algebr8660921524188924756oprime @ int @ P3 @ Q4 ) ) ).
% quotient_of_coprime
thf(fact_6237_IdE,axiom,
! [A: $tType,P3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P3 @ ( id2 @ A ) )
=> ~ ! [X3: A] :
( P3
!= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ).
% IdE
thf(fact_6238_BNF__Greatest__Fixpoint_OIdD,axiom,
! [A: $tType,A3: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
=> ( A3 = B2 ) ) ).
% BNF_Greatest_Fixpoint.IdD
thf(fact_6239_normalize__coprime,axiom,
! [R3: product_prod @ int @ int,P3: int,Q4: int] :
( ( ( normalize @ R3 )
= ( product_Pair @ int @ int @ P3 @ Q4 ) )
=> ( algebr8660921524188924756oprime @ int @ P3 @ Q4 ) ) ).
% normalize_coprime
thf(fact_6240_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% coprime_dvd_mult_right_iff
thf(fact_6241_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% coprime_dvd_mult_left_iff
thf(fact_6242_divides__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ C2 )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% divides_mult
thf(fact_6243_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [A3: A,N2: A,M: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ N2 ) @ M )
= ( modulo_modulo @ A @ ( times_times @ A @ B2 @ N2 ) @ M ) )
=> ( ( algebr8660921524188924756oprime @ A @ M @ N2 )
=> ( ( modulo_modulo @ A @ A3 @ M )
= ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% mult_mod_cancel_right
thf(fact_6244_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [N2: A,A3: A,M: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ N2 @ A3 ) @ M )
= ( modulo_modulo @ A @ ( times_times @ A @ N2 @ B2 ) @ M ) )
=> ( ( algebr8660921524188924756oprime @ A @ M @ N2 )
=> ( ( modulo_modulo @ A @ A3 @ M )
= ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% mult_mod_cancel_left
thf(fact_6245_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P6: product_prod @ A @ A] :
? [X4: A] :
( P6
= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% Id_def
thf(fact_6246_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_6247_invertible__coprime,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ C2 ) ) ) ).
% invertible_coprime
thf(fact_6248_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,A4: A,B3: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
= ( times_times @ A @ A4 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( ( B2
= ( times_times @ A @ B3 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) ) ) ).
% gcd_coprime
thf(fact_6249_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ? [A11: A,B10: A] :
( ( A3
= ( times_times @ A @ A11 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( B2
= ( times_times @ A @ B10 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( algebr8660921524188924756oprime @ A @ A11 @ B10 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_6250_Rat__induct,axiom,
! [P: rat > $o,Q4: rat] :
( ! [A6: int,B5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( algebr8660921524188924756oprime @ int @ A6 @ B5 )
=> ( P @ ( fract @ A6 @ B5 ) ) ) )
=> ( P @ Q4 ) ) ).
% Rat_induct
thf(fact_6251_Rat__cases,axiom,
! [Q4: rat] :
~ ! [A6: int,B5: int] :
( ( Q4
= ( fract @ A6 @ B5 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ~ ( algebr8660921524188924756oprime @ int @ A6 @ B5 ) ) ) ).
% Rat_cases
thf(fact_6252_rtrancl__trancl__reflcl,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R4 ) @ ( id2 @ A ) ) ) ) ).
% rtrancl_trancl_reflcl
thf(fact_6253_rtrancl__unfold,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ ( relcomp @ A @ A @ A @ ( transitive_rtrancl @ A @ R4 ) @ R4 ) ) ) ) ).
% rtrancl_unfold
thf(fact_6254_reflcl__set__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 )
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_6255_Rat__cases__nonzero,axiom,
! [Q4: rat] :
( ! [A6: int,B5: int] :
( ( Q4
= ( fract @ A6 @ B5 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( A6
!= ( zero_zero @ int ) )
=> ~ ( algebr8660921524188924756oprime @ int @ A6 @ B5 ) ) ) )
=> ( Q4
= ( zero_zero @ rat ) ) ) ).
% Rat_cases_nonzero
thf(fact_6256_rtrancl__Int__subset,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ S2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ S2 ) @ R3 ) @ S2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ S2 ) ) ) ).
% rtrancl_Int_subset
thf(fact_6257_quotient__of__unique,axiom,
! [R3: rat] :
? [X3: product_prod @ int @ int] :
( ( R3
= ( fract @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ X3 ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) )
& ! [Y6: product_prod @ int @ int] :
( ( ( R3
= ( fract @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ Y6 ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Y6 ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ Y6 ) ) )
=> ( Y6 = X3 ) ) ) ).
% quotient_of_unique
thf(fact_6258_relInvImage__Id__on,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B4: set @ B] :
( ! [A14: A,A24: A] :
( ( ( F3 @ A14 )
= ( F3 @ A24 ) )
= ( A14 = A24 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Gr7122648621184425601vImage @ A @ B @ A5 @ ( id_on @ B @ B4 ) @ F3 ) @ ( id2 @ A ) ) ) ).
% relInvImage_Id_on
thf(fact_6259_Rats__cases_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( field_char_0_Rats @ A ) )
=> ~ ! [A6: int,B5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( algebr8660921524188924756oprime @ int @ A6 @ B5 )
=> ( X
!= ( divide_divide @ A @ ( ring_1_of_int @ A @ A6 ) @ ( ring_1_of_int @ A @ B5 ) ) ) ) ) ) ) ).
% Rats_cases'
thf(fact_6260_Rats__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( field_char_0_Rats @ A ) )
=> ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% Rats_mult
thf(fact_6261_relInvImage__mono,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ A @ A ),A5: set @ B,F3: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A5 @ R1 @ F3 ) @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A5 @ R22 @ F3 ) ) ) ).
% relInvImage_mono
thf(fact_6262_Ints__subset__Rats,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ord_less_eq @ ( set @ A ) @ ( ring_1_Ints @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Ints_subset_Rats
thf(fact_6263_coprime__diff__one__left__nat,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ N2 ) ) ).
% coprime_diff_one_left_nat
thf(fact_6264_coprime__diff__one__right__nat,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( algebr8660921524188924756oprime @ nat @ N2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_6265_mult__inj__if__coprime__nat,axiom,
! [B: $tType,A: $tType,F3: A > nat,A5: set @ A,G3: B > nat,B4: set @ B] :
( ( inj_on @ A @ nat @ F3 @ A5 )
=> ( ( inj_on @ B @ nat @ G3 @ B4 )
=> ( ! [A6: A,B5: B] :
( ( member @ A @ A6 @ A5 )
=> ( ( member @ B @ B5 @ B4 )
=> ( algebr8660921524188924756oprime @ nat @ ( F3 @ A6 ) @ ( G3 @ B5 ) ) ) )
=> ( inj_on @ ( product_prod @ A @ B ) @ nat
@ ( product_case_prod @ A @ B @ nat
@ ^ [A7: A,B6: B] : ( times_times @ nat @ ( F3 @ A7 ) @ ( G3 @ B6 ) ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) ) ) ) ) ).
% mult_inj_if_coprime_nat
thf(fact_6266_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr7122648621184425601vImage @ A @ B )
= ( ^ [A8: set @ A,R6: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu2: product_prod @ A @ A] :
? [A12: A,A23: A] :
( ( Uu2
= ( product_Pair @ A @ A @ A12 @ A23 ) )
& ( member @ A @ A12 @ A8 )
& ( member @ A @ A23 @ A8 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ A12 ) @ ( F4 @ A23 ) ) @ R6 ) ) ) ) ) ).
% relInvImage_def
thf(fact_6267_relImage__relInvImage,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F3 @ A5 )
@ ^ [Uu2: A] : ( image2 @ B @ A @ F3 @ A5 ) ) )
=> ( ( bNF_Gr4221423524335903396lImage @ B @ A @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A5 @ R2 @ F3 ) @ F3 )
= R2 ) ) ).
% relImage_relInvImage
thf(fact_6268_relInvImage__UNIV__relImage,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( bNF_Gr7122648621184425601vImage @ A @ B @ ( top_top @ ( set @ A ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R2 @ F3 ) @ F3 ) ) ).
% relInvImage_UNIV_relImage
thf(fact_6269_relImage__mono,axiom,
! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ A @ A ),F3: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R1 @ F3 ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R22 @ F3 ) ) ) ).
% relImage_mono
thf(fact_6270_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr4221423524335903396lImage @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ B ),F4: B > A] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu2: product_prod @ A @ A] :
? [A12: B,A23: B] :
( ( Uu2
= ( product_Pair @ A @ A @ ( F4 @ A12 ) @ ( F4 @ A23 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A12 @ A23 ) @ R6 ) ) ) ) ) ).
% relImage_def
thf(fact_6271_card__quotient__disjoint,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A] : ( equiv_quotient @ A @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
@ A5 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R3 ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_6272_eq__f__restr__ss__eq,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: ( A > ( option @ B ) ) > A > ( option @ B ),A5: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ ( dom @ A @ B @ ( F3 @ A5 ) ) )
=> ( ( A5
= ( restrict_map @ A @ B @ ( F3 @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ S2 ) ) )
= ( ( map_le @ A @ B @ A5 @ ( F3 @ A5 ) )
& ( S2
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F3 @ A5 ) ) @ ( dom @ A @ B @ A5 ) ) ) ) ) ) ).
% eq_f_restr_ss_eq
thf(fact_6273_quotient__empty,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( bot_bot @ ( set @ A ) ) @ R3 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% quotient_empty
thf(fact_6274_quotient__is__empty,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ( equiv_quotient @ A @ A5 @ R3 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty
thf(fact_6275_quotient__is__empty2,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ( bot_bot @ ( set @ ( set @ A ) ) )
= ( equiv_quotient @ A @ A5 @ R3 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty2
thf(fact_6276_map__leI,axiom,
! [B: $tType,A: $tType,M14: A > ( option @ B ),M24: A > ( option @ B )] :
( ! [X3: A,V4: B] :
( ( ( M14 @ X3 )
= ( some @ B @ V4 ) )
=> ( ( M24 @ X3 )
= ( some @ B @ V4 ) ) )
=> ( map_le @ A @ B @ M14 @ M24 ) ) ).
% map_leI
thf(fact_6277_map__leD,axiom,
! [A: $tType,B: $tType,M14: A > ( option @ B ),M24: A > ( option @ B ),K: A,V2: B] :
( ( map_le @ A @ B @ M14 @ M24 )
=> ( ( ( M14 @ K )
= ( some @ B @ V2 ) )
=> ( ( M24 @ K )
= ( some @ B @ V2 ) ) ) ) ).
% map_leD
thf(fact_6278_map__le__empty,axiom,
! [B: $tType,A: $tType,G3: A > ( option @ B )] :
( map_le @ A @ B
@ ^ [X4: A] : ( none @ B )
@ G3 ) ).
% map_le_empty
thf(fact_6279_map__le__implies__dom__le,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B ),G3: A > ( option @ B )] :
( ( map_le @ A @ B @ F3 @ G3 )
=> ( ord_less_eq @ ( set @ A ) @ ( dom @ A @ B @ F3 ) @ ( dom @ A @ B @ G3 ) ) ) ).
% map_le_implies_dom_le
thf(fact_6280_quotient__diff1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,A3: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A7: A] : ( equiv_quotient @ A @ ( insert @ A @ A7 @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
@ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ R3 )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A5 @ R3 ) @ ( equiv_quotient @ A @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) @ R3 ) ) ) ) ) ).
% quotient_diff1
thf(fact_6281_finite__equiv__class,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( finite_finite2 @ A @ X6 ) ) ) ) ).
% finite_equiv_class
thf(fact_6282_map__le__imp__upd__le,axiom,
! [A: $tType,B: $tType,M14: A > ( option @ B ),M24: A > ( option @ B ),X: A,Y: B] :
( ( map_le @ A @ B @ M14 @ M24 )
=> ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M14 @ X @ ( none @ B ) ) @ ( fun_upd @ A @ ( option @ B ) @ M24 @ X @ ( some @ B @ Y ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_6283_finite__quotient,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
=> ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R3 ) ) ) ) ).
% finite_quotient
thf(fact_6284_eq__f__restr__conv,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: ( A > ( option @ B ) ) > A > ( option @ B ),A5: A > ( option @ B )] :
( ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( dom @ A @ B @ ( F3 @ A5 ) ) )
& ( A5
= ( restrict_map @ A @ B @ ( F3 @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ S2 ) ) ) )
= ( ( map_le @ A @ B @ A5 @ ( F3 @ A5 ) )
& ( S2
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F3 @ A5 ) ) @ ( dom @ A @ B @ A5 ) ) ) ) ) ).
% eq_f_restr_conv
thf(fact_6285_le__map__mmupd__not__dom,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),K5: set @ A,V2: B] : ( map_le @ A @ B @ M @ ( map_mmupd @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ K5 @ ( dom @ A @ B @ M ) ) @ V2 ) ) ).
% le_map_mmupd_not_dom
thf(fact_6286_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A] : ( insert @ ( set @ A ) @ ( image @ A @ A @ R4 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A8 ) ) ) ) ).
% quotient_def
thf(fact_6287_mmupd__notin__upd,axiom,
! [B: $tType,A: $tType,K: A,K5: set @ A,M: A > ( option @ B ),V2: B] :
( ~ ( member @ A @ K @ K5 )
=> ( ( map_mmupd @ A @ B @ M @ K5 @ V2 @ K )
= ( M @ K ) ) ) ).
% mmupd_notin_upd
thf(fact_6288_ImageI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( ( member @ A @ A3 @ A5 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% ImageI
thf(fact_6289_Image__empty2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( image @ B @ A @ R2 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty2
thf(fact_6290_Image__Id,axiom,
! [A: $tType,A5: set @ A] :
( ( image @ A @ A @ ( id2 @ A ) @ A5 )
= A5 ) ).
% Image_Id
thf(fact_6291_map__mmupd__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),V2: B] :
( ( map_mmupd @ A @ B @ M @ ( bot_bot @ ( set @ A ) ) @ V2 )
= M ) ).
% map_mmupd_empty
thf(fact_6292_mmupd__in__upd,axiom,
! [A: $tType,B: $tType,K: A,K5: set @ A,M: A > ( option @ B ),V2: B] :
( ( member @ A @ K @ K5 )
=> ( ( map_mmupd @ A @ B @ M @ K5 @ V2 @ K )
= ( some @ B @ V2 ) ) ) ).
% mmupd_in_upd
thf(fact_6293_Image__empty1,axiom,
! [B: $tType,A: $tType,X6: set @ B] :
( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty1
thf(fact_6294_Image__Id__on,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( image @ A @ A @ ( id_on @ A @ A5 ) @ B4 )
= ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ).
% Image_Id_on
thf(fact_6295_dom__mmupd,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K5: set @ A,V2: B] :
( ( dom @ A @ B @ ( map_mmupd @ A @ B @ M @ K5 @ V2 ) )
= ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ M ) @ K5 ) ) ).
% dom_mmupd
thf(fact_6296_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A3: B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_6297_pair__vimage__is__Image,axiom,
! [A: $tType,B: $tType,U: B,E6: set @ ( product_prod @ B @ A )] :
( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ U ) @ E6 )
= ( image @ B @ A @ E6 @ ( insert @ B @ U @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% pair_vimage_is_Image
thf(fact_6298_Image__Int__subset,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),A5: set @ B,B4: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R2 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R2 @ A5 ) @ ( image @ B @ A @ R2 @ B4 ) ) ) ).
% Image_Int_subset
thf(fact_6299_Image__mono,axiom,
! [B: $tType,A: $tType,R7: set @ ( product_prod @ A @ B ),R3: set @ ( product_prod @ A @ B ),A17: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R7 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A17 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R7 @ A17 ) @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% Image_mono
thf(fact_6300_Image__closed__trancl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ X6 ) @ X6 )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ X6 )
= X6 ) ) ).
% Image_closed_trancl
thf(fact_6301_rtrancl__image__unfold__right,axiom,
! [A: $tType,E6: set @ ( product_prod @ A @ A ),V: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E6 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ V ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ V ) ) ).
% rtrancl_image_unfold_right
thf(fact_6302_rtrancl__reachable__induct,axiom,
! [A: $tType,I5: set @ A,INV: set @ A,E6: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ I5 @ INV )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E6 @ INV ) @ INV )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ I5 ) @ INV ) ) ) ).
% rtrancl_reachable_induct
thf(fact_6303_Un__Image,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A ),A5: set @ B] :
( ( image @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R2 @ S ) @ A5 )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R2 @ A5 ) @ ( image @ B @ A @ S @ A5 ) ) ) ).
% Un_Image
thf(fact_6304_Image__Un,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),A5: set @ B,B4: set @ B] :
( ( image @ B @ A @ R2 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ R2 @ A5 ) @ ( image @ B @ A @ R2 @ B4 ) ) ) ).
% Image_Un
thf(fact_6305_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q4: A,R2: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
( ( member @ A @ Q4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ X ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_6306_rtrancl__image__advance,axiom,
! [A: $tType,Q4: A,R2: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
( ( member @ A @ Q4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q4 @ X ) @ R2 )
=> ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_6307_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A5 ) )
=> ~ ! [X3: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B2 ) @ R3 )
=> ~ ( member @ B @ X3 @ A5 ) ) ) ).
% ImageE
thf(fact_6308_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R3: set @ ( product_prod @ B @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R3 @ A5 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ B2 ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_6309_rev__ImageI,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ A5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R3 @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_6310_relcomp__Image,axiom,
! [A: $tType,C: $tType,B: $tType,X6: set @ ( product_prod @ B @ C ),Y7: set @ ( product_prod @ C @ A ),Z5: set @ B] :
( ( image @ B @ A @ ( relcomp @ B @ C @ A @ X6 @ Y7 ) @ Z5 )
= ( image @ C @ A @ Y7 @ ( image @ B @ C @ X6 @ Z5 ) ) ) ).
% relcomp_Image
thf(fact_6311_trancl__Image__unfold__left,axiom,
! [A: $tType,E6: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E6 ) @ S )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ ( image @ A @ A @ E6 @ S ) ) ) ).
% trancl_Image_unfold_left
thf(fact_6312_trancl__Image__unfold__right,axiom,
! [A: $tType,E6: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E6 ) @ S )
= ( image @ A @ A @ E6 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ S ) ) ) ).
% trancl_Image_unfold_right
thf(fact_6313_map__mmupdE,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),K5: set @ B,V2: A,K: B,X: A] :
( ( ( map_mmupd @ B @ A @ M @ K5 @ V2 @ K )
= ( some @ A @ X ) )
=> ( ( ~ ( member @ B @ K @ K5 )
=> ( ( M @ K )
!= ( some @ A @ X ) ) )
=> ~ ( ( member @ B @ K @ K5 )
=> ( X != V2 ) ) ) ) ).
% map_mmupdE
thf(fact_6314_map__mmupd__def,axiom,
! [A: $tType,B: $tType] :
( ( map_mmupd @ B @ A )
= ( ^ [M3: B > ( option @ A ),K7: set @ B,V3: A,K3: B] : ( if @ ( option @ A ) @ ( member @ B @ K3 @ K7 ) @ ( some @ A @ V3 ) @ ( M3 @ K3 ) ) ) ) ).
% map_mmupd_def
thf(fact_6315_finite__Image,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),A5: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
=> ( finite_finite2 @ B @ ( image @ A @ B @ R2 @ A5 ) ) ) ).
% finite_Image
thf(fact_6316_finite__Image__subset,axiom,
! [A: $tType,B: $tType,A5: set @ ( product_prod @ B @ A ),B4: set @ B,C4: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ ( image @ B @ A @ A5 @ B4 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ C4 @ A5 )
=> ( finite_finite2 @ A @ ( image @ B @ A @ C4 @ B4 ) ) ) ) ).
% finite_Image_subset
thf(fact_6317_Image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,R3: set @ ( product_prod @ B @ A ),B4: C > ( set @ B ),A5: set @ C] :
( ( image @ B @ A @ R3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R3 @ ( B4 @ X4 ) )
@ A5 ) ) ) ).
% Image_UN
thf(fact_6318_Image__empty__rtrancl__Image__id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),V2: A] :
( ( ( image @ A @ A @ R2 @ ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_6319_Image__empty__trancl__Image__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),V2: A] :
( ( ( image @ A @ A @ R2 @ ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_trancl @ A @ R2 ) @ ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_6320_reachable__mono,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A @ A ),X6: set @ A,X16: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R9 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ X16 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ X6 ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R9 ) @ X16 ) ) ) ) ).
% reachable_mono
thf(fact_6321_quotientI,axiom,
! [A: $tType,X: A,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ A5 )
=> ( member @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( equiv_quotient @ A @ A5 @ R3 ) ) ) ).
% quotientI
thf(fact_6322_quotientE,axiom,
! [A: $tType,X6: set @ A,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ~ ! [X3: A] :
( ( X6
= ( image @ A @ A @ R3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ~ ( member @ A @ X3 @ A5 ) ) ) ).
% quotientE
thf(fact_6323_Image__subset__snd__image,axiom,
! [A: $tType,B: $tType,A5: set @ ( product_prod @ B @ A ),B4: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ A5 @ B4 ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A5 ) ) ).
% Image_subset_snd_image
thf(fact_6324_trancl__image__by__rtrancl,axiom,
! [A: $tType,E6: set @ ( product_prod @ A @ A ),Vi: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E6 ) @ Vi ) @ Vi )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ Vi ) ) ).
% trancl_image_by_rtrancl
thf(fact_6325_Image__singleton,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ A ),A3: B] :
( ( image @ B @ A @ R3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B6: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B6 ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_6326_Image__subset,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),A5: set @ A,B4: set @ B,C4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R3 @ C4 ) @ B4 ) ) ).
% Image_subset
thf(fact_6327_Image__INT__subset,axiom,
! [A: $tType,B: $tType,C: $tType,R3: set @ ( product_prod @ B @ A ),B4: C > ( set @ B ),A5: set @ C] :
( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
@ ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R3 @ ( B4 @ X4 ) )
@ A5 ) ) ) ).
% Image_INT_subset
thf(fact_6328_rtrancl__apply__insert,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,S: set @ A] :
( ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ ( insert @ A @ X @ S ) )
= ( insert @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ ( sup_sup @ ( set @ A ) @ S @ ( image @ A @ A @ R2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_6329_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V2: A,E6: set @ ( product_prod @ A @ A ),R2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_rtrancl @ A @ E6 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E6 @ R2 ) @ R2 )
=> ( ~ ( member @ A @ V2 @ R2 )
=> ~ ( ~ ( member @ A @ U @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E6 @ R2 ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_6330_rel__restrict__tranclI,axiom,
! [A: $tType,X: A,Y: A,E6: set @ ( product_prod @ A @ A ),R2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ E6 ) )
=> ( ~ ( member @ A @ X @ R2 )
=> ( ~ ( member @ A @ Y @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E6 @ R2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E6 @ R2 ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_6331_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A ),B7: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] : ( image @ B @ A @ R4 @ ( insert @ B @ Y4 @ ( bot_bot @ ( set @ B ) ) ) )
@ B7 ) ) ) ) ).
% Image_eq_UN
thf(fact_6332_Sigma__Image,axiom,
! [A: $tType,B: $tType,A5: set @ B,B4: B > ( set @ A ),X6: set @ B] :
( ( image @ B @ A @ ( product_Sigma @ B @ A @ A5 @ B4 ) @ X6 )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ B ) @ X6 @ A5 ) ) ) ) ).
% Sigma_Image
thf(fact_6333_UN__Image,axiom,
! [A: $tType,B: $tType,C: $tType,X6: C > ( set @ ( product_prod @ B @ A ) ),I5: set @ C,S: set @ B] :
( ( image @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ X6 @ I5 ) ) @ S )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [I4: C] : ( image @ B @ A @ ( X6 @ I4 ) @ S )
@ I5 ) ) ) ).
% UN_Image
thf(fact_6334_finite__reachable__advance,axiom,
! [A: $tType,E6: set @ ( product_prod @ A @ A ),V0: A,V2: A] :
( ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ ( insert @ A @ V0 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V0 @ V2 ) @ ( transitive_rtrancl @ A @ E6 ) )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_6335_rtrancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V2: A,E6: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ E6 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_6336_trancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V2: A,E6: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ E6 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E6 ) @ ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E6 ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_6337_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V2: A,E6: set @ ( product_prod @ A @ A ),S: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_trancl @ A @ E6 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E6 @ 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 ) ) @ E6
@ ( product_Sigma @ A @ A @ S
@ ^ [Uu2: A] : S ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_6338_singleton__quotient,axiom,
! [A: $tType,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
= ( insert @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% singleton_quotient
thf(fact_6339_Image__fold,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
=> ( ( image @ A @ B @ R2 @ S )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X4: A,Y4: B,A8: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X4 @ S ) @ ( insert @ B @ Y4 @ A8 ) @ A8 ) )
@ ( bot_bot @ ( set @ B ) )
@ R2 ) ) ) ).
% Image_fold
thf(fact_6340_map__mmupd__update__less,axiom,
! [A: $tType,B: $tType,K5: set @ A,K8: set @ A,M: A > ( option @ B ),V2: B] :
( ( ord_less_eq @ ( set @ A ) @ K5 @ K8 )
=> ( map_le @ A @ B @ ( map_mmupd @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ K5 @ ( dom @ A @ B @ M ) ) @ V2 ) @ ( map_mmupd @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ K8 @ ( dom @ A @ B @ M ) ) @ V2 ) ) ) ).
% map_mmupd_update_less
thf(fact_6341_listrel__Cons,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A ),X: B,Xs: list @ B] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R3 ) @ ( insert @ ( list @ B ) @ ( cons @ B @ X @ Xs ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( set_Cons @ A @ ( image @ B @ A @ R3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) @ ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R3 ) @ ( insert @ ( list @ B ) @ Xs @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) ) ) ) ).
% listrel_Cons
thf(fact_6342_positive_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [X4: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ X4 ) ) )
@ ^ [X4: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ X4 ) ) ) ) ).
% positive.rsp
thf(fact_6343_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) ) ).
% listrel_rtrancl_refl
thf(fact_6344_listrel__Nil,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ A )] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R3 ) @ ( insert @ ( list @ B ) @ ( nil @ B ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listrel_Nil
thf(fact_6345_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R2
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ R2 )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_6346_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R3 ) )
=> ( Xs
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_6347_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R3 ) )
=> ( Xs
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_6348_listrel_ONil,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R3 ) ) ).
% listrel.Nil
thf(fact_6349_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys3 ) @ ( listrel @ A @ B @ R3 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) ) ) ).
% listrel_eq_len
thf(fact_6350_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A ),Zs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys3 @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_6351_listrel__mono,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( listrel @ A @ B @ R3 ) @ ( listrel @ A @ B @ S2 ) ) ) ).
% listrel_mono
thf(fact_6352_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys3: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys3 ) ) @ ( listrel @ A @ B @ R3 ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys3 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_6353_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys3: list @ A,Xs: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys3 ) @ Xs ) @ ( listrel @ A @ B @ R3 ) )
=> ~ ! [Y3: B,Ys4: list @ B] :
( ( Xs
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys3 @ Ys4 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_6354_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R3: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys3: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys3 ) @ ( listrel @ A @ B @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys3 ) ) @ ( listrel @ A @ B @ R3 ) ) ) ) ).
% listrel.Cons
thf(fact_6355_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel1 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_6356_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs: list @ A,Ys3: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( listrel @ A @ A @ R3 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ).
% rtrancl_listrel1_if_listrel
thf(fact_6357_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R3 ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X3: A,Y3: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys4: list @ B] :
( ( A22
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys4 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_6358_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X4: A,Y4: B,Xs3: list @ A,Ys2: list @ B] :
( ( A1
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y4 @ Ys2 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys2 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ) ).
% listrel.simps
thf(fact_6359_rel__fun__Collect__case__prodD,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,A5: A > B > $o,B4: C > D > $o,F3: A > C,G3: B > D,X6: set @ ( product_prod @ A @ B ),X: product_prod @ A @ B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B4 @ F3 @ G3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) )
=> ( ( member @ ( product_prod @ A @ B ) @ X @ X6 )
=> ( B4 @ ( comp @ A @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_fst @ A @ B ) @ X ) @ ( comp @ B @ D @ ( product_prod @ A @ B ) @ G3 @ ( product_snd @ A @ B ) @ X ) ) ) ) ) ).
% rel_fun_Collect_case_prodD
thf(fact_6360_listrel__subset__rtrancl__listrel1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R3 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ).
% listrel_subset_rtrancl_listrel1
thf(fact_6361_of__rat_Orsp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ A @ A @ ratrel
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ^ [X4: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X4 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X4 ) ) )
@ ^ [X4: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X4 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X4 ) ) ) ) ) ).
% of_rat.rsp
thf(fact_6362_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys3: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys3 ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [N: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N ) @ ( nth @ B @ Ys3 @ N ) ) @ R3 ) ) ) ) ).
% listrel_iff_nth
thf(fact_6363_fun_Oin__rel,axiom,
! [B: $tType,A: $tType,D: $tType,R2: A > B > $o,A3: D > A,B2: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R2
@ A3
@ B2 )
= ( ? [Z7: D > ( product_prod @ A @ B )] :
( ( member @ ( D > ( product_prod @ A @ B ) ) @ Z7
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X4: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X4 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) ) )
& ( ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) @ Z7 )
= A3 )
& ( ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) @ Z7 )
= B2 ) ) ) ) ).
% fun.in_rel
thf(fact_6364_fun_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: C > B > $o,I2: A > C,X: D > A,Y: D > B] :
( ( bNF_rel_fun @ D @ D @ C @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ Sb
@ ( comp @ A @ C @ D @ I2 @ X )
@ Y )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ^ [X4: A] : ( Sb @ ( I2 @ X4 ) )
@ X
@ Y ) ) ).
% fun.rel_map(1)
thf(fact_6365_sub_Orsp,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ( bNF_rel_fun @ num @ num @ int @ int
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ^ [M3: num,N: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M3 ) @ ( numeral_numeral @ int @ N ) )
@ ^ [M3: num,N: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M3 ) @ ( numeral_numeral @ int @ N ) ) ) ).
% sub.rsp
thf(fact_6366_Fract_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ratrel )
@ ^ [A7: int,B6: int] :
( if @ ( product_prod @ int @ int )
@ ( B6
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A7 @ B6 ) )
@ ^ [A7: int,B6: int] :
( if @ ( product_prod @ int @ int )
@ ( B6
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A7 @ B6 ) ) ) ).
% Fract.rsp
thf(fact_6367_dup_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 )
@ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ).
% dup.rsp
thf(fact_6368_less__eq__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less_eq @ int )
@ ( ord_less_eq @ int ) ) ).
% less_eq_integer.rsp
thf(fact_6369_less__eq__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less_eq @ nat )
@ ( ord_less_eq @ nat ) ) ).
% less_eq_natural.rsp
thf(fact_6370_less__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less @ nat )
@ ( ord_less @ nat ) ) ).
% less_natural.rsp
thf(fact_6371_less__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less @ int )
@ ( ord_less @ int ) ) ).
% less_integer.rsp
thf(fact_6372_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ^ [X4: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ).
% times_rat.rsp
thf(fact_6373_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ^ [X4: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ).
% plus_rat.rsp
thf(fact_6374_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T6: D > A] :
( ( comp @ A @ A @ D
@ ^ [X4: A] : X4
@ T6 )
= T6 ) ).
% fun.map_ident
thf(fact_6375_predicate2__transferD,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > B > $o,R22: C > D > $o,P: A > C > $o,Q: B > D > $o,A3: product_prod @ A @ B,A5: set @ ( product_prod @ A @ B ),B2: product_prod @ C @ D,B4: set @ ( product_prod @ C @ D )] :
( ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ R1
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ R22
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ P
@ Q )
=> ( ( member @ ( product_prod @ A @ B ) @ A3 @ A5 )
=> ( ( member @ ( product_prod @ C @ D ) @ B2 @ B4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R1 ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ B4 @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ R22 ) ) )
=> ( ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_fst @ C @ D @ B2 ) )
= ( Q @ ( product_snd @ A @ B @ A3 ) @ ( product_snd @ C @ D @ B2 ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_6376_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T6: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [X4: B] : X4
@ T6 )
= T6 ) ).
% prod.map_ident
thf(fact_6377_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_6378_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_6379_fun_Orel__mono,axiom,
! [D: $tType,B: $tType,A: $tType,R2: A > B > $o,Ra2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ Ra2 )
=> ( ord_less_eq @ ( ( D > A ) > ( D > B ) > $o )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R2 )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ Ra2 ) ) ) ).
% fun.rel_mono
thf(fact_6380_fun_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: A > C > $o,X: D > A,G3: B > C,Y: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ Sa
@ X
@ ( comp @ B @ C @ D @ G3 @ Y ) )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ^ [X4: A,Y4: B] : ( Sa @ X4 @ ( G3 @ Y4 ) )
@ X
@ Y ) ) ).
% fun.rel_map(2)
thf(fact_6381_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ R2
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_6382_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ( plus_plus @ rat ) ) ).
% plus_rat.transfer
thf(fact_6383_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% one_rat.transfer
thf(fact_6384_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% zero_rat.transfer
thf(fact_6385_Fract_Otransfer,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ pcr_rat )
@ ^ [A7: int,B6: int] :
( if @ ( product_prod @ int @ int )
@ ( B6
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A7 @ B6 ) )
@ fract ) ).
% Fract.transfer
thf(fact_6386_of__rat_Otransfer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ A @ A @ pcr_rat
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ^ [X4: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X4 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X4 ) ) )
@ ( field_char_0_of_rat @ A ) ) ) ).
% of_rat.transfer
thf(fact_6387_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_6388_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,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ( times_times @ rat ) ) ).
% times_rat.transfer
thf(fact_6389_positive_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [X4: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X4 ) @ ( product_snd @ int @ int @ X4 ) ) )
@ positive ) ).
% positive.transfer
thf(fact_6390_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_6391_fun__mono,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,C4: A > B > $o,A5: A > B > $o,B4: C > D > $o,D4: C > D > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ C4 @ A5 )
=> ( ( ord_less_eq @ ( C > D > $o ) @ B4 @ D4 )
=> ( ord_less_eq @ ( ( A > C ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B4 ) @ ( bNF_rel_fun @ A @ B @ C @ D @ C4 @ D4 ) ) ) ) ).
% fun_mono
thf(fact_6392_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y4 @ V3 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V3 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_6393_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_6394_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ pcr_int
@ ^ [N: nat] : ( product_Pair @ nat @ nat @ N @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_6395_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,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X4 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_6396_nat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ nat @ nat @ pcr_int
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ nat2 ) ).
% nat.transfer
thf(fact_6397_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_6398_of__int_Otransfer,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( ring_1_of_int @ A ) ) ) ).
% of_int.transfer
thf(fact_6399_less__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_6400_less__eq__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_6401_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y4 @ V3 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_6402_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_6403_rel__pred__comp__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_pred_comp @ A @ B )
= ( ^ [R6: A > B > $o,P4: B > $o,X4: A] :
? [Y4: B] :
( ( R6 @ X4 @ Y4 )
& ( P4 @ Y4 ) ) ) ) ).
% rel_pred_comp_def
thf(fact_6404_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys3: list @ B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys3 ) @ ( listrel @ A @ B @ R3 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [X4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys3 ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y4: A,Z7: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y4 @ Z7 ) @ R3 )
@ X4 ) ) ) ) ).
% listrel_iff_zip
thf(fact_6405_ball__empty,axiom,
! [A: $tType,P: A > $o,X7: A] :
( ( member @ A @ X7 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X7 ) ) ).
% ball_empty
thf(fact_6406_INF__bool__eq,axiom,
! [A: $tType] :
( ( ^ [A8: set @ A,F4: A > $o] : ( complete_Inf_Inf @ $o @ ( image2 @ A @ $o @ F4 @ A8 ) ) )
= ( ball @ A ) ) ).
% INF_bool_eq
thf(fact_6407_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A8: set @ A,P4: A > $o] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( P4 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_6408_INTER__eq,axiom,
! [B: $tType,A: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) )
= ( collect @ A
@ ^ [Y4: A] :
! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( member @ A @ Y4 @ ( B4 @ X4 ) ) ) ) ) ).
% INTER_eq
thf(fact_6409_Collect__ball__eq,axiom,
! [A: $tType,B: $tType,A5: set @ B,P: A > B > $o] :
( ( collect @ A
@ ^ [X4: A] :
! [Y4: B] :
( ( member @ B @ Y4 @ A5 )
=> ( P @ X4 @ Y4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] :
( collect @ A
@ ^ [X4: A] : ( P @ X4 @ Y4 ) )
@ A5 ) ) ) ).
% Collect_ball_eq
thf(fact_6410_Ball__fold,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( P @ X4 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S6: $o] :
( S6
& ( P @ K3 ) )
@ $true
@ A5 ) ) ) ).
% Ball_fold
thf(fact_6411_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A8: set @ A,P4: A > $o] : ( ord_less_eq @ ( set @ A ) @ A8 @ ( collect @ A @ P4 ) ) ) ) ).
% Ball_Collect
thf(fact_6412_Ball__comp__iff,axiom,
! [C: $tType,B: $tType,A: $tType,A5: B > ( set @ C ),F3: C > $o,G3: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X4: B] :
! [Y4: C] :
( ( member @ C @ Y4 @ ( A5 @ X4 ) )
=> ( F3 @ Y4 ) )
@ G3 )
= ( ^ [X4: A] :
! [Y4: C] :
( ( member @ C @ Y4 @ ( comp @ B @ ( set @ C ) @ A @ A5 @ G3 @ X4 ) )
=> ( F3 @ Y4 ) ) ) ) ).
% Ball_comp_iff
thf(fact_6413_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A8: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B6: A] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ord_less_eq @ A @ B6 @ X4 ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_6414_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A8: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B6: A] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ord_less_eq @ A @ X4 @ B6 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_6415_takeWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys3: list @ A] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys3 ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys3 ) ) ) )
& ( ~ ! [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ Xs ) )
=> ( P @ X7 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys3 ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% takeWhile_append
thf(fact_6416_dropWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys3: list @ A] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys3 ) )
= ( dropWhile @ A @ P @ Ys3 ) ) )
& ( ~ ! [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ Xs ) )
=> ( P @ X7 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys3 ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys3 ) ) ) ) ).
% dropWhile_append
thf(fact_6417_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z4: list @ A] : Y5 = Z4 )
= ( ^ [Xs3: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs3 @ Ys2 ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X4 ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_6418_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys3: list @ ( list @ A )] :
( ! [X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X3 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y4: list @ A,Z7: list @ A] :
( ( size_size @ ( list @ A ) @ Y4 )
= ( size_size @ ( list @ A ) @ Z7 ) )
@ X3 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys3 ) )
=> ( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys3 ) )
= ( Xs = Ys3 ) ) ) ) ).
% concat_eq_concat_iff
thf(fact_6419_concat__injective,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys3: list @ ( list @ A )] :
( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys3 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys3 ) )
=> ( ! [X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X3 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y4: list @ A,Z7: list @ A] :
( ( size_size @ ( list @ A ) @ Y4 )
= ( size_size @ ( list @ A ) @ Z7 ) )
@ X3 ) )
=> ( Xs = Ys3 ) ) ) ) ).
% concat_injective
thf(fact_6420_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Ys4 ) )
=> ( ( Y
= ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X3 @ Y4 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ Ys4 ) ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
thf(fact_6421_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
=> ~ ! [X3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Ys4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ Ys4 ) ) )
=> ~ ( ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X3 @ Xa2 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
thf(fact_6422_Inter__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X4: A] :
! [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ A8 )
=> ( member @ A @ X4 @ Y4 ) ) ) ) ) ).
% Inter_eq
thf(fact_6423_Sup__int__def,axiom,
( ( complete_Sup_Sup @ int )
= ( ^ [X11: set @ int] :
( the @ int
@ ^ [X4: int] :
( ( member @ int @ X4 @ X11 )
& ! [Y4: int] :
( ( member @ int @ Y4 @ X11 )
=> ( ord_less_eq @ int @ Y4 @ X4 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_6424_Union__maximal__sets,axiom,
! [A: $tType,F14: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F14 )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [T8: set @ A] :
( ( member @ ( set @ A ) @ T8 @ F14 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ F14 )
=> ~ ( ord_less @ ( set @ A ) @ T8 @ X4 ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ F14 ) ) ) ).
% Union_maximal_sets
thf(fact_6425_Sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A5: set @ ( set @ A )] :
( ( complete_Sup_Sup @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) @ A5 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu2: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu2
= ( image2 @ ( set @ A ) @ A @ F4 @ A5 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Sup_Inf
thf(fact_6426_Inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A5: set @ ( set @ A )] :
( ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu2: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu2
= ( image2 @ ( set @ A ) @ A @ F4 @ A5 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Inf_Sup
thf(fact_6427_Inf__filter__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( filter @ A ) )
= ( ^ [S5: set @ ( filter @ A )] :
( complete_Sup_Sup @ ( filter @ A )
@ ( collect @ ( filter @ A )
@ ^ [F7: filter @ A] :
! [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ S5 )
=> ( ord_less_eq @ ( filter @ A ) @ F7 @ X4 ) ) ) ) ) ) ).
% Inf_filter_def
thf(fact_6428_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ ( set @ A )] :
( ord_less_eq @ A
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu2: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu2
= ( image2 @ ( set @ A ) @ A @ F4 @ A5 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) ) ) ) ).
% Sup_Inf_le
thf(fact_6429_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A5: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu2: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu2
= ( image2 @ ( set @ A ) @ A @ F4 @ A5 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_6430_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A5: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu2: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu2
= ( image2 @ ( set @ A ) @ A @ F4 @ A5 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_6431_SUP__INF__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G3: B > A,A5: set @ ( set @ B )] :
( ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [X4: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ X4 ) )
@ A5 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [X4: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ X4 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu2: set @ B] :
? [F4: ( set @ B ) > B] :
( ( Uu2
= ( image2 @ ( set @ B ) @ B @ F4 @ A5 ) )
& ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ A5 )
=> ( member @ B @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% SUP_INF_set
thf(fact_6432_INF__SUP__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G3: B > A,A5: set @ ( set @ B )] :
( ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [B7: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B7 ) )
@ A5 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [B7: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B7 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu2: set @ B] :
? [F4: ( set @ B ) > B] :
( ( Uu2
= ( image2 @ ( set @ B ) @ B @ F4 @ A5 ) )
& ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ A5 )
=> ( member @ B @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% INF_SUP_set
thf(fact_6433_option__Inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A5: set @ ( set @ ( option @ A ) )] :
( ord_less_eq @ ( option @ A ) @ ( complete_Inf_Inf @ ( option @ A ) @ ( image2 @ ( set @ ( option @ A ) ) @ ( option @ A ) @ ( complete_Sup_Sup @ ( option @ A ) ) @ A5 ) )
@ ( complete_Sup_Sup @ ( option @ A )
@ ( image2 @ ( set @ ( option @ A ) ) @ ( option @ A ) @ ( complete_Inf_Inf @ ( option @ A ) )
@ ( collect @ ( set @ ( option @ A ) )
@ ^ [Uu2: set @ ( option @ A )] :
? [F4: ( set @ ( option @ A ) ) > ( option @ A )] :
( ( Uu2
= ( image2 @ ( set @ ( option @ A ) ) @ ( option @ A ) @ F4 @ A5 ) )
& ! [X4: set @ ( option @ A )] :
( ( member @ ( set @ ( option @ A ) ) @ X4 @ A5 )
=> ( member @ ( option @ A ) @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% option_Inf_Sup
thf(fact_6434_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ! [X3: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Ys4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ Ys4 ) ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X3 @ Xa3 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
thf(fact_6435_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P6: A > $o,X4: A] :
( ? [Y4: A] :
( ( X4
= ( F3 @ Y4 ) )
& ( P6 @ Y4 ) )
| ? [M10: set @ A] :
( ( X4
= ( complete_Sup_Sup @ A @ M10 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M10 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ M10 )
=> ( P6 @ Y4 ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_6436_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y4 @ V3 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V3 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y4 @ V3 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V3 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_6437_intrel__iff,axiom,
! [X: nat,Y: nat,U: nat,V2: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ U @ V2 ) )
= ( ( plus_plus @ nat @ X @ V2 )
= ( plus_plus @ nat @ U @ Y ) ) ) ).
% intrel_iff
thf(fact_6438_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_6439_chain__empty,axiom,
! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ).
% chain_empty
thf(fact_6440_chain__compr,axiom,
! [A: $tType,Ord: A > A > $o,A5: set @ A,P: A > $o] :
( ( comple1602240252501008431_chain @ A @ Ord @ A5 )
=> ( comple1602240252501008431_chain @ A @ Ord
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) ) ) ) ) ).
% chain_compr
thf(fact_6441_ccpo__Sup__mono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A5 )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ B4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ B4 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B4 ) ) ) ) ) ) ).
% ccpo_Sup_mono
thf(fact_6442_ccpo__Sup__upper,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A5: set @ A,X: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_6443_ccpo__Sup__least,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A5: set @ A,Z2: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ Z2 ) ) ) ) ).
% ccpo_Sup_least
thf(fact_6444_chain__subset,axiom,
! [A: $tType,Ord: A > A > $o,A5: set @ A,B4: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( comple1602240252501008431_chain @ A @ Ord @ B4 ) ) ) ).
% chain_subset
thf(fact_6445_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,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X4 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X4 ) ) ) ).
% uminus_int.rsp
thf(fact_6446_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_6447_nat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ nat @ nat @ intrel
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ).
% nat.rsp
thf(fact_6448_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_6449_of__int_Orsp,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ).
% of_int.rsp
thf(fact_6450_intrel__def,axiom,
( intrel
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] :
( ( plus_plus @ nat @ X4 @ V3 )
= ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ).
% intrel_def
thf(fact_6451_less__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ).
% less_int.rsp
thf(fact_6452_less__eq__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_6453_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_6454_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,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y4 @ V3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y4 @ V3 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_6455_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A5: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A5 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A5 ) @ A5 ) ) ) ) ) ).
% in_chain_finite
thf(fact_6456_iteratesp__def,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F4: A > A] :
( complete_lattice_lfp @ ( A > $o )
@ ^ [P6: A > $o,X4: A] :
( ? [Y4: A] :
( ( X4
= ( F4 @ Y4 ) )
& ( P6 @ Y4 ) )
| ? [M10: set @ A] :
( ( X4
= ( complete_Sup_Sup @ A @ M10 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M10 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ M10 )
=> ( P6 @ Y4 ) ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_6457_iteratesp_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M6: set @ A,F3: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ M6 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X3 ) )
=> ( comple7512665784863727008ratesp @ A @ F3 @ ( complete_Sup_Sup @ A @ M6 ) ) ) ) ) ).
% iteratesp.Sup
thf(fact_6458_iteratesp_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A,A3: A] :
( ( comple7512665784863727008ratesp @ A @ F3 @ A3 )
=> ( ! [X3: A] :
( ( A3
= ( F3 @ X3 ) )
=> ~ ( comple7512665784863727008ratesp @ A @ F3 @ X3 ) )
=> ~ ! [M13: set @ A] :
( ( A3
= ( complete_Sup_Sup @ A @ M13 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M13 )
=> ~ ! [X7: A] :
( ( member @ A @ X7 @ M13 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X7 ) ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_6459_iteratesp_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F4: A > A,A7: A] :
( ? [X4: A] :
( ( A7
= ( F4 @ X4 ) )
& ( comple7512665784863727008ratesp @ A @ F4 @ X4 ) )
| ? [M10: set @ A] :
( ( A7
= ( complete_Sup_Sup @ A @ M10 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M10 )
& ! [X4: A] :
( ( member @ A @ X4 @ M10 )
=> ( comple7512665784863727008ratesp @ A @ F4 @ X4 ) ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_6460_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B6: A,A8: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A8 @ ( insert @ A @ B6 @ ( bot_bot @ ( set @ A ) ) ) ) @ B6
@ ( the @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ B6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_6461_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R4 ) ) ) ) ) ) ).
% listrel_def
thf(fact_6462_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R3 ) )
= ( ^ [X4: list @ A,Y4: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X4 @ Y4 ) @ ( listrel @ A @ B @ R3 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_6463_listrel1__subset__listrel,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R7 )
=> ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R7 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R3 ) @ ( listrel @ A @ A @ R7 ) ) ) ) ).
% listrel1_subset_listrel
thf(fact_6464_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu2: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X11: set @ A,Y10: set @ A] :
( ( Uu2
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X11 @ Y10 ) )
& ( X11
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ Y10 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ X11 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R4 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_6465_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X7: A] :
( ( member @ A @ X7 @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X7 ) ) ).
% bex_empty
thf(fact_6466_finite__Collect__bex,axiom,
! [B: $tType,A: $tType,A5: set @ A,Q: B > A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
? [Y4: A] :
( ( member @ A @ Y4 @ A5 )
& ( Q @ X4 @ Y4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Y4: B] : ( Q @ Y4 @ X4 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_6467_bex__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) )
& ( P @ X4 ) ) )
= ( ? [X11: A] : ( P @ X11 ) ) ) ).
% bex_UNIV
thf(fact_6468_Image__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o,A5: set @ B] :
( ( image @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) @ A5 )
= ( collect @ A
@ ^ [Y4: A] :
? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( P @ X4 @ Y4 ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_6469_SUP__bool__eq,axiom,
! [A: $tType] :
( ( ^ [A8: set @ A,F4: A > $o] : ( complete_Sup_Sup @ $o @ ( image2 @ A @ $o @ F4 @ A8 ) ) )
= ( bex @ A ) ) ).
% SUP_bool_eq
thf(fact_6470_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A] :
( ( vimage @ A @ B @ F3 @ ( image2 @ A @ B @ F3 @ A5 ) )
= ( collect @ A
@ ^ [Y4: A] :
? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ( F3 @ X4 )
= ( F3 @ Y4 ) ) ) ) ) ).
% vimage_image_eq
thf(fact_6471_Collect__bex__eq,axiom,
! [A: $tType,B: $tType,A5: set @ B,P: A > B > $o] :
( ( collect @ A
@ ^ [X4: A] :
? [Y4: B] :
( ( member @ B @ Y4 @ A5 )
& ( P @ X4 @ Y4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] :
( collect @ A
@ ^ [X4: A] : ( P @ X4 @ Y4 ) )
@ A5 ) ) ) ).
% Collect_bex_eq
thf(fact_6472_UNION__eq,axiom,
! [B: $tType,A: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) )
= ( collect @ A
@ ^ [Y4: A] :
? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( member @ A @ Y4 @ ( B4 @ X4 ) ) ) ) ) ).
% UNION_eq
thf(fact_6473_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ A @ B )
= ( ^ [F4: A > B,A8: set @ A] :
( collect @ B
@ ^ [Y4: B] :
? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ( Y4
= ( F4 @ X4 ) ) ) ) ) ) ).
% image_def
thf(fact_6474_Union__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X4: A] :
? [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ A8 )
& ( member @ A @ X4 @ Y4 ) ) ) ) ) ).
% Union_eq
thf(fact_6475_Bex__def__raw,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A8: set @ A,P4: A > $o] :
? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ( P4 @ X4 ) ) ) ) ).
% Bex_def_raw
thf(fact_6476_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B ),S6: set @ A] :
( collect @ B
@ ^ [Y4: B] :
? [X4: A] :
( ( member @ A @ X4 @ S6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R4 ) ) ) ) ) ).
% Image_def
thf(fact_6477_refl__on__Un,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),B4: set @ A,S2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A5 @ R3 )
=> ( ( refl_on @ A @ B4 @ S2 )
=> ( refl_on @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 ) ) ) ) ).
% refl_on_Un
thf(fact_6478_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_6479_refl__onD,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ A5 @ R3 )
=> ( ( member @ A @ A3 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R3 ) ) ) ).
% refl_onD
thf(fact_6480_refl__onD1,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A5 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ X @ A5 ) ) ) ).
% refl_onD1
thf(fact_6481_refl__onD2,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A5 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ Y @ A5 ) ) ) ).
% refl_onD2
thf(fact_6482_Bex__def,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A8: set @ A,P4: A > $o] :
? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ( P4 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_6483_refl__on__Id__on,axiom,
! [A: $tType,A5: set @ A] : ( refl_on @ A @ A5 @ ( id_on @ A @ A5 ) ) ).
% refl_on_Id_on
thf(fact_6484_refl__Id,axiom,
! [A: $tType] : ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ ( id2 @ A ) ) ).
% refl_Id
thf(fact_6485_refl__on__Int,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),B4: set @ A,S2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A5 @ R3 )
=> ( ( refl_on @ A @ B4 @ S2 )
=> ( refl_on @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 ) ) ) ) ).
% refl_on_Int
thf(fact_6486_Bex__fold,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S6: $o] :
( S6
| ( P @ K3 ) )
@ $false
@ A5 ) ) ) ).
% Bex_fold
thf(fact_6487_nths__nths,axiom,
! [A: $tType,Xs: list @ A,A5: set @ nat,B4: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs @ A5 ) @ B4 )
= ( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I4: nat] :
( ( member @ nat @ I4 @ A5 )
& ( member @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I8: nat] :
( ( member @ nat @ I8 @ A5 )
& ( ord_less @ nat @ I8 @ I4 ) ) ) )
@ B4 ) ) ) ) ) ).
% nths_nths
thf(fact_6488_refl__reflcl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ).
% refl_reflcl
thf(fact_6489_refl__onI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 ) )
=> ( refl_on @ A @ A5 @ R3 ) ) ) ).
% refl_onI
thf(fact_6490_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu2: A] : A8 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R4 ) ) ) ) ) ).
% refl_on_def
thf(fact_6491_max__extp_Ocases,axiom,
! [A: $tType,R2: A > A > $o,A1: set @ A,A22: set @ A] :
( ( max_extp @ A @ R2 @ A1 @ A22 )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X7: A] :
( ( member @ A @ X7 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( R2 @ X7 @ Xa3 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_6492_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R6: A > A > $o,A12: set @ A,A23: set @ A] :
( ( finite_finite2 @ A @ A12 )
& ( finite_finite2 @ A @ A23 )
& ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A12 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A23 )
& ( R6 @ X4 @ Y4 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_6493_max__extp_Omax__extI,axiom,
! [A: $tType,X6: set @ A,Y7: set @ A,R2: A > A > $o] :
( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( Y7
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y7 )
& ( R2 @ X3 @ Xa2 ) ) )
=> ( max_extp @ A @ R2 @ X6 @ Y7 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_6494_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,Z7: A] :
( ( member @ A @ Y4 @ A8 )
& ( member @ A @ Z7 @ A8 ) )
@ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R4 ) ) ) ) ) ).
% refl_on_def'
thf(fact_6495_refl__on__reflcl__Image,axiom,
! [A: $tType,B4: set @ A,A5: set @ ( product_prod @ A @ A ),C4: set @ A] :
( ( refl_on @ A @ B4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ B4 )
=> ( ( image @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ ( id2 @ A ) ) @ C4 )
= ( image @ A @ A @ A5 @ C4 ) ) ) ) ).
% refl_on_reflcl_Image
thf(fact_6496_refl__on__UNION,axiom,
! [B: $tType,A: $tType,S: set @ A,A5: A > ( set @ B ),R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( refl_on @ B @ ( A5 @ X3 ) @ ( R3 @ X3 ) ) )
=> ( refl_on @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ S ) ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ S ) ) ) ) ).
% refl_on_UNION
thf(fact_6497_refl__on__INTER,axiom,
! [B: $tType,A: $tType,S: set @ A,A5: A > ( set @ B ),R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( refl_on @ B @ ( A5 @ X3 ) @ ( R3 @ X3 ) ) )
=> ( refl_on @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ S ) ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ S ) ) ) ) ).
% refl_on_INTER
thf(fact_6498_refl__on__singleton,axiom,
! [A: $tType,X: A] : ( refl_on @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_6499_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F4: A > ( option @ B ),A8: set @ A] :
( collect @ B
@ ^ [B6: B] :
? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ( ( F4 @ X4 )
= ( some @ B @ B6 ) ) ) ) ) ) ).
% map_project_def
thf(fact_6500_refl__on__domain,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ A5 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ A @ A3 @ A5 )
& ( member @ A @ B2 @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_6501_linear__order__on__singleton,axiom,
! [A: $tType,X: A] : ( order_679001287576687338der_on @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_6502_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X11: set @ A,Y10: set @ A] :
( ( finite_finite2 @ A @ X11 )
& ( finite_finite2 @ A @ Y10 )
& ( Y10
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ X11 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ Y10 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_6503_ball__UNIV,axiom,
! [A: $tType,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) )
=> ( P @ X4 ) ) )
= ( ! [X11: A] : ( P @ X11 ) ) ) ).
% ball_UNIV
thf(fact_6504_Ball__def__raw,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A8: set @ A,P4: A > $o] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( P4 @ X4 ) ) ) ) ).
% Ball_def_raw
thf(fact_6505_rel__fun__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ C @ B @ D )
= ( ^ [A8: A > C > $o,B7: B > D > $o,F4: A > B,G4: C > D] :
! [X4: A,Y4: C] :
( ( A8 @ X4 @ Y4 )
=> ( B7 @ ( F4 @ X4 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% rel_fun_def
thf(fact_6506_rel__fun__eq__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R2: B > C > $o] :
( ( bNF_rel_fun @ A @ A @ B @ C
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ R2 )
= ( ^ [F4: A > B,G4: A > C] :
! [X4: A] : ( R2 @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ).
% rel_fun_eq_rel
thf(fact_6507_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_6508_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B,D3: B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F4: A > B] :
! [X4: A] :
( ( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F4 @ X4 ) @ B4 ) )
& ( ~ ( member @ A @ X4 @ A5 )
=> ( ( F4 @ X4 )
= D3 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_6509_Collect__all__eq,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ( collect @ A
@ ^ [X4: A] :
! [X11: B] : ( P @ X4 @ X11 ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] :
( collect @ A
@ ^ [X4: A] : ( P @ X4 @ Y4 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_all_eq
thf(fact_6510_Func__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_Func @ A @ B )
= ( ^ [A8: set @ A,B7: set @ B] :
( collect @ ( A > B )
@ ^ [F4: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ B @ ( F4 @ X4 ) @ B7 ) )
& ! [A7: A] :
( ~ ( member @ A @ A7 @ A8 )
=> ( ( F4 @ A7 )
= ( undefined @ B ) ) ) ) ) ) ) ).
% Func_def
thf(fact_6511_linear__order__on__Restr,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( order_679001287576687338der_on @ A @ A5 @ R3 )
=> ( order_679001287576687338der_on @ A @ ( inf_inf @ ( set @ A ) @ A5 @ ( order_above @ A @ R3 @ X ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_above @ A @ R3 @ X )
@ ^ [Uu2: A] : ( order_above @ A @ R3 @ X ) ) ) ) ) ).
% linear_order_on_Restr
thf(fact_6512_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P4: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P4 @ X4 )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_6513_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_6514_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_6515_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_6516_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_6517_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_6518_above__def,axiom,
! [A: $tType] :
( ( order_above @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B6 ) @ R4 ) ) ) ) ).
% above_def
thf(fact_6519_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq2: A > A > $o,P4: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P4 @ X4 )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( Less_eq2 @ X4 @ Y4 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_6520_transfer__bforall__def,axiom,
! [A: $tType] :
( ( transfer_bforall @ A )
= ( ^ [P4: A > $o,Q3: A > $o] :
! [X4: A] :
( ( P4 @ X4 )
=> ( Q3 @ X4 ) ) ) ) ).
% transfer_bforall_def
thf(fact_6521_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_6522_subset__mset_OLeast__def,axiom,
! [A: $tType,P: ( multiset @ A ) > $o] :
( ( least @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ P )
= ( the @ ( multiset @ A )
@ ^ [X4: multiset @ A] :
( ( P @ X4 )
& ! [Y4: multiset @ A] :
( ( P @ Y4 )
=> ( subseteq_mset @ A @ X4 @ Y4 ) ) ) ) ) ).
% subset_mset.Least_def
thf(fact_6523_Total__subset__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) )
=> ( ( R3
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
| ? [A6: A] :
( R3
= ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_6524_lists__empty,axiom,
! [A: $tType] :
( ( lists @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% lists_empty
thf(fact_6525_lists__Int__eq,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( lists @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ A5 ) @ ( lists @ A @ B4 ) ) ) ).
% lists_Int_eq
thf(fact_6526_Field__square,axiom,
! [A: $tType,X: set @ A] :
( ( field2 @ A
@ ( product_Sigma @ A @ A @ X
@ ^ [Uu2: A] : X ) )
= X ) ).
% Field_square
thf(fact_6527_Field__empty,axiom,
! [A: $tType] :
( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_6528_finite__Field__eq__finite,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ ( field2 @ A @ R2 ) )
= ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 ) ) ).
% finite_Field_eq_finite
thf(fact_6529_Field__Un,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 ) )
= ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R3 ) @ ( field2 @ A @ S2 ) ) ) ).
% Field_Un
thf(fact_6530_Field__Union,axiom,
! [A: $tType,R2: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( field2 @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ A ) @ ( field2 @ A ) @ R2 ) ) ) ).
% Field_Union
thf(fact_6531_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) )
= ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) ) ) ).
% Field_insert
thf(fact_6532_lists__IntI,axiom,
! [A: $tType,L: list @ A,A5: set @ A,B4: set @ A] :
( ( member @ ( list @ A ) @ L @ ( lists @ A @ A5 ) )
=> ( ( member @ ( list @ A ) @ L @ ( lists @ A @ B4 ) )
=> ( member @ ( list @ A ) @ L @ ( lists @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ).
% lists_IntI
thf(fact_6533_FieldI2,axiom,
! [A: $tType,I2: A,J: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R2 )
=> ( member @ A @ J @ ( field2 @ A @ R2 ) ) ) ).
% FieldI2
thf(fact_6534_FieldI1,axiom,
! [A: $tType,I2: A,J: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R2 )
=> ( member @ A @ I2 @ ( field2 @ A @ R2 ) ) ) ).
% FieldI1
thf(fact_6535_mono__Field,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 )
=> ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ R3 ) @ ( field2 @ A @ S2 ) ) ) ).
% mono_Field
thf(fact_6536_lists__image__witness,axiom,
! [A: $tType,B: $tType,X: list @ A,F3: B > A,Q: set @ B] :
( ( member @ ( list @ A ) @ X @ ( lists @ A @ ( image2 @ B @ A @ F3 @ Q ) ) )
=> ~ ! [Xo2: list @ B] :
( ( member @ ( list @ B ) @ Xo2 @ ( lists @ B @ Q ) )
=> ( X
!= ( map @ B @ A @ F3 @ Xo2 ) ) ) ) ).
% lists_image_witness
thf(fact_6537_finite__Field,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( finite_finite2 @ A @ ( field2 @ A @ R3 ) ) ) ).
% finite_Field
thf(fact_6538_R__subset__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R2 )
@ ^ [Uu2: A] : ( field2 @ A @ R2 ) ) ) ).
% R_subset_Field
thf(fact_6539_Restr__Field,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R3 )
@ ^ [Uu2: A] : ( field2 @ A @ R3 ) ) )
= R3 ) ).
% Restr_Field
thf(fact_6540_lists__eq__set,axiom,
! [A: $tType] :
( ( lists @ A )
= ( ^ [A8: set @ A] :
( collect @ ( list @ A )
@ ^ [Xs3: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A8 ) ) ) ) ).
% lists_eq_set
thf(fact_6541_Field__not__elem,axiom,
! [A: $tType,V2: A,R2: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ V2 @ ( field2 @ A @ R2 ) )
=> ! [X7: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X7 @ R2 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,Z7: A] :
( ( Y4 != V2 )
& ( Z7 != V2 ) )
@ X7 ) ) ) ).
% Field_not_elem
thf(fact_6542_fst__in__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_fst @ A @ A ) @ R2 ) @ ( field2 @ A @ R2 ) ) ).
% fst_in_Field
thf(fact_6543_snd__in__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_snd @ A @ A ) @ R2 ) @ ( field2 @ A @ R2 ) ) ).
% snd_in_Field
thf(fact_6544_lists__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( lists @ A @ A5 ) @ ( lists @ A @ B4 ) ) ) ).
% lists_mono
thf(fact_6545_rel__restrict__Int__empty,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ ( field2 @ A @ R2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( rel_restrict @ A @ R2 @ A5 )
= R2 ) ) ).
% rel_restrict_Int_empty
thf(fact_6546_Field__rel__restrict,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ ( rel_restrict @ A @ R2 @ A5 ) ) @ ( minus_minus @ ( set @ A ) @ ( field2 @ A @ R2 ) @ A5 ) ) ).
% Field_rel_restrict
thf(fact_6547_Field__Restr__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ord_less_eq @ ( set @ A )
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
@ A5 ) ).
% Field_Restr_subset
thf(fact_6548_trancl__subset__Field2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 )
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R3 )
@ ^ [Uu2: A] : ( field2 @ A @ R3 ) ) ) ).
% trancl_subset_Field2
thf(fact_6549_Field__natLeq__on,axiom,
! [N2: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y4: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y4 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) )
= ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N2 ) ) ) ).
% Field_natLeq_on
thf(fact_6550_Refl__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( refl_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ) ).
% Refl_Restr
thf(fact_6551_Total__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( total_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( total_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ) ).
% Total_Restr
thf(fact_6552_total__on__imp__Total__Restr,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A5 @ R3 )
=> ( total_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ) ).
% total_on_imp_Total_Restr
thf(fact_6553_Linear__order__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_679001287576687338der_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ) ).
% Linear_order_Restr
thf(fact_6554_lists__of__len__fin1,axiom,
! [A: $tType,P: set @ A,N2: nat] :
( ( finite_finite2 @ A @ P )
=> ( finite_finite2 @ ( list @ A )
@ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
@ ( collect @ ( list @ A )
@ ^ [L2: list @ A] :
( ( size_size @ ( list @ A ) @ L2 )
= N2 ) ) ) ) ) ).
% lists_of_len_fin1
thf(fact_6555_lists__of__len__fin2,axiom,
! [A: $tType,P: set @ A,N2: nat] :
( ( finite_finite2 @ A @ P )
=> ( finite_finite2 @ ( list @ A )
@ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
@ ( collect @ ( list @ A )
@ ^ [L2: list @ A] :
( N2
= ( size_size @ ( list @ A ) @ L2 ) ) ) ) ) ) ).
% lists_of_len_fin2
thf(fact_6556_rtrancl__Image__in__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),V: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ V ) @ ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R2 ) @ V ) ) ).
% rtrancl_Image_in_Field
thf(fact_6557_Linear__order__in__diff__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_6558_Total__Id__Field,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) )
=> ( ( field2 @ A @ R3 )
= ( field2 @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ) ).
% Total_Id_Field
thf(fact_6559_Refl__Field__Restr2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R3 ) )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
= A5 ) ) ) ).
% Refl_Field_Restr2
thf(fact_6560_Refl__Field__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
= ( inf_inf @ ( set @ A ) @ ( field2 @ A @ R3 ) @ A5 ) ) ) ).
% Refl_Field_Restr
thf(fact_6561_listrel__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R3 )
@ ( product_Sigma @ ( list @ A ) @ ( list @ A ) @ ( lists @ A @ A5 )
@ ^ [Uu2: list @ A] : ( lists @ A @ A5 ) ) ) ) ).
% listrel_subset
thf(fact_6562_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B6: A] :
( ( member @ A @ B6 @ ( field2 @ A @ R4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ( B6 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ X4 ) @ R4 ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_6563_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B6: A] :
( ( member @ A @ B6 @ ( field2 @ A @ R4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ X4 ) @ R4 ) ) ) ) ) ) ).
% Under_def
thf(fact_6564_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B6: A] :
( ( member @ A @ B6 @ ( field2 @ A @ R4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B6 ) @ R4 ) ) ) ) ) ) ).
% Above_def
thf(fact_6565_subset__mset_OGreatest__def,axiom,
! [A: $tType,P: ( multiset @ A ) > $o] :
( ( greatest @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ P )
= ( the @ ( multiset @ A )
@ ^ [X4: multiset @ A] :
( ( P @ X4 )
& ! [Y4: multiset @ A] :
( ( P @ Y4 )
=> ( subseteq_mset @ A @ Y4 @ X4 ) ) ) ) ) ).
% subset_mset.Greatest_def
thf(fact_6566_order_OGreatest_Ocong,axiom,
! [A: $tType] :
( ( greatest @ A )
= ( greatest @ A ) ) ).
% order.Greatest.cong
thf(fact_6567_subset__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_6568_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [A12: A,A23: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B17: A,B24: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A12 @ ( insert @ A @ A23 @ ( insert @ A @ B17 @ ( insert @ A @ B24 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R4 ) )
& ( ( ( A12 = B17 )
& ( A23 = B24 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B17 @ B24 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B17 @ B24 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B17 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B17 @ B24 ) )
& ( A12 = B17 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B24 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_6569_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_6570_Field__bsqr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( field2 @ ( product_prod @ A @ A ) @ ( bNF_Wellorder_bsqr @ A @ R3 ) )
= ( product_Sigma @ A @ A @ ( field2 @ A @ R3 )
@ ^ [Uu2: A] : ( field2 @ A @ R3 ) ) ) ).
% Field_bsqr
thf(fact_6571_Preorder__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_preorder_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ) ).
% Preorder_Restr
thf(fact_6572_subset__Image__Image__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ A5 ) @ ( image @ A @ A @ R3 @ B4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ B4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R3 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_6573_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R4 ) )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A8 )
& ( X4 != Y4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R4 ) ) ) ) ) ).
% cofinal_def
thf(fact_6574_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
= ( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R3 ) )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_6575_wf__empty,axiom,
! [A: $tType] : ( wf @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% wf_empty
thf(fact_6576_brk__rel__wf,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( wf @ ( product_prod @ $o @ A ) @ ( brk_rel @ A @ A @ R2 ) ) ) ).
% brk_rel_wf
thf(fact_6577_wf__insert,axiom,
! [A: $tType,Y: A,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R3 ) )
= ( ( wf @ A @ R3 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% wf_insert
thf(fact_6578_wf__Int2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( wf @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R3 ) ) ) ).
% wf_Int2
thf(fact_6579_wf__Int1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( wf @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R7 ) ) ) ).
% wf_Int1
thf(fact_6580_wf__if__measure,axiom,
! [A: $tType,P: A > $o,F3: A > nat,G3: A > A] :
( ! [X3: A] :
( ( P @ X3 )
=> ( ord_less @ nat @ ( F3 @ ( G3 @ X3 ) ) @ ( F3 @ X3 ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,X4: A] :
( ( P @ X4 )
& ( Y4
= ( G3 @ X4 ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_6581_wf__subset__mset__rel,axiom,
! [A: $tType] : ( wf @ ( multiset @ A ) @ ( collect @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_case_prod @ ( multiset @ A ) @ ( multiset @ A ) @ $o @ ( subset_mset @ A ) ) ) ) ).
% wf_subset_mset_rel
thf(fact_6582_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_6583_wf__no__loop,axiom,
! [B: $tType,R2: set @ ( product_prod @ B @ B )] :
( ( ( relcomp @ B @ B @ B @ R2 @ R2 )
= ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
=> ( wf @ B @ R2 ) ) ).
% wf_no_loop
thf(fact_6584_wfE__min_H,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Q: set @ A] :
( ( wf @ A @ R2 )
=> ( ( Q
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R2 )
=> ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_6585_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
~ ? [F4: nat > A] :
! [I4: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ ( suc @ I4 ) ) @ ( F4 @ I4 ) ) @ R4 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_6586_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),F3: nat > A] :
( ( wf @ A @ R3 )
=> ~ ! [K2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ ( suc @ K2 ) ) @ ( F3 @ K2 ) ) @ R3 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_6587_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [P4: A > $o] :
( ! [X4: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R4 )
=> ( P4 @ Y4 ) )
=> ( P4 @ X4 ) )
=> ! [X11: A] : ( P4 @ X11 ) ) ) ) ).
% wf_def
thf(fact_6588_wfE__min,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Q: set @ A] :
( ( wf @ A @ R2 )
=> ( ( member @ A @ X @ Q )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R2 )
=> ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_6589_wfI__min,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Q2: set @ A] :
( ( member @ A @ X3 @ Q2 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Q2 )
& ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa2 ) @ R2 )
=> ~ ( member @ A @ Y3 @ Q2 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wfI_min
thf(fact_6590_wfUNIVI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [P2: A > $o,X3: A] :
( ! [Xa2: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa2 ) @ R3 )
=> ( P2 @ Y3 ) )
=> ( P2 @ Xa2 ) )
=> ( P2 @ X3 ) )
=> ( wf @ A @ R3 ) ) ).
% wfUNIVI
thf(fact_6591_wf__asym,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,X: A] :
( ( wf @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R3 ) ) ) ).
% wf_asym
thf(fact_6592_wf__induct,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct
thf(fact_6593_wf__irrefl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R3 ) ) ).
% wf_irrefl
thf(fact_6594_wf__not__sym,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,X: A] :
( ( wf @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R3 ) ) ) ).
% wf_not_sym
thf(fact_6595_wf__not__refl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R3 ) ) ).
% wf_not_refl
thf(fact_6596_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [Q3: set @ A] :
( ? [X4: A] : ( member @ A @ X4 @ Q3 )
=> ? [X4: A] :
( ( member @ A @ X4 @ Q3 )
& ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R4 )
=> ~ ( member @ A @ Y4 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_6597_wf__induct__rule,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct_rule
thf(fact_6598_wf__union__merge,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) )
= ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ R2 ) @ ( relcomp @ A @ A @ A @ S @ R2 ) ) @ S ) ) ) ).
% wf_union_merge
thf(fact_6599_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_6600_wf__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ P3 @ R3 )
=> ( wf @ A @ P3 ) ) ) ).
% wf_subset
thf(fact_6601_wf__relcomp__compatible,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S ) @ ( relcomp @ A @ A @ A @ S @ R2 ) )
=> ( wf @ A @ ( relcomp @ A @ A @ A @ S @ R2 ) ) ) ) ).
% wf_relcomp_compatible
thf(fact_6602_wf__bounded__measure,axiom,
! [A: $tType,R3: 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 ) @ R3 )
=> ( ( 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 @ R3 ) ) ).
% wf_bounded_measure
thf(fact_6603_wfE__pf,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( wf @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( image @ A @ A @ R2 @ A5 ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% wfE_pf
thf(fact_6604_wfI__pf,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [A13: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A13 @ ( image @ A @ A @ R2 @ A13 ) )
=> ( A13
= ( bot_bot @ ( set @ A ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wfI_pf
thf(fact_6605_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M: B > A] :
( ( wf @ A @ R3 )
=> ( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) )
=> ( ( P @ K )
=> ? [X3: B] :
( ( P @ X3 )
& ! [Y6: B] :
( ( P @ Y6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X3 ) @ ( M @ Y6 ) ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_6606_wf__union__compatible,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( ( wf @ A @ S )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S ) @ R2 )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) ) ) ) ) ).
% wf_union_compatible
thf(fact_6607_wfI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : B4 ) )
=> ( ! [X3: A,P2: A > $o] :
( ! [Xa2: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa2 ) @ R3 )
=> ( P2 @ Y3 ) )
=> ( P2 @ Xa2 ) )
=> ( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ X3 @ B4 )
=> ( P2 @ X3 ) ) ) )
=> ( wf @ A @ R3 ) ) ) ).
% wfI
thf(fact_6608_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [A8: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R4 ) )
& ( A8
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A8 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_6609_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R3: 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 ) @ R3 )
=> ( ( 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 @ R3 ) ) ).
% wf_bounded_set
thf(fact_6610_qc__wf__relto__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S ) @ ( relcomp @ A @ A @ A @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) ) @ R2 ) )
=> ( ( wf @ A @ ( relcomp @ A @ A @ A @ ( transitive_rtrancl @ A @ S ) @ ( relcomp @ A @ A @ A @ R2 @ ( transitive_rtrancl @ A @ S ) ) ) )
= ( wf @ A @ R2 ) ) ) ).
% qc_wf_relto_iff
thf(fact_6611_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
@ ^ [Q10: set @ A,Q3: set @ A] :
( ( ord_less @ ( set @ A ) @ Q3 @ Q10 )
& ( ord_less_eq @ ( set @ A ) @ Q10 @ S ) ) ) ) ) ) ).
% wf_bounded_supset
thf(fact_6612_finite__subset__wf,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X11: set @ A,Y10: set @ A] :
( ( ord_less @ ( set @ A ) @ X11 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A5 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_6613_reduction__pairI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S ) @ R2 )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) ) ) ) ).
% reduction_pairI
thf(fact_6614_reduction__pair__lemma,axiom,
! [A: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( fun_reduction_pair @ A @ P )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
=> ( ( wf @ A @ S )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S ) ) ) ) ) ) ).
% reduction_pair_lemma
thf(fact_6615_reduction__pair__def,axiom,
! [A: $tType] :
( ( fun_reduction_pair @ A )
= ( ^ [P4: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )] :
( ( wf @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) ) ) ) ).
% reduction_pair_def
thf(fact_6616_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
( ( wf @ A @ R2 )
=> ( ! [F2: A > B,G2: A > B,X3: A,R: B] :
( ! [Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ X3 ) @ R2 )
=> ( ( F2 @ Z6 )
= ( G2 @ Z6 ) ) )
=> ( ( P @ F2 @ X3 @ R )
= ( P @ G2 @ X3 @ R ) ) )
=> ( ! [X3: A,F2: A > B] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R2 )
=> ( P @ F2 @ Y6 @ ( F2 @ Y6 ) ) )
=> ? [X_12: B] : ( P @ F2 @ X3 @ X_12 ) )
=> ? [F2: A > B] :
! [X7: A] : ( P @ F2 @ X7 @ ( F2 @ X7 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_6617_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( antisym @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B2 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_6618_antisym__reflcl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( antisym @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
= ( antisym @ A @ R3 ) ) ).
% antisym_reflcl
thf(fact_6619_antisymD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( antisym @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 )
=> ( A3 = B2 ) ) ) ) ).
% antisymD
thf(fact_6620_antisymI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
=> ( X3 = Y3 ) ) )
=> ( antisym @ A @ R3 ) ) ).
% antisymI
thf(fact_6621_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R4 )
=> ( X4 = Y4 ) ) ) ) ) ).
% antisym_def
thf(fact_6622_antisym__Id,axiom,
! [A: $tType] : ( antisym @ A @ ( id2 @ A ) ) ).
% antisym_Id
thf(fact_6623_antisym__empty,axiom,
! [A: $tType] : ( antisym @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% antisym_empty
thf(fact_6624_antisym__Id__on,axiom,
! [A: $tType,A5: set @ A] : ( antisym @ A @ ( id_on @ A @ A5 ) ) ).
% antisym_Id_on
thf(fact_6625_antisym__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 )
=> ( ( antisym @ A @ S2 )
=> ( antisym @ A @ R3 ) ) ) ).
% antisym_subset
thf(fact_6626_antisym__singleton,axiom,
! [A: $tType,X: product_prod @ A @ A] : ( antisym @ A @ ( insert @ ( product_prod @ A @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% antisym_singleton
thf(fact_6627_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R: B,F2: A > B,G2: A > B,X3: A] :
( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X3 )
=> ( ( F2 @ Y6 )
= ( G2 @ Y6 ) ) )
=> ( ( P @ F2 @ X3 @ R )
= ( P @ G2 @ X3 @ R ) ) )
=> ( ! [X3: A,F2: A > B] :
( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X3 )
=> ( P @ F2 @ Y6 @ ( F2 @ Y6 ) ) )
=> ? [X_12: B] : ( P @ F2 @ X3 @ X_12 ) )
=> ? [F2: A > B] :
! [X7: A] : ( P @ F2 @ X7 @ ( F2 @ X7 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_6628_antisym__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( antisym @ A @ R3 )
=> ( antisym @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ) ).
% antisym_Restr
thf(fact_6629_chains__extend,axiom,
! [A: $tType,C2: set @ ( set @ A ),S: set @ ( set @ A ),Z2: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains2 @ A @ S ) )
=> ( ( member @ ( set @ A ) @ Z2 @ S )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ X3 @ Z2 ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z2 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C2 ) @ ( chains2 @ A @ S ) ) ) ) ) ).
% chains_extend
thf(fact_6630_rp__inv__image__rp,axiom,
! [A: $tType,B: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F3: B > A] :
( ( fun_reduction_pair @ A @ P )
=> ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P @ F3 ) ) ) ).
% rp_inv_image_rp
thf(fact_6631_chainsD,axiom,
! [A: $tType,C2: set @ ( set @ A ),S: set @ ( set @ A ),X: set @ A,Y: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains2 @ A @ S ) )
=> ( ( member @ ( set @ A ) @ X @ C2 )
=> ( ( member @ ( set @ A ) @ Y @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ X @ Y )
| ( ord_less_eq @ ( set @ A ) @ Y @ X ) ) ) ) ) ).
% chainsD
thf(fact_6632_Zorn__Lemma2,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ! [X3: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X3 @ ( chains2 @ A @ A5 ) )
=> ? [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ A5 )
& ! [Xb3: set @ A] :
( ( member @ ( set @ A ) @ Xb3 @ X3 )
=> ( ord_less_eq @ ( set @ A ) @ Xb3 @ Xa2 ) ) ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A5 )
& ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa2 )
=> ( Xa2 = X3 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_6633_chainsD2,axiom,
! [A: $tType,C2: set @ ( set @ A ),S: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ C2 @ ( chains2 @ A @ S ) )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ C2 @ S ) ) ).
% chainsD2
thf(fact_6634_Zorn__Lemma,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ! [X3: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X3 @ ( chains2 @ A @ A5 ) )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ X3 ) @ A5 ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A5 )
& ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa2 )
=> ( Xa2 = X3 ) ) ) ) ) ).
% Zorn_Lemma
thf(fact_6635_chains__def,axiom,
! [A: $tType] :
( ( chains2 @ A )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ ( set @ ( set @ A ) )
@ ^ [C8: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C8 @ A8 )
& ( chain_subset @ A @ C8 ) ) ) ) ) ).
% chains_def
thf(fact_6636_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R6: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F4 ) @ ( inv_image @ A @ B @ S5 @ F4 ) ) ) ) ).
% rp_inv_image_def
thf(fact_6637_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R3: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R3 @ F3 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) @ R3 ) ) ).
% in_inv_image
thf(fact_6638_chain__subset__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( ^ [C8: set @ ( set @ A )] :
! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C8 )
=> ! [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ C8 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y4 )
| ( ord_less_eq @ ( set @ A ) @ Y4 @ X4 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_6639_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) @ R4 ) ) ) ) ) ).
% inv_image_def
thf(fact_6640_total__inv__image,axiom,
! [B: $tType,A: $tType,F3: A > B,R3: set @ ( product_prod @ B @ B )] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( total_on @ B @ ( top_top @ ( set @ B ) ) @ R3 )
=> ( total_on @ A @ ( top_top @ ( set @ A ) ) @ ( inv_image @ B @ A @ R3 @ F3 ) ) ) ) ).
% total_inv_image
thf(fact_6641_lenlex__def,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R4 ) )
@ ^ [Xs3: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs3 ) @ Xs3 ) ) ) ) ).
% lenlex_def
thf(fact_6642_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_antimono @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ Q @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_6643_less__than__iff,axiom,
! [X: nat,Y: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ less_than )
= ( ord_less @ nat @ X @ Y ) ) ).
% less_than_iff
thf(fact_6644_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoD
thf(fact_6645_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoE
thf(fact_6646_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ Y3 ) @ ( F3 @ X3 ) ) )
=> ( order_antimono @ A @ B @ F3 ) ) ) ).
% antimonoI
thf(fact_6647_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F4: A > B] :
! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F4 @ Y4 ) @ ( F4 @ X4 ) ) ) ) ) ) ).
% antimono_def
thf(fact_6648_antimono__iff__le__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N: nat] : ( ord_less_eq @ A @ ( F4 @ ( suc @ N ) ) @ ( F4 @ N ) ) ) ) ) ).
% antimono_iff_le_Suc
thf(fact_6649_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_max @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( F3 @ ( ord_min @ A @ X @ Y ) ) ) ) ) ).
% max_of_antimono
thf(fact_6650_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_min @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( F3 @ ( ord_max @ A @ X @ Y ) ) ) ) ) ).
% min_of_antimono
thf(fact_6651_mlex__prod__def,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F4: A > nat,R6: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R6 )
@ ^ [X4: A] : ( product_Pair @ nat @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ).
% mlex_prod_def
thf(fact_6652_sum__mset__constant,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [Y: B,A5: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [X4: A] : Y
@ A5 ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( multiset @ A ) @ A5 ) ) @ Y ) ) ) ).
% sum_mset_constant
thf(fact_6653_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,A5: set @ A,Z2: B,Y: B,A3: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A5 @ Y )
=> ( ( member @ A @ A3 @ A5 )
=> ? [Y11: B] :
( ( Y
= ( F3 @ A3 @ Y11 ) )
& ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ Y11 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
thf(fact_6654_Union__image__single__mset,axiom,
! [A: $tType,M: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ ( multiset @ A )
@ ( image_mset @ A @ ( multiset @ A )
@ ^ [X4: A] : ( add_mset @ A @ X4 @ ( zero_zero @ ( multiset @ A ) ) )
@ M ) )
= M ) ).
% Union_image_single_mset
thf(fact_6655_sum__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [Uu2: B] : ( zero_zero @ A )
@ A5 ) )
= ( zero_zero @ A ) ) ) ).
% sum_mset.neutral_const
thf(fact_6656_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 ) )
=> ~ ! [X3: A,A13: set @ A] :
( ( A1
= ( insert @ A @ X3 @ A13 ) )
=> ! [Y3: B] :
( ( A22
= ( F3 @ X3 @ Y3 ) )
=> ( ~ ( member @ A @ X3 @ A13 )
=> ~ ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A13 @ Y3 ) ) ) ) ) ) ).
% fold_graph.cases
thf(fact_6657_fold__graph_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold_graph @ A @ B )
= ( ^ [F4: A > B > B,Z7: B,A12: set @ A,A23: B] :
( ( ( A12
= ( bot_bot @ ( set @ A ) ) )
& ( A23 = Z7 ) )
| ? [X4: A,A8: set @ A,Y4: B] :
( ( A12
= ( insert @ A @ X4 @ A8 ) )
& ( A23
= ( F4 @ X4 @ Y4 ) )
& ~ ( member @ A @ X4 @ A8 )
& ( finite_fold_graph @ A @ B @ F4 @ Z7 @ A8 @ Y4 ) ) ) ) ) ).
% fold_graph.simps
thf(fact_6658_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_6659_empty__fold__graphE,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z2: B,X: B] :
( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( bot_bot @ ( set @ A ) ) @ X )
=> ( X = Z2 ) ) ).
% empty_fold_graphE
thf(fact_6660_sum__mset_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > C > A,B4: multiset @ C,A5: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [I4: B] : ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ C @ A @ ( G3 @ I4 ) @ B4 ) )
@ A5 ) )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ C @ A
@ ^ [J3: C] :
( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [I4: B] : ( G3 @ I4 @ J3 )
@ A5 ) )
@ B4 ) ) ) ) ).
% sum_mset.swap
thf(fact_6661_comp__fun__commute__on_Ofold__graph__determ,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,A5: set @ A,Z2: B,X: B,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A5 @ X )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A5 @ Y )
=> ( Y = X ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_determ
thf(fact_6662_sum__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,H: B > A,A5: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( G3 @ X4 ) @ ( H @ X4 ) )
@ A5 ) )
= ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G3 @ A5 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ H @ A5 ) ) ) ) ) ).
% sum_mset.distrib
thf(fact_6663_sum__mset__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F3: B > A,M6: multiset @ B] :
( ( times_times @ A @ C2 @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F3 @ M6 ) ) )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C2 @ ( F3 @ X4 ) )
@ M6 ) ) ) ) ).
% sum_mset_distrib_left
thf(fact_6664_sum__mset__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F3: B > A,M6: multiset @ B,C2: A] :
( ( times_times @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F3 @ M6 ) ) @ C2 )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C2 )
@ M6 ) ) ) ) ).
% sum_mset_distrib_right
thf(fact_6665_sum__mset__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( times @ A )
& ( semiring_0 @ B ) )
=> ! [F3: A > B,A5: multiset @ A,G3: C > B,B4: multiset @ C] :
( ( times_times @ B @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ A @ B @ F3 @ A5 ) ) @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ C @ B @ G3 @ B4 ) ) )
= ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [I4: A] :
( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F3 @ I4 ) @ ( G3 @ J3 ) )
@ B4 ) )
@ A5 ) ) ) ) ).
% sum_mset_product
thf(fact_6666_size__eq__sum__mset,axiom,
! [A: $tType] :
( ( size_size @ ( multiset @ A ) )
= ( ^ [M10: multiset @ A] :
( comm_m7189776963980413722m_mset @ nat
@ ( image_mset @ A @ nat
@ ^ [A7: A] : ( one_one @ nat )
@ M10 ) ) ) ) ).
% size_eq_sum_mset
thf(fact_6667_comp__fun__commute__on_Ofold__graph__insertE,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,X: A,A5: set @ A,Z2: B,V2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ ( insert @ A @ X @ A5 ) @ V2 )
=> ( ~ ( member @ A @ X @ A5 )
=> ~ ! [Y3: B] :
( ( V2
= ( F3 @ X @ Y3 ) )
=> ~ ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A5 @ Y3 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE
thf(fact_6668_comp__fun__commute__on_Ofold__equality,axiom,
! [A: $tType,B: $tType,S: set @ A,F3: A > B > B,A5: set @ A,Z2: B,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A5 @ Y )
=> ( ( finite_fold @ A @ B @ F3 @ Z2 @ A5 )
= Y ) ) ) ) ).
% comp_fun_commute_on.fold_equality
thf(fact_6669_Finite__Set_Ofold__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold @ A @ B )
= ( ^ [F4: A > B > B,Z7: B,A8: set @ A] : ( if @ B @ ( finite_finite2 @ A @ A8 ) @ ( the @ B @ ( finite_fold_graph @ A @ B @ F4 @ Z7 @ A8 ) ) @ Z7 ) ) ) ).
% Finite_Set.fold_def
thf(fact_6670_comp__fun__commute__on_Ofold__graph__fold,axiom,
! [B: $tType,A: $tType,S: set @ A,F3: A > B > B,A5: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S )
=> ( ( finite_finite2 @ A @ A5 )
=> ( finite_fold_graph @ A @ B @ F3 @ Z2 @ A5 @ ( finite_fold @ A @ B @ F3 @ Z2 @ A5 ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_fold
thf(fact_6671_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X212: A,X223: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X212 @ X223 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X212 ) @ ( size_list @ A @ X @ X223 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_6672_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_6673_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_6674_foldl__conv__fold,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F4: A > B > A,S6: A,Xs3: list @ B] :
( fold @ B @ A
@ ^ [X4: B,T4: A] : ( F4 @ T4 @ X4 )
@ Xs3
@ S6 ) ) ) ).
% foldl_conv_fold
thf(fact_6675_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y @ ( F3 @ X ) )
=> ( ord_less @ nat @ Y @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_6676_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F3: A > nat,G3: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F3 @ Xs ) @ ( size_list @ A @ G3 @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_6677_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F3 @ X ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_6678_union__set__fold,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
= ( fold @ A @ ( set @ A ) @ ( insert @ A ) @ Xs @ A5 ) ) ).
% union_set_fold
thf(fact_6679_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_6680_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_6681_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_6682_Inf__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic7752659483105999362nf_fin @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs @ X ) ) ) ).
% Inf_fin.set_eq_fold
thf(fact_6683_Sup__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic5882676163264333800up_fin @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs @ X ) ) ) ).
% Sup_fin.set_eq_fold
thf(fact_6684_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_6685_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_6686_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_6687_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N2: nat,S2: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ N2 @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ S @ ( set_ord_lessThan @ A @ S2 ) ) ) )
=> ( ( infini527867602293511546merate @ A @ ( inf_inf @ ( set @ A ) @ S @ ( set_ord_lessThan @ A @ S2 ) ) @ N2 )
= ( infini527867602293511546merate @ A @ S @ N2 ) ) ) ) ) ).
% finite_enumerate_initial_segment
thf(fact_6688_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N2: nat,J: nat,I2: nat] :
( ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ J @ ( suc @ I2 ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I2 @ J ) ) @ N2 )
= ( suc @ ( plus_plus @ nat @ I2 @ N2 ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_6689_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I2 )
& ( ord_less @ A @ I2 @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_6690_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or5935395276787703475ssThan @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_6691_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_6692_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_6693_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_6694_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_6695_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_6696_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_6697_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_6698_enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,M: nat,N2: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M ) @ ( infini527867602293511546merate @ A @ S @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ) ) ).
% enumerate_mono_iff
thf(fact_6699_Int__greaterThanLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D3 ) )
= ( set_or5935395276787703475ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanLessThan
thf(fact_6700_finite__enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,M: nat,N2: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ M @ ( finite_card @ A @ S ) )
=> ( ( ord_less @ nat @ N2 @ ( finite_card @ A @ S ) )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M ) @ ( infini527867602293511546merate @ A @ S @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_6701_le__enumerate,axiom,
! [S: set @ nat,N2: nat] :
( ~ ( finite_finite2 @ nat @ S )
=> ( ord_less_eq @ nat @ N2 @ ( infini527867602293511546merate @ nat @ S @ N2 ) ) ) ).
% le_enumerate
thf(fact_6702_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_6703_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_6704_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(5)
thf(fact_6705_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(4)
thf(fact_6706_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(1)
thf(fact_6707_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_6708_sorted__list__of__set__greaterThanLessThan,axiom,
! [I2: nat,J: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I2 @ J ) )
= ( cons @ nat @ ( suc @ I2 ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_6709_enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N2: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N2 ) @ ( infini527867602293511546merate @ A @ S @ ( suc @ N2 ) ) ) ) ) ).
% enumerate_step
thf(fact_6710_enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M: nat,N2: nat,S: set @ A] :
( ( ord_less @ nat @ M @ N2 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M ) @ ( infini527867602293511546merate @ A @ S @ N2 ) ) ) ) ) ).
% enumerate_mono
thf(fact_6711_finite__enum__ext,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X6: set @ A,Y7: set @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( finite_card @ A @ X6 ) )
=> ( ( infini527867602293511546merate @ A @ X6 @ I3 )
= ( infini527867602293511546merate @ A @ Y7 @ I3 ) ) )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( ( finite_card @ A @ X6 )
= ( finite_card @ A @ Y7 ) )
=> ( X6 = Y7 ) ) ) ) ) ) ).
% finite_enum_ext
thf(fact_6712_finite__enumerate__Ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,S2: A] :
( ( finite_finite2 @ A @ S )
=> ( ( member @ A @ S2 @ S )
=> ? [N5: nat] :
( ( ord_less @ nat @ N5 @ ( finite_card @ A @ S ) )
& ( ( infini527867602293511546merate @ A @ S @ N5 )
= S2 ) ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_6713_finite__enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N2: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ N2 @ ( finite_card @ A @ S ) )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S @ N2 ) @ S ) ) ) ) ).
% finite_enumerate_in_set
thf(fact_6714_greaterThanLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L2 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% greaterThanLessThan_def
thf(fact_6715_greaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [A7: A,B6: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A7 ) @ ( set_ord_lessThan @ A @ B6 ) ) ) ) ) ).
% greaterThanLessThan_eq
thf(fact_6716_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_6717_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_6718_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_6719_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_6720_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_6721_finite__enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M: nat,N2: nat,S: set @ A] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ N2 @ ( finite_card @ A @ S ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ M ) @ ( infini527867602293511546merate @ A @ S @ N2 ) ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_6722_finite__le__enumerate,axiom,
! [S: set @ nat,N2: nat] :
( ( finite_finite2 @ nat @ S )
=> ( ( ord_less @ nat @ N2 @ ( finite_card @ nat @ S ) )
=> ( ord_less_eq @ nat @ N2 @ ( infini527867602293511546merate @ nat @ S @ N2 ) ) ) ) ).
% finite_le_enumerate
thf(fact_6723_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_6724_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_6725_finite__enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N2: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ ( suc @ N2 ) @ ( finite_card @ A @ S ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N2 ) @ ( infini527867602293511546merate @ A @ S @ ( suc @ N2 ) ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_6726_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N2: nat] :
( ( infini527867602293511546merate @ A @ S @ ( suc @ N2 ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ ( infini527867602293511546merate @ A @ S @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N2 ) ) ) ).
% enumerate_Suc'
thf(fact_6727_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X6: set @ A,Y7: set @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( finite_card @ A @ X6 ) )
=> ( ( infini527867602293511546merate @ A @ X6 @ I3 )
= ( infini527867602293511546merate @ A @ Y7 @ I3 ) ) )
=> ( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X6 ) @ ( finite_card @ A @ Y7 ) )
=> ( ord_less_eq @ ( set @ A ) @ X6 @ Y7 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_6728_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N2: nat] :
( ( finite_finite2 @ A @ S )
=> ( ( ord_less @ nat @ ( suc @ N2 ) @ ( finite_card @ A @ S ) )
=> ( ( infini527867602293511546merate @ A @ S @ ( suc @ N2 ) )
= ( ord_Least @ A
@ ^ [S6: A] :
( ( member @ A @ S6 @ S )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N2 ) @ S6 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_6729_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N2: nat,J: nat,I2: nat] :
( ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ J @ I2 ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I2 @ J ) ) @ N2 )
= ( suc @ ( plus_plus @ nat @ I2 @ N2 ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_6730_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_6731_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_6732_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I2 )
& ( ord_less_eq @ A @ I2 @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_6733_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_6734_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_6735_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K @ L ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_6736_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_6737_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_6738_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_6739_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_6740_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_6741_Int__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanAtMost
thf(fact_6742_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_6743_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ K ) ) ) ).
% Least_le
thf(fact_6744_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B12: A] :
( ( P @ B12 )
=> ( ord_less_eq @ A @ A6 @ B12 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_6745_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [A6: A] :
( ( P @ A6 )
=> ( ! [B12: A] :
( ( P @ B12 )
=> ( ord_less_eq @ A @ A6 @ B12 ) )
=> ( Q @ A6 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_6746_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( ord_Least @ A @ P )
= X ) ) ) ) ).
% Least_equality
thf(fact_6747_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ X3 @ Y6 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_6748_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z2: A] :
( ? [X7: A] :
( ( P @ X7 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X7 @ Y3 ) )
& ! [Y3: A] :
( ( ( P @ Y3 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y3 @ Ya2 ) ) )
=> ( Y3 = X7 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z2 ) ) ) ) ).
% Least1_le
thf(fact_6749_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X7: A] :
( ( P @ X7 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ X7 @ Y3 ) )
& ! [Y3: A] :
( ( ( P @ Y3 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y3 @ Ya2 ) ) )
=> ( Y3 = X7 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_6750_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A3 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ D3 @ C2 ) )
| ( ( A3 = C2 )
& ( B2 = D3 ) ) ) ) ) ).
% Ioc_inj
thf(fact_6751_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_6752_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_6753_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K: A,P: A > $o] :
( ( ord_less @ A @ K @ ( ord_Least @ A @ P ) )
=> ~ ( P @ K ) ) ) ).
% not_less_Least
thf(fact_6754_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_6755_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_6756_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_6757_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_6758_Inf__nat__def,axiom,
( ( complete_Inf_Inf @ nat )
= ( ^ [X11: set @ nat] :
( ord_Least @ nat
@ ^ [N: nat] : ( member @ nat @ N @ X11 ) ) ) ) ).
% Inf_nat_def
thf(fact_6759_Least__Suc2,axiom,
! [P: nat > $o,N2: nat,Q: nat > $o,M: nat] :
( ( P @ N2 )
=> ( ( Q @ M )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least @ nat @ P )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_6760_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(6)
thf(fact_6761_Least__Suc,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M3: nat] : ( P @ ( suc @ M3 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_6762_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ord_less_eq @ A @ D3 @ C2 )
| ( ord_less_eq @ A @ B2 @ C2 )
| ( ord_less_eq @ A @ D3 @ A3 ) ) ) ) ).
% Ioc_disjoint
thf(fact_6763_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_6764_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(8)
thf(fact_6765_ivl__disj__un__one_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_6766_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_6767_Least__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ? [X_12: A] : ( P @ X_12 )
=> ( ( ord_Least @ A @ P )
= ( lattic643756798350308766er_Min @ A @ ( collect @ A @ P ) ) ) ) ) ) ).
% Least_Min
thf(fact_6768_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(2)
thf(fact_6769_Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( bleast @ A )
= ( ^ [S5: set @ A,P4: A > $o] :
( ord_Least @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S5 )
& ( P4 @ X4 ) ) ) ) ) ) ).
% Bleast_def
thf(fact_6770_abort__Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( abort_Bleast @ A )
= ( ^ [S5: set @ A,P4: A > $o] :
( ord_Least @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S5 )
& ( P4 @ X4 ) ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_6771_enumerate__0,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A] :
( ( infini527867602293511546merate @ A @ S @ ( zero_zero @ nat ) )
= ( ord_Least @ A
@ ^ [N: A] : ( member @ A @ N @ S ) ) ) ) ).
% enumerate_0
thf(fact_6772_ivl__disj__un__one_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_6773_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_6774_greaterThanAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L2 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% greaterThanAtMost_def
thf(fact_6775_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( plus_plus @ A @ ( G3 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or3652927894154168847AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% sum.head
thf(fact_6776_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_6777_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M @ N2 ) )
= ( times_times @ A @ ( G3 @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or3652927894154168847AtMost @ nat @ M @ N2 ) ) ) ) ) ) ).
% prod.head
thf(fact_6778_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_6779_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_6780_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A7: A,B6: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A7 @ B6 ) @ ( insert @ A @ A7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_6781_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_6782_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_6783_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_6784_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_6785_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,S: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ? [X7: A] :
( ( member @ A @ X7 @ S )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ S )
=> ( ord_less_eq @ A @ X7 @ Xa3 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y4: B] : ( member @ B @ Y4 @ ( image2 @ A @ B @ F3 @ S ) ) )
= ( F3
@ ( ord_Least @ A
@ ^ [X4: A] : ( member @ A @ X4 @ S ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_6786_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P4: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P4 @ X4 )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( ord_less_eq @ A @ X4 @ Y4 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_6787_sorted__list__of__set__greaterThanAtMost,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq @ nat @ ( suc @ I2 ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I2 @ J ) )
= ( cons @ nat @ ( suc @ I2 ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_6788_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 ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_6789_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 ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_6790_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_6791_enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N2: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( ( infini527867602293511546merate @ A @ S @ ( suc @ N2 ) )
= ( ord_Least @ A
@ ^ [S6: A] :
( ( member @ A @ S6 @ S )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S @ N2 ) @ S6 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_6792_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S: set @ A,N2: nat] :
( ( infini527867602293511546merate @ A @ S @ ( suc @ N2 ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S
@ ( insert @ A
@ ( ord_Least @ A
@ ^ [N: A] : ( member @ A @ N @ S ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N2 ) ) ) ).
% enumerate_Suc
thf(fact_6793_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S: set @ A] :
( ! [A6: A,B5: A,X3: A] :
( ( member @ A @ A6 @ S )
=> ( ( member @ A @ B5 @ S )
=> ( ( ord_less_eq @ A @ A6 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B5 )
=> ( member @ A @ X3 @ S ) ) ) ) )
=> ? [A6: A,B5: A] :
( ( S
= ( bot_bot @ ( set @ A ) ) )
| ( S
= ( top_top @ ( set @ A ) ) )
| ( S
= ( set_ord_lessThan @ A @ B5 ) )
| ( S
= ( set_ord_atMost @ A @ B5 ) )
| ( S
= ( set_ord_greaterThan @ A @ A6 ) )
| ( S
= ( set_ord_atLeast @ A @ A6 ) )
| ( S
= ( set_or5935395276787703475ssThan @ A @ A6 @ B5 ) )
| ( S
= ( set_or3652927894154168847AtMost @ A @ A6 @ B5 ) )
| ( S
= ( set_or7035219750837199246ssThan @ A @ A6 @ B5 ) )
| ( S
= ( set_or1337092689740270186AtMost @ A @ A6 @ B5 ) ) ) ) ) ).
% interval_cases
thf(fact_6794_list_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R6: A > B > $o,A7: list @ A,B6: list @ B] :
? [Z7: list @ ( product_prod @ A @ B )] :
( ( member @ ( list @ ( product_prod @ A @ B ) ) @ Z7
@ ( 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 @ R6 ) ) ) ) )
& ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z7 )
= A7 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z7 )
= B6 ) ) ) ) ).
% list.in_rel
thf(fact_6795_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_6796_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K: A] :
( ( member @ A @ I2 @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I2 ) ) ) ).
% atLeast_iff
thf(fact_6797_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_6798_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% atLeast_subset_iff
thf(fact_6799_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H: A,L3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H ) @ ( set_ord_atLeast @ A @ L3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ L3 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_6800_Int__atLeastAtMostR2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ D3 ) ) ) ).
% Int_atLeastAtMostR2
thf(fact_6801_Int__atLeastAtMostL2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atLeast @ A @ C2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% Int_atLeastAtMostL2
thf(fact_6802_list_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R2 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [List: list @ A] :
( List
= ( nil @ A ) )
@ ^ [List: list @ B] :
( List
= ( nil @ B ) ) ) ).
% list.disc_transfer(1)
thf(fact_6803_list_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R2 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [List: list @ A] :
( List
!= ( nil @ A ) )
@ ^ [List: list @ B] :
( List
!= ( nil @ B ) ) ) ).
% list.disc_transfer(2)
thf(fact_6804_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ( set_ord_atLeast @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( bot_bot @ A ) ) ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_6805_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_6806_Misc_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: A > B > $o,L: list @ A,L3: list @ B,Q: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ P @ L @ L3 )
=> ( ( Q @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,X9: B,Ls: list @ A,Ls3: list @ B] :
( ( P @ X3 @ X9 )
=> ( ( list_all2 @ A @ B @ P @ Ls @ Ls3 )
=> ( ( Q @ Ls @ Ls3 )
=> ( Q @ ( cons @ A @ X3 @ Ls ) @ ( cons @ B @ X9 @ Ls3 ) ) ) ) )
=> ( Q @ L @ L3 ) ) ) ) ).
% Misc.list_all2_induct
thf(fact_6807_list__all2__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As: list @ A,F3: C > B,Bs: list @ C] :
( ( list_all2 @ A @ B @ P @ As @ ( map @ C @ B @ F3 @ Bs ) )
= ( list_all2 @ A @ C
@ ^ [X4: A,Y4: C] : ( P @ X4 @ ( F3 @ Y4 ) )
@ As
@ Bs ) ) ).
% list_all2_map2
thf(fact_6808_list__all2__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F3: C > A,As: list @ C,Bs: list @ B] :
( ( list_all2 @ A @ B @ P @ ( map @ C @ A @ F3 @ As ) @ Bs )
= ( list_all2 @ C @ B
@ ^ [X4: C] : ( P @ ( F3 @ X4 ) )
@ As
@ Bs ) ) ).
% list_all2_map1
thf(fact_6809_list_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I2: A > C,X: list @ A,Y: list @ B] :
( ( list_all2 @ C @ B @ Sb @ ( map @ A @ C @ I2 @ X ) @ Y )
= ( list_all2 @ A @ B
@ ^ [X4: A] : ( Sb @ ( I2 @ X4 ) )
@ X
@ Y ) ) ).
% list.rel_map(1)
thf(fact_6810_list_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: list @ A,G3: B > C,Y: list @ B] :
( ( list_all2 @ A @ C @ Sa @ X @ ( map @ B @ C @ G3 @ Y ) )
= ( list_all2 @ A @ B
@ ^ [X4: A,Y4: B] : ( Sa @ X4 @ ( G3 @ Y4 ) )
@ X
@ Y ) ) ).
% list.rel_map(2)
thf(fact_6811_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_Ici_le_Iic
thf(fact_6812_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H: A,L3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_Iic_le_Ici
thf(fact_6813_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,L3: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_Ici_le_Icc
thf(fact_6814_list_Orel__mono,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,Ra2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ Ra2 )
=> ( ord_less_eq @ ( ( list @ A ) > ( list @ B ) > $o ) @ ( list_all2 @ A @ B @ R2 ) @ ( list_all2 @ A @ B @ Ra2 ) ) ) ).
% list.rel_mono
thf(fact_6815_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ).
% atLeast_def
thf(fact_6816_not__UNIV__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L ) ) ) ).
% not_UNIV_le_Ici
thf(fact_6817_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% Ioi_le_Ico
thf(fact_6818_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys3: list @ B,P3: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys3 )
=> ( ( ord_less @ nat @ P3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ P3 ) @ ( nth @ B @ Ys3 @ P3 ) ) ) ) ).
% list_all2_nthD
thf(fact_6819_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys3: list @ B,P3: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys3 )
=> ( ( ord_less @ nat @ P3 @ ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( P @ ( nth @ A @ Xs @ P3 ) @ ( nth @ B @ Ys3 @ P3 ) ) ) ) ).
% list_all2_nthD2
thf(fact_6820_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ A3 ) )
=> ( P @ ( nth @ A @ A3 @ N5 ) @ ( nth @ B @ B2 @ N5 ) ) )
=> ( list_all2 @ A @ B @ P @ A3 @ B2 ) ) ) ).
% list_all2_all_nthI
thf(fact_6821_list__all2__conv__all__nth,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P4: A > B > $o,Xs3: list @ A,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P4 @ ( nth @ A @ Xs3 @ I4 ) @ ( nth @ B @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_6822_ivl__disj__un__one_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_6823_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_6824_atLeastAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or1337092689740270186AtMost @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% atLeastAtMost_def
thf(fact_6825_atLeastLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% atLeastLessThan_def
thf(fact_6826_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_6827_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_6828_product__lists__set,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) )
= ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( list_all2 @ A @ ( list @ A )
@ ^ [X4: A,Ys2: list @ A] : ( member @ A @ X4 @ ( set2 @ A @ Ys2 ) )
@ Xs3
@ Xss ) ) ) ).
% product_lists_set
thf(fact_6829_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N2 ) @ ( set_ord_atLeast @ A @ N2 ) )
= ( insert @ A @ N2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_6830_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) )
= ( set_ord_atLeast @ A @ L ) ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_6831_ivl__disj__un__one_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_6832_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_6833_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_6834_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_6835_UN__atLeast__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atLeast @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atLeast_UNIV
thf(fact_6836_list__all2__iff,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P4: A > B > $o,Xs3: list @ A,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [X4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs3 @ Ys2 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P4 @ X4 ) ) ) ) ) ).
% list_all2_iff
thf(fact_6837_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A5: A > B > $o,B4: C > D > $o] :
( ( A5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A5 @ ( bNF_rel_fun @ A @ B @ A @ B @ A5 @ A5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A5 @ ( bNF_rel_fun @ A @ B @ A @ B @ A5 @ A5 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B4 @ A5 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A5 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B4 ) @ A5 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_6838_at__top__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K3 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_top_def
thf(fact_6839_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_mult @ B )
& ( monoid_mult @ A ) )
=> ! [A5: A > B > $o] :
( ( A5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A5 @ ( bNF_rel_fun @ A @ B @ A @ B @ A5 @ A5 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A5 ) @ A5 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% prod_list_transfer
thf(fact_6840_prod__list_OCons,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Xs: list @ A] :
( ( groups5270119922927024881d_list @ A @ ( cons @ A @ X @ Xs ) )
= ( times_times @ A @ X @ ( groups5270119922927024881d_list @ A @ Xs ) ) ) ) ).
% prod_list.Cons
thf(fact_6841_prod__list_Oappend,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Xs: list @ A,Ys3: list @ A] :
( ( groups5270119922927024881d_list @ A @ ( append @ A @ Xs @ Ys3 ) )
= ( times_times @ A @ ( groups5270119922927024881d_list @ A @ Xs ) @ ( groups5270119922927024881d_list @ A @ Ys3 ) ) ) ) ).
% prod_list.append
thf(fact_6842_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_top @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_top_linorder
thf(fact_6843_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_6844_at__top__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K3 ) )
@ ( set_ord_atLeast @ A @ C2 ) ) ) ) ) ).
% at_top_sub
thf(fact_6845_finite__refines__card__le,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( equiv_equiv @ A @ A5 @ S )
=> ( ord_less_eq @ nat @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ S ) ) @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R2 ) ) ) ) ) ) ) ).
% finite_refines_card_le
thf(fact_6846_congruent__def,axiom,
! [B: $tType,A: $tType] :
( ( equiv_congruent @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),F4: A > B] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,Z7: A] :
( ( F4 @ Y4 )
= ( F4 @ Z7 ) )
@ X4 ) ) ) ) ).
% congruent_def
thf(fact_6847_in__quotient__imp__subset,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ord_less_eq @ ( set @ A ) @ X6 @ A5 ) ) ) ).
% in_quotient_imp_subset
thf(fact_6848_equiv__type,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ).
% equiv_type
thf(fact_6849_quotient__eqI,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,Y7: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y7 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( member @ A @ X @ X6 )
=> ( ( member @ A @ Y @ Y7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( X6 = Y7 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_6850_quotient__eq__iff,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,Y7: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y7 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( member @ A @ X @ X6 )
=> ( ( member @ A @ Y @ Y7 )
=> ( ( X6 = Y7 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_6851_in__quotient__imp__closed,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( member @ A @ X @ X6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( member @ A @ Y @ X6 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_6852_congruentI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F3: A > B] :
( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R3 )
=> ( ( F3 @ Y3 )
= ( F3 @ Z3 ) ) )
=> ( equiv_congruent @ A @ B @ R3 @ F3 ) ) ).
% congruentI
thf(fact_6853_congruentD,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F3: A > B,Y: A,Z2: A] :
( ( equiv_congruent @ A @ B @ R3 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R3 )
=> ( ( F3 @ Y )
= ( F3 @ Z2 ) ) ) ) ).
% congruentD
thf(fact_6854_UN__equiv__class__type,axiom,
! [A: $tType,B: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),F3: A > ( set @ B ),X6: set @ A,B4: set @ ( set @ B )] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R3 @ F3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ ( set @ B ) @ ( F3 @ X3 ) @ B4 ) )
=> ( member @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ X6 ) ) @ B4 ) ) ) ) ) ).
% UN_equiv_class_type
thf(fact_6855_in__quotient__imp__non__empty,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( X6
!= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_6856_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),F3: A > ( set @ B ),X6: set @ A,Y7: set @ A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R3 @ F3 )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ X6 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ Y7 ) ) )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y7 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( ( F3 @ X3 )
= ( F3 @ Y3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) )
=> ( X6 = Y7 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_6857_equiv__class__self,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ A @ A3 @ A5 )
=> ( member @ A @ A3 @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_self
thf(fact_6858_quotient__disj,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,Y7: set @ A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( member @ ( set @ A ) @ Y7 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( X6 = Y7 )
| ( ( inf_inf @ ( set @ A ) @ X6 @ Y7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% quotient_disj
thf(fact_6859_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),F3: A > ( set @ B ),A3: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R3 @ F3 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F3 @ A3 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_6860_finite__refines__finite,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( equiv_equiv @ A @ A5 @ S )
=> ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ S ) ) ) ) ) ) ).
% finite_refines_finite
thf(fact_6861_eq__equiv__class,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,A5: set @ A] :
( ( ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ A @ B2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% eq_equiv_class
thf(fact_6862_equiv__class__eq,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_6863_eq__equiv__class__iff,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y @ A5 )
=> ( ( ( image @ A @ A @ R3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_6864_equiv__class__eq__iff,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
= ( ( ( image @ A @ A @ R3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X @ A5 )
& ( member @ A @ Y @ A5 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_6865_eq__equiv__class__iff2,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y @ A5 )
=> ( ( ( equiv_quotient @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R3 )
= ( equiv_quotient @ A @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R3 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_6866_refines__equiv__class__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A5: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( equiv_equiv @ A @ A5 @ S )
=> ( ( image @ A @ A @ R2 @ ( image @ A @ A @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_6867_refines__equiv__class__eq2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A5: set @ A,A3: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( equiv_equiv @ A @ A5 @ S )
=> ( ( image @ A @ A @ S @ ( image @ A @ A @ R2 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_6868_refines__equiv__image__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( equiv_equiv @ A @ A5 @ S )
=> ( ( image2 @ ( set @ A ) @ ( set @ A ) @ ( image @ A @ A @ S ) @ ( equiv_quotient @ A @ A5 @ R2 ) )
= ( equiv_quotient @ A @ A5 @ S ) ) ) ) ) ).
% refines_equiv_image_eq
thf(fact_6869_equiv__class__subset,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_6870_subset__equiv__class,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% subset_equiv_class
thf(fact_6871_equiv__class__nondisjoint,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_6872_in__quotient__imp__in__rel,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X6: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_6873_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A18: set @ A,R12: set @ ( product_prod @ A @ A ),A25: set @ B,R23: set @ ( product_prod @ B @ B ),F3: A > B > ( set @ C ),A3: B] :
( ( equiv_equiv @ A @ A18 @ R12 )
=> ( ( equiv_equiv @ B @ A25 @ R23 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R12 @ R23 @ F3 )
=> ( ( member @ B @ A3 @ A25 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R12
@ ^ [X13: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F3 @ X13 ) @ ( image @ B @ B @ R23 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_6874_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A18: set @ A,R12: set @ ( product_prod @ A @ A ),A25: 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 @ A25 @ R23 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R12 @ R23 @ F3 )
=> ( ( member @ A @ A1 @ A18 )
=> ( ( member @ B @ A22 @ A25 )
=> ( ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X13: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F3 @ X13 ) @ ( image @ B @ B @ R23 @ ( insert @ B @ A22 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R12 @ ( insert @ A @ A1 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F3 @ A1 @ A22 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_6875_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,Z1: A,Y2: B,Z22: B] :
( ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z1 ) @ R12 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z22 ) @ R23 )
=> ( ( F3 @ Y1 @ Y2 )
= ( F3 @ Z1 @ Z22 ) ) ) ) ) ).
% congruent2D
thf(fact_6876_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,Z12: A,Y22: B,Z23: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z12 ) @ R12 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z23 ) @ R23 )
=> ( ( F3 @ Y12 @ Y22 )
= ( F3 @ Z12 @ Z23 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 ) ) ).
% congruent2I'
thf(fact_6877_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A18: set @ A,R12: set @ ( product_prod @ A @ A ),A25: set @ B,R23: set @ ( product_prod @ B @ B ),F3: A > B > C] :
( ( equiv_equiv @ A @ A18 @ R12 )
=> ( ( equiv_equiv @ B @ A25 @ R23 )
=> ( ! [Y3: A,Z3: A,W: B] :
( ( member @ B @ W @ A25 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R12 )
=> ( ( F3 @ Y3 @ W )
= ( F3 @ Z3 @ W ) ) ) )
=> ( ! [Y3: B,Z3: B,W: A] :
( ( member @ A @ W @ A18 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Z3 ) @ R23 )
=> ( ( F3 @ W @ Y3 )
= ( F3 @ W @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F3 ) ) ) ) ) ).
% congruent2I
thf(fact_6878_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),F3: A > A > B] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ! [Y3: A,Z3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ( member @ A @ Z3 @ A5 )
=> ( ( F3 @ Y3 @ Z3 )
= ( F3 @ Z3 @ Y3 ) ) ) )
=> ( ! [Y3: A,Z3: A,W: A] :
( ( member @ A @ W @ A5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R3 )
=> ( ( F3 @ W @ Y3 )
= ( F3 @ W @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R3 @ R3 @ F3 ) ) ) ) ).
% congruent2_commuteI
thf(fact_6879_congruent2__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( equiv_congruent2 @ A @ B @ C )
= ( ^ [R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),F4: A > B > C] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R13 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y13: A,Z13: A] :
! [Y4: product_prod @ B @ B] :
( ( member @ ( product_prod @ B @ B ) @ Y4 @ R24 )
=> ( product_case_prod @ B @ B @ $o
@ ^ [Y23: B,Z24: B] :
( ( F4 @ Y13 @ Y23 )
= ( F4 @ Z13 @ Z24 ) )
@ Y4 ) )
@ X4 ) ) ) ) ).
% congruent2_def
thf(fact_6880_UN__equiv__class__type2,axiom,
! [A: $tType,B: $tType,C: $tType,A18: set @ A,R12: set @ ( product_prod @ A @ A ),A25: set @ B,R23: set @ ( product_prod @ B @ B ),F3: A > B > ( set @ C ),X17: set @ A,X24: set @ B,B4: set @ ( set @ C )] :
( ( equiv_equiv @ A @ A18 @ R12 )
=> ( ( equiv_equiv @ B @ A25 @ R23 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R12 @ R23 @ F3 )
=> ( ( member @ ( set @ A ) @ X17 @ ( equiv_quotient @ A @ A18 @ R12 ) )
=> ( ( member @ ( set @ B ) @ X24 @ ( equiv_quotient @ B @ A25 @ R23 ) )
=> ( ! [X12: A,X22: B] :
( ( member @ A @ X12 @ A18 )
=> ( ( member @ B @ X22 @ A25 )
=> ( member @ ( set @ C ) @ ( F3 @ X12 @ X22 ) @ B4 ) ) )
=> ( member @ ( set @ C )
@ ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X13: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F3 @ X13 ) @ X24 ) )
@ X17 ) )
@ B4 ) ) ) ) ) ) ) ).
% UN_equiv_class_type2
thf(fact_6881_proj__iff,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 )
=> ( ( ( equiv_proj @ A @ A @ R3 @ X )
= ( equiv_proj @ A @ A @ R3 @ Y ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 ) ) ) ) ).
% proj_iff
thf(fact_6882_relImage__proj,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ ( set @ A ) @ R2 @ ( equiv_proj @ A @ A @ R2 ) ) @ ( id_on @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R2 ) ) ) ) ).
% relImage_proj
thf(fact_6883_proj__def,axiom,
! [A: $tType,B: $tType] :
( ( equiv_proj @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A ),X4: B] : ( image @ B @ A @ R4 @ ( insert @ B @ X4 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% proj_def
thf(fact_6884_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X4: A,A8: set @ A] : ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_6885_lenlex__append2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Us: list @ A,Xs: list @ A,Ys3: list @ A] :
( ( irrefl @ A @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs ) @ ( append @ A @ Us @ Ys3 ) ) @ ( lenlex @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys3 ) @ ( lenlex @ A @ R2 ) ) ) ) ).
% lenlex_append2
thf(fact_6886_member__remove,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( member @ A @ X @ ( remove @ A @ Y @ A5 ) )
= ( ( member @ A @ X @ A5 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_6887_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys3: list @ A,Zs: list @ A] :
( ( irrefl @ A @ R3 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys3 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R3 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys3 @ Zs ) @ ( lexord @ A @ R3 ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_6888_irreflI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R2 )
=> ( irrefl @ A @ R2 ) ) ).
% irreflI
thf(fact_6889_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [A7: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A7 ) @ R4 ) ) ) ).
% irrefl_def
thf(fact_6890_lexl__not__refl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( irrefl @ A @ R3 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ X ) @ ( lex @ A @ R3 ) ) ) ).
% lexl_not_refl
thf(fact_6891_irrefl__diff__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( irrefl @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ).
% irrefl_diff_Id
thf(fact_6892_irrefl__distinct,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [A7: A,B6: A] : A7 != B6
@ X4 ) ) ) ) ).
% irrefl_distinct
thf(fact_6893_inter__coset__fold,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( coset @ A @ Xs ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs @ A5 ) ) ).
% inter_coset_fold
thf(fact_6894_relImage__Gr,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
=> ( ( bNF_Gr4221423524335903396lImage @ A @ B @ R2 @ F3 )
= ( relcomp @ B @ A @ B @ ( converse @ A @ B @ ( bNF_Gr @ A @ B @ A5 @ F3 ) ) @ ( relcomp @ A @ A @ B @ R2 @ ( bNF_Gr @ A @ B @ A5 @ F3 ) ) ) ) ) ).
% relImage_Gr
thf(fact_6895_converse__converse,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( converse @ B @ A @ ( converse @ A @ B @ R3 ) )
= R3 ) ).
% converse_converse
thf(fact_6896_converse__inject,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
( ( ( converse @ B @ A @ R3 )
= ( converse @ B @ A @ S2 ) )
= ( R3 = S2 ) ) ).
% converse_inject
thf(fact_6897_converse__iff,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R3 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R3 ) ) ).
% converse_iff
thf(fact_6898_Field__converse,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( converse @ A @ A @ R3 ) )
= ( field2 @ A @ R3 ) ) ).
% Field_converse
thf(fact_6899_converse__mono,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R3 ) @ ( converse @ B @ A @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R3 @ S2 ) ) ).
% converse_mono
thf(fact_6900_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_6901_converse__Id,axiom,
! [A: $tType] :
( ( converse @ A @ A @ ( id2 @ A ) )
= ( id2 @ A ) ) ).
% converse_Id
thf(fact_6902_refl__on__converse,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A5 @ ( converse @ A @ A @ R3 ) )
= ( refl_on @ A @ A5 @ R3 ) ) ).
% refl_on_converse
thf(fact_6903_finite__converse,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( converse @ B @ A @ R3 ) )
= ( finite_finite2 @ ( product_prod @ B @ A ) @ R3 ) ) ).
% finite_converse
thf(fact_6904_total__on__converse,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A5 @ ( converse @ A @ A @ R3 ) )
= ( total_on @ A @ A5 @ R3 ) ) ).
% total_on_converse
thf(fact_6905_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_6906_antisym__converse,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( antisym @ A @ ( converse @ A @ A @ R3 ) )
= ( antisym @ A @ R3 ) ) ).
% antisym_converse
thf(fact_6907_converse__Id__on,axiom,
! [A: $tType,A5: set @ A] :
( ( converse @ A @ A @ ( id_on @ A @ A5 ) )
= ( id_on @ A @ A5 ) ) ).
% converse_Id_on
thf(fact_6908_converse__inv__image,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( converse @ A @ A @ ( inv_image @ B @ A @ R2 @ F3 ) )
= ( inv_image @ B @ A @ ( converse @ B @ B @ R2 ) @ F3 ) ) ).
% converse_inv_image
thf(fact_6909_card__inverse,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( finite_card @ ( product_prod @ A @ B ) @ ( converse @ B @ A @ R2 ) )
= ( finite_card @ ( product_prod @ B @ A ) @ R2 ) ) ).
% card_inverse
thf(fact_6910_pair__set__inverse,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( converse @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [B6: A,A7: B] : ( P @ A7 @ B6 ) ) ) ) ).
% pair_set_inverse
thf(fact_6911_below__Id__inv,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R2 ) @ ( id2 @ A ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ).
% below_Id_inv
thf(fact_6912_converse__subset__swap,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ ( converse @ B @ A @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R3 ) @ S2 ) ) ).
% converse_subset_swap
thf(fact_6913_converse__unfold,axiom,
! [A: $tType,B: $tType] :
( ( converse @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [Y4: A,X4: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y4 ) @ R4 ) ) ) ) ) ).
% converse_unfold
thf(fact_6914_trancl__converseI,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R3 ) ) ) ) ).
% trancl_converseI
thf(fact_6915_trancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% trancl_converseD
thf(fact_6916_in__listrel1__converse,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( listrel1 @ A @ ( converse @ A @ A @ R3 ) ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R3 ) ) ) ) ).
% in_listrel1_converse
thf(fact_6917_converse_Ocases,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R3 ) ) ).
% converse.cases
thf(fact_6918_converse_Osimps,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R3 ) )
= ( ? [A7: A,B6: B] :
( ( A1 = B6 )
& ( A22 = A7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B6 ) @ R3 ) ) ) ) ).
% converse.simps
thf(fact_6919_converseD,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R3 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R3 ) ) ).
% converseD
thf(fact_6920_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R3: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R3 ) )
=> ~ ! [X3: B,Y3: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y3 @ X3 ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R3 ) ) ) ).
% converseE
thf(fact_6921_converseI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ ( converse @ A @ B @ R3 ) ) ) ).
% converseI
thf(fact_6922_rtrancl__converseI,axiom,
! [A: $tType,Y: A,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R3 ) ) ) ) ).
% rtrancl_converseI
thf(fact_6923_rtrancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ).
% rtrancl_converseD
thf(fact_6924_finite__wf__eq__wf__converse,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( wf @ A @ ( converse @ A @ A @ R2 ) )
= ( wf @ A @ R2 ) ) ) ).
% finite_wf_eq_wf_converse
thf(fact_6925_converse__Times,axiom,
! [B: $tType,A: $tType,A5: set @ B,B4: set @ A] :
( ( converse @ B @ A
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu2: B] : B4 ) )
= ( product_Sigma @ A @ B @ B4
@ ^ [Uu2: A] : A5 ) ) ).
% converse_Times
thf(fact_6926_converse__Int,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ R3 @ S2 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R3 ) @ ( converse @ B @ A @ S2 ) ) ) ).
% converse_Int
thf(fact_6927_converse__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ B @ C ),S2: set @ ( product_prod @ C @ A )] :
( ( converse @ B @ A @ ( relcomp @ B @ C @ A @ R3 @ S2 ) )
= ( relcomp @ A @ C @ B @ ( converse @ C @ A @ S2 ) @ ( converse @ B @ C @ R3 ) ) ) ).
% converse_relcomp
thf(fact_6928_converse__Un,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A ),S2: set @ ( product_prod @ B @ A )] :
( ( converse @ B @ A @ ( sup_sup @ ( set @ ( product_prod @ B @ A ) ) @ R3 @ S2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R3 ) @ ( converse @ B @ A @ S2 ) ) ) ).
% converse_Un
thf(fact_6929_converse__UNION,axiom,
! [B: $tType,A: $tType,C: $tType,R3: C > ( set @ ( product_prod @ B @ A ) ),S: set @ C] :
( ( converse @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R3 @ S ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: C] : ( converse @ B @ A @ ( R3 @ X4 ) )
@ S ) ) ) ).
% converse_UNION
thf(fact_6930_converse__INTER,axiom,
! [B: $tType,A: $tType,C: $tType,R3: C > ( set @ ( product_prod @ B @ A ) ),S: set @ C] :
( ( converse @ B @ A @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R3 @ S ) ) )
= ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: C] : ( converse @ B @ A @ ( R3 @ X4 ) )
@ S ) ) ) ).
% converse_INTER
thf(fact_6931_Image__subset__eq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ B @ A ),A5: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R3 @ A5 ) @ B4 )
= ( ord_less_eq @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ ( converse @ B @ A @ R3 ) @ ( uminus_uminus @ ( set @ A ) @ B4 ) ) ) ) ) ).
% Image_subset_eq
thf(fact_6932_refl__on__comp__subset,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A5 @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( relcomp @ A @ A @ A @ ( converse @ A @ A @ R3 ) @ R3 ) ) ) ).
% refl_on_comp_subset
thf(fact_6933_subset__code_I2_J,axiom,
! [B: $tType,A5: set @ B,Ys3: list @ B] :
( ( ord_less_eq @ ( set @ B ) @ A5 @ ( coset @ B @ Ys3 ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Ys3 ) )
=> ~ ( member @ B @ X4 @ A5 ) ) ) ) ).
% subset_code(2)
thf(fact_6934_union__coset__filter,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( coset @ A @ Xs ) @ A5 )
= ( coset @ A
@ ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A5 )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_6935_irrefl__tranclI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A] :
( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R3 ) @ ( transitive_rtrancl @ A @ R3 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R3 ) ) ) ).
% irrefl_tranclI
thf(fact_6936_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_6937_minus__coset__filter,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( coset @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_6938_relInvImage__Gr,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ A ),B4: set @ A,A5: set @ B,F3: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu2: A] : B4 ) )
=> ( ( bNF_Gr7122648621184425601vImage @ B @ A @ A5 @ R2 @ F3 )
= ( relcomp @ B @ A @ B @ ( bNF_Gr @ B @ A @ A5 @ F3 ) @ ( relcomp @ A @ A @ B @ R2 @ ( converse @ B @ A @ ( bNF_Gr @ B @ A @ A5 @ F3 ) ) ) ) ) ) ).
% relInvImage_Gr
thf(fact_6939_trans__wf__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( ( wf @ A @ R3 )
= ( ! [A7: A] :
( wf @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( image @ A @ A @ ( converse @ A @ A @ R3 ) @ ( insert @ A @ A7 @ ( bot_bot @ ( set @ A ) ) ) )
@ ^ [Uu2: A] : ( image @ A @ A @ ( converse @ A @ A @ R3 ) @ ( insert @ A @ A7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% trans_wf_iff
thf(fact_6940_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R3: set @ ( product_prod @ B @ A ),A5: set @ C,B4: C > ( set @ B )] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R3 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( image @ B @ A @ R3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R3 @ ( B4 @ X4 ) )
@ A5 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_6941_trans__converse,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ ( converse @ A @ A @ R3 ) )
= ( trans @ A @ R3 ) ) ).
% trans_converse
thf(fact_6942_single__valued__Id__on,axiom,
! [A: $tType,A5: set @ A] : ( single_valued @ A @ A @ ( id_on @ A @ A5 ) ) ).
% single_valued_Id_on
thf(fact_6943_trans__Id__on,axiom,
! [A: $tType,A5: set @ A] : ( trans @ A @ ( id_on @ A @ A5 ) ) ).
% trans_Id_on
thf(fact_6944_trans__inv__image,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),F3: B > A] :
( ( trans @ A @ R3 )
=> ( trans @ B @ ( inv_image @ A @ B @ R3 @ F3 ) ) ) ).
% trans_inv_image
thf(fact_6945_single__valued__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( single_valued @ A @ B @ R3 )
=> ( ( single_valued @ B @ C @ S2 )
=> ( single_valued @ A @ C @ ( relcomp @ A @ B @ C @ R3 @ S2 ) ) ) ) ).
% single_valued_relcomp
thf(fact_6946_single__valued__empty,axiom,
! [B: $tType,A: $tType] : ( single_valued @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% single_valued_empty
thf(fact_6947_trans__empty,axiom,
! [A: $tType] : ( trans @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trans_empty
thf(fact_6948_single__valued__Id,axiom,
! [A: $tType] : ( single_valued @ A @ A @ ( id2 @ A ) ) ).
% single_valued_Id
thf(fact_6949_trans__Id,axiom,
! [A: $tType] : ( trans @ A @ ( id2 @ A ) ) ).
% trans_Id
thf(fact_6950_trans__reflclI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( trans @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ).
% trans_reflclI
thf(fact_6951_single__valued__inter1,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R2 )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ).
% single_valued_inter1
thf(fact_6952_single__valued__inter2,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R2 )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ S @ R2 ) ) ) ).
% single_valued_inter2
thf(fact_6953_trans__Int,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( ( trans @ A @ S2 )
=> ( trans @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 ) ) ) ) ).
% trans_Int
thf(fact_6954_trans__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( trans @ A @ R3 )
=> ( trans @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ) ).
% trans_Restr
thf(fact_6955_lenlex__trans,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lenlex @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lenlex @ A @ R3 ) )
=> ( ( trans @ A @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lenlex @ A @ R3 ) ) ) ) ) ).
% lenlex_trans
thf(fact_6956_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
! [X4: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R4 )
=> ! [Z7: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z7 ) @ R4 )
=> ( Y4 = Z7 ) ) ) ) ) ).
% single_valued_def
thf(fact_6957_single__valuedI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B,Z3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z3 ) @ R3 )
=> ( Y3 = Z3 ) ) )
=> ( single_valued @ A @ B @ R3 ) ) ).
% single_valuedI
thf(fact_6958_single__valuedD,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B ),X: A,Y: B,Z2: B] :
( ( single_valued @ A @ B @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ R3 )
=> ( Y = Z2 ) ) ) ) ).
% single_valuedD
thf(fact_6959_transD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R3 ) ) ) ) ).
% transD
thf(fact_6960_transE,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R3 ) ) ) ) ).
% transE
thf(fact_6961_transI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ R3 ) ) )
=> ( trans @ A @ R3 ) ) ).
% transI
thf(fact_6962_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X4: A,Y4: A,Z7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z7 ) @ R4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z7 ) @ R4 ) ) ) ) ) ).
% trans_def
thf(fact_6963_lexord__trans,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R3: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lexord @ A @ R3 ) )
=> ( ( trans @ A @ R3 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lexord @ A @ R3 ) ) ) ) ) ).
% lexord_trans
thf(fact_6964_trans__INTER,axiom,
! [B: $tType,A: $tType,S: set @ A,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( trans @ B @ ( R3 @ X3 ) ) )
=> ( trans @ B @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ S ) ) ) ) ).
% trans_INTER
thf(fact_6965_trans__O__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R3 @ R3 ) @ R3 ) ) ).
% trans_O_subset
thf(fact_6966_single__valued__subset,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 )
=> ( ( single_valued @ A @ B @ S2 )
=> ( single_valued @ A @ B @ R3 ) ) ) ).
% single_valued_subset
thf(fact_6967_single__valued__confluent,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( single_valued @ A @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R3 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ Y ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_6968_single__valued__below__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) )
=> ( single_valued @ A @ A @ R2 ) ) ).
% single_valued_below_Id
thf(fact_6969_trans__singleton,axiom,
! [A: $tType,A3: A] : ( trans @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_6970_trans__rtrancl__eq__reflcl,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ A5 )
=> ( ( transitive_rtrancl @ A @ A5 )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ ( id2 @ A ) ) ) ) ).
% trans_rtrancl_eq_reflcl
thf(fact_6971_trans__diff__Id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R3 )
=> ( ( antisym @ A @ R3 )
=> ( trans @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ).
% trans_diff_Id
thf(fact_6972_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,Y13: A] :
! [Z7: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z7 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y23: A,Aa2: A] :
( ( Y13 = Y23 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Aa2 ) @ R4 ) )
@ Z7 ) )
@ X4 ) ) ) ) ).
% trans_join
thf(fact_6973_bijective__alt,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R6 )
& ( single_valued @ B @ A @ ( converse @ A @ B @ R6 ) ) ) ) ) ).
% bijective_alt
thf(fact_6974_Image__absorb__rtrancl,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B4: set @ A,C4: set @ A] :
( ( trans @ A @ A5 )
=> ( ( refl_on @ A @ B4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C4 @ B4 )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ A5 ) @ C4 )
= ( image @ A @ A @ A5 @ C4 ) ) ) ) ) ).
% Image_absorb_rtrancl
thf(fact_6975_Image__Int__eq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),A5: set @ B,B4: set @ B] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R2 ) )
=> ( ( image @ B @ A @ R2 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ R2 @ A5 ) @ ( image @ B @ A @ R2 @ B4 ) ) ) ) ).
% Image_Int_eq
thf(fact_6976_wf__finite__segments,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R3 )
=> ( ( trans @ A @ R3 )
=> ( ! [X3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X3 ) @ R3 ) ) )
=> ( wf @ A @ R3 ) ) ) ) ).
% wf_finite_segments
thf(fact_6977_relation__of__def,axiom,
! [A: $tType] :
( ( order_relation_of @ A )
= ( ^ [P4: A > A > $o,A8: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A7: A,B6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B6 )
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu2: A] : A8 ) )
& ( P4 @ A7 @ B6 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_6978_size__diff__se,axiom,
! [A: $tType,T6: A,S: multiset @ A] :
( ( member @ A @ T6 @ ( set_mset @ A @ S ) )
=> ( ( size_size @ ( multiset @ A ) @ S )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T6 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% size_diff_se
thf(fact_6979_set__mset__empty,axiom,
! [A: $tType] :
( ( set_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% set_mset_empty
thf(fact_6980_set__mset__eq__empty__iff,axiom,
! [A: $tType,M6: multiset @ A] :
( ( ( set_mset @ A @ M6 )
= ( bot_bot @ ( set @ A ) ) )
= ( M6
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% set_mset_eq_empty_iff
thf(fact_6981_set__mset__union,axiom,
! [A: $tType,M6: multiset @ A,N7: multiset @ A] :
( ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M6 @ N7 ) )
= ( sup_sup @ ( set @ A ) @ ( set_mset @ A @ M6 ) @ ( set_mset @ A @ N7 ) ) ) ).
% set_mset_union
thf(fact_6982_mset__diff__cancel1elem,axiom,
! [A: $tType,A3: A,B4: multiset @ A] :
( ~ ( member @ A @ A3 @ ( set_mset @ A @ B4 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B4 )
= ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% mset_diff_cancel1elem
thf(fact_6983_set__mset__Union__mset,axiom,
! [A: $tType,MM: multiset @ ( multiset @ A )] :
( ( set_mset @ A @ ( comm_m7189776963980413722m_mset @ ( multiset @ A ) @ MM ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( multiset @ A ) @ ( set @ A ) @ ( set_mset @ A ) @ ( set_mset @ ( multiset @ A ) @ MM ) ) ) ) ).
% set_mset_Union_mset
thf(fact_6984_set__mset__mono,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A] :
( ( subseteq_mset @ A @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( set_mset @ A @ A5 ) @ ( set_mset @ A @ B4 ) ) ) ).
% set_mset_mono
thf(fact_6985_image__mset__cong,axiom,
! [B: $tType,A: $tType,M6: multiset @ A,F3: A > B,G3: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ M6 ) )
=> ( ( F3 @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( image_mset @ A @ B @ F3 @ M6 )
= ( image_mset @ A @ B @ G3 @ M6 ) ) ) ).
% image_mset_cong
thf(fact_6986_in__image__mset,axiom,
! [A: $tType,B: $tType,Y: A,F3: B > A,M6: multiset @ B] :
( ( member @ A @ Y @ ( set_mset @ A @ ( image_mset @ B @ A @ F3 @ M6 ) ) )
= ( member @ A @ Y @ ( image2 @ B @ A @ F3 @ ( set_mset @ B @ M6 ) ) ) ) ).
% in_image_mset
thf(fact_6987_image__mset__cong__pair,axiom,
! [C: $tType,B: $tType,A: $tType,M6: multiset @ ( product_prod @ A @ B ),F3: A > B > C,G3: A > B > C] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( set_mset @ ( product_prod @ A @ B ) @ M6 ) )
=> ( ( F3 @ X3 @ Y3 )
= ( G3 @ X3 @ Y3 ) ) )
=> ( ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ M6 )
= ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ G3 ) @ M6 ) ) ) ).
% image_mset_cong_pair
thf(fact_6988_in__Inf__multiset__iff,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: A] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( member @ A @ X @ ( set_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) ) )
= ( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ A5 )
=> ( member @ A @ X @ ( set_mset @ A @ X4 ) ) ) ) ) ) ).
% in_Inf_multiset_iff
thf(fact_6989_mset__right__cancel__union,axiom,
! [A: $tType,A3: A,A5: multiset @ A,B4: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) ) )
=> ( ~ ( member @ A @ A3 @ ( set_mset @ A @ B4 ) )
=> ( member @ A @ A3 @ ( set_mset @ A @ A5 ) ) ) ) ).
% mset_right_cancel_union
thf(fact_6990_mset__left__cancel__union,axiom,
! [A: $tType,A3: A,A5: multiset @ A,B4: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) ) )
=> ( ~ ( member @ A @ A3 @ ( set_mset @ A @ A5 ) )
=> ( member @ A @ A3 @ ( set_mset @ A @ B4 ) ) ) ) ).
% mset_left_cancel_union
thf(fact_6991_mset__un__cases,axiom,
! [A: $tType,A3: A,A5: multiset @ A,B4: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) ) )
=> ( ~ ( member @ A @ A3 @ ( set_mset @ A @ A5 ) )
=> ( member @ A @ A3 @ ( set_mset @ A @ B4 ) ) ) ) ).
% mset_un_cases
thf(fact_6992_ex__Melem__conv,axiom,
! [A: $tType,A5: multiset @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ ( set_mset @ A @ A5 ) ) )
= ( A5
!= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% ex_Melem_conv
thf(fact_6993_multiset__induct__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( multiset @ A ) > $o,M6: multiset @ A] :
( ( P @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A,M13: multiset @ A] :
( ( P @ M13 )
=> ( ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ M13 ) )
=> ( ord_less_eq @ A @ X3 @ Xa2 ) )
=> ( P @ ( add_mset @ A @ X3 @ M13 ) ) ) )
=> ( P @ M6 ) ) ) ) ).
% multiset_induct_min
thf(fact_6994_multiset__induct__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( multiset @ A ) > $o,M6: multiset @ A] :
( ( P @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A,M13: multiset @ A] :
( ( P @ M13 )
=> ( ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ M13 ) )
=> ( ord_less_eq @ A @ Xa2 @ X3 ) )
=> ( P @ ( add_mset @ A @ X3 @ M13 ) ) ) )
=> ( P @ M6 ) ) ) ) ).
% multiset_induct_max
thf(fact_6995_mset__right__cancel__elem,axiom,
! [A: $tType,A3: A,A5: multiset @ A,B2: A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( A3 != B2 )
=> ( member @ A @ A3 @ ( set_mset @ A @ A5 ) ) ) ) ).
% mset_right_cancel_elem
thf(fact_6996_mset__left__cancel__elem,axiom,
! [A: $tType,A3: A,B2: A,A5: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A5 ) ) )
=> ( ( A3 != B2 )
=> ( member @ A @ A3 @ ( set_mset @ A @ A5 ) ) ) ) ).
% mset_left_cancel_elem
thf(fact_6997_sum__mset__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K5: multiset @ B,F3: B > A,G3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ ( set_mset @ B @ K5 ) )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F3 @ K5 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G3 @ K5 ) ) ) ) ) ).
% sum_mset_mono
thf(fact_6998_set__mset__Inf,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( set_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( multiset @ A ) @ ( set @ A ) @ ( set_mset @ A ) @ A5 ) ) ) ) ).
% set_mset_Inf
thf(fact_6999_set__mset__single,axiom,
! [A: $tType,B2: A] :
( ( set_mset @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_mset_single
thf(fact_7000_mset__2dist2__cases,axiom,
! [A: $tType,A3: A,B2: A,A5: multiset @ A,B4: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ A5 )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ B4 )
=> ( ( ( member @ A @ A3 @ ( set_mset @ A @ A5 ) )
=> ~ ( member @ A @ B2 @ ( set_mset @ A @ B4 ) ) )
=> ~ ( ( member @ A @ A3 @ ( set_mset @ A @ B4 ) )
=> ~ ( member @ A @ B2 @ ( set_mset @ A @ A5 ) ) ) ) ) ) ) ).
% mset_2dist2_cases
thf(fact_7001_mset__union__subset__s,axiom,
! [A: $tType,A3: A,B4: multiset @ A,C4: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B4 ) @ C4 )
=> ( ( member @ A @ A3 @ ( set_mset @ A @ C4 ) )
& ( subseteq_mset @ A @ B4 @ C4 ) ) ) ).
% mset_union_subset_s
thf(fact_7002_mset__le__mono__add__single,axiom,
! [A: $tType,A3: A,Ys3: multiset @ A,B2: A,Ws2: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ Ys3 ) )
=> ( ( member @ A @ B2 @ ( set_mset @ A @ Ws2 ) )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ Ys3 @ Ws2 ) ) ) ) ).
% mset_le_mono_add_single
thf(fact_7003_nth__mem__mset,axiom,
! [A: $tType,I2: nat,Ls2: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ls2 ) )
=> ( member @ A @ ( nth @ A @ Ls2 @ I2 ) @ ( set_mset @ A @ ( mset @ A @ Ls2 ) ) ) ) ).
% nth_mem_mset
thf(fact_7004_mset__un__single__un__cases,axiom,
! [A: $tType,A3: A,A5: multiset @ A,B4: multiset @ A,C4: multiset @ A] :
( ( ( add_mset @ A @ A3 @ A5 )
= ( plus_plus @ ( multiset @ A ) @ B4 @ C4 ) )
=> ( ( ( member @ A @ A3 @ ( set_mset @ A @ B4 ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C4 ) ) )
=> ~ ( ( member @ A @ A3 @ ( set_mset @ A @ C4 ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ B4 @ ( minus_minus @ ( multiset @ A ) @ C4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_un_single_un_cases
thf(fact_7005_diff__union__single__conv2,axiom,
! [A: $tType,A3: A,J5: multiset @ A,I5: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ J5 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ J5 @ I5 ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ J5 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ I5 ) ) ) ).
% diff_union_single_conv2
thf(fact_7006_mset__union__diff__comm,axiom,
! [A: $tType,T6: A,S: multiset @ A,T2: multiset @ A] :
( ( member @ A @ T6 @ ( set_mset @ A @ S ) )
=> ( ( plus_plus @ ( multiset @ A ) @ T2 @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T6 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ T2 @ S ) @ ( add_mset @ A @ T6 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_union_diff_comm
thf(fact_7007_mset__contains__eq,axiom,
! [A: $tType,M: A,M6: multiset @ A] :
( ( member @ A @ M @ ( set_mset @ A @ M6 ) )
= ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ M @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ M6 @ ( add_mset @ A @ M @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= M6 ) ) ).
% mset_contains_eq
thf(fact_7008_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_7009_size__Diff1__less,axiom,
! [A: $tType,X: A,M6: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M6 ) )
=> ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M6 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( size_size @ ( multiset @ A ) @ M6 ) ) ) ).
% size_Diff1_less
thf(fact_7010_size__Diff2__less,axiom,
! [A: $tType,X: A,M6: multiset @ A,Y: A] :
( ( member @ A @ X @ ( set_mset @ A @ M6 ) )
=> ( ( member @ A @ Y @ ( set_mset @ A @ M6 ) )
=> ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M6 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( size_size @ ( multiset @ A ) @ M6 ) ) ) ) ).
% size_Diff2_less
thf(fact_7011_at__most__one__mset__mset__diff,axiom,
! [A: $tType,A3: A,M6: multiset @ A] :
( ~ ( member @ A @ A3 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M6 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M6 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( set_mset @ A @ M6 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_7012_mult1__def,axiom,
! [A: $tType] :
( ( mult1 @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) )
@ ( product_case_prod @ ( multiset @ A ) @ ( multiset @ A ) @ $o
@ ^ [N8: multiset @ A,M10: multiset @ A] :
? [A7: A,M0: multiset @ A,K7: multiset @ A] :
( ( M10
= ( add_mset @ A @ A7 @ M0 ) )
& ( N8
= ( plus_plus @ ( multiset @ A ) @ M0 @ K7 ) )
& ! [B6: A] :
( ( member @ A @ B6 @ ( set_mset @ A @ K7 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ A7 ) @ R4 ) ) ) ) ) ) ) ).
% mult1_def
thf(fact_7013_sum__wcount__Int,axiom,
! [A: $tType,A5: set @ A,F3: A > nat,N7: multiset @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( wcount @ A @ F3 @ N7 ) @ ( inf_inf @ ( set @ A ) @ A5 @ ( set_mset @ A @ N7 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ ( wcount @ A @ F3 @ N7 ) @ A5 ) ) ) ).
% sum_wcount_Int
thf(fact_7014_not__less__empty,axiom,
! [A: $tType,M6: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M6 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R3 ) ) ).
% not_less_empty
thf(fact_7015_mult1__union,axiom,
! [A: $tType,B4: multiset @ A,D4: multiset @ A,R3: set @ ( product_prod @ A @ A ),C4: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ B4 @ D4 ) @ ( mult1 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ B4 ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ D4 ) ) @ ( mult1 @ A @ R3 ) ) ) ).
% mult1_union
thf(fact_7016_mono__mult1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R7 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R3 ) @ ( mult1 @ A @ R7 ) ) ) ).
% mono_mult1
thf(fact_7017_less__add,axiom,
! [A: $tType,N7: multiset @ A,A3: A,M02: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ ( add_mset @ A @ A3 @ M02 ) ) @ ( mult1 @ A @ R3 ) )
=> ( ? [M13: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M13 @ M02 ) @ ( mult1 @ A @ R3 ) )
& ( N7
= ( add_mset @ A @ A3 @ M13 ) ) )
| ? [K9: multiset @ A] :
( ! [B12: A] :
( ( member @ A @ B12 @ ( set_mset @ A @ K9 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B12 @ A3 ) @ R3 ) )
& ( N7
= ( plus_plus @ ( multiset @ A ) @ M02 @ K9 ) ) ) ) ) ).
% less_add
thf(fact_7018_mult1I,axiom,
! [A: $tType,M6: multiset @ A,A3: A,M02: multiset @ A,N7: multiset @ A,K5: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( M6
= ( add_mset @ A @ A3 @ M02 ) )
=> ( ( N7
= ( plus_plus @ ( multiset @ A ) @ M02 @ K5 ) )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ ( set_mset @ A @ K5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ M6 ) @ ( mult1 @ A @ R3 ) ) ) ) ) ).
% mult1I
thf(fact_7019_mult1E,axiom,
! [A: $tType,N7: multiset @ A,M6: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ M6 ) @ ( mult1 @ A @ R3 ) )
=> ~ ! [A6: A,M03: multiset @ A] :
( ( M6
= ( add_mset @ A @ A6 @ M03 ) )
=> ! [K9: multiset @ A] :
( ( N7
= ( plus_plus @ ( multiset @ A ) @ M03 @ K9 ) )
=> ~ ! [B12: A] :
( ( member @ A @ B12 @ ( set_mset @ A @ K9 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B12 @ A6 ) @ R3 ) ) ) ) ) ).
% mult1E
thf(fact_7020_mult1__lessE,axiom,
! [A: $tType,N7: multiset @ A,M6: multiset @ A,R3: A > A > $o] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ M6 ) @ ( mult1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) )
=> ( ( asymp @ A @ R3 )
=> ~ ! [A6: A,M03: multiset @ A] :
( ( M6
= ( add_mset @ A @ A6 @ M03 ) )
=> ! [K9: multiset @ A] :
( ( N7
= ( plus_plus @ ( multiset @ A ) @ M03 @ K9 ) )
=> ( ~ ( member @ A @ A6 @ ( set_mset @ A @ K9 ) )
=> ~ ! [B12: A] :
( ( member @ A @ B12 @ ( set_mset @ A @ K9 ) )
=> ( R3 @ B12 @ A6 ) ) ) ) ) ) ) ).
% mult1_lessE
thf(fact_7021_mult__implies__one__step,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),M6: multiset @ A,N7: multiset @ A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M6 @ N7 ) @ ( mult @ A @ R3 ) )
=> ? [I9: multiset @ A,J6: multiset @ A] :
( ( N7
= ( plus_plus @ ( multiset @ A ) @ I9 @ J6 ) )
& ? [K9: multiset @ A] :
( ( M6
= ( plus_plus @ ( multiset @ A ) @ I9 @ K9 ) )
& ( J6
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X7: A] :
( ( member @ A @ X7 @ ( set_mset @ A @ K9 ) )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ ( set_mset @ A @ J6 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_7022_Multiset_Omono__mult,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R7 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R3 ) @ ( mult @ A @ R7 ) ) ) ).
% Multiset.mono_mult
thf(fact_7023_subset__implies__mult,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,R3: set @ ( product_prod @ A @ A )] :
( ( subset_mset @ A @ A5 @ B4 )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ A5 @ B4 ) @ ( mult @ A @ R3 ) ) ) ).
% subset_implies_mult
thf(fact_7024_asymp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( asymp @ A
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ) ).
% asymp_greater
thf(fact_7025_asympD,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A] :
( ( asymp @ A @ R2 )
=> ( ( R2 @ X @ Y )
=> ~ ( R2 @ Y @ X ) ) ) ).
% asympD
thf(fact_7026_asymp_Ointros,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [A6: A,B5: A] :
( ( R2 @ A6 @ B5 )
=> ~ ( R2 @ B5 @ A6 ) )
=> ( asymp @ A @ R2 ) ) ).
% asymp.intros
thf(fact_7027_asymp_Osimps,axiom,
! [A: $tType] :
( ( asymp @ A )
= ( ^ [A7: A > A > $o] :
? [R6: A > A > $o] :
( ( A7 = R6 )
& ! [X4: A,Y4: A] :
( ( R6 @ X4 @ Y4 )
=> ~ ( R6 @ Y4 @ X4 ) ) ) ) ) ).
% asymp.simps
thf(fact_7028_asymp_Ocases,axiom,
! [A: $tType,A3: A > A > $o] :
( ( asymp @ A @ A3 )
=> ! [A15: A,B12: A] :
( ( A3 @ A15 @ B12 )
=> ~ ( A3 @ B12 @ A15 ) ) ) ).
% asymp.cases
thf(fact_7029_asymp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( asymp @ A @ ( ord_less @ A ) ) ) ).
% asymp_less
thf(fact_7030_subset__mset_Oasymp__greater,axiom,
! [A: $tType] :
( asymp @ ( multiset @ A )
@ ^ [X4: multiset @ A,Y4: multiset @ A] : ( subset_mset @ A @ Y4 @ X4 ) ) ).
% subset_mset.asymp_greater
thf(fact_7031_mult__cancel__add__mset,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A ),Uu: A,X6: multiset @ A,Y7: multiset @ A] :
( ( trans @ A @ S2 )
=> ( ( irrefl @ A @ S2 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( add_mset @ A @ Uu @ X6 ) @ ( add_mset @ A @ Uu @ Y7 ) ) @ ( mult @ A @ S2 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X6 @ Y7 ) @ ( mult @ A @ S2 ) ) ) ) ) ).
% mult_cancel_add_mset
thf(fact_7032_mult__cancel,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A ),X6: multiset @ A,Z5: multiset @ A,Y7: multiset @ A] :
( ( trans @ A @ S2 )
=> ( ( irrefl @ A @ S2 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X6 @ Z5 ) @ ( plus_plus @ ( multiset @ A ) @ Y7 @ Z5 ) ) @ ( mult @ A @ S2 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X6 @ Y7 ) @ ( mult @ A @ S2 ) ) ) ) ) ).
% mult_cancel
thf(fact_7033_mult__cancel__max,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A ),X6: multiset @ A,Y7: multiset @ A] :
( ( trans @ A @ S2 )
=> ( ( irrefl @ A @ S2 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X6 @ Y7 ) @ ( mult @ A @ S2 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X6 @ Y7 ) @ ( minus_minus @ ( multiset @ A ) @ Y7 @ X6 ) ) @ ( mult @ A @ S2 ) ) ) ) ) ).
% mult_cancel_max
thf(fact_7034_one__step__implies__mult,axiom,
! [A: $tType,J5: multiset @ A,K5: multiset @ A,R3: set @ ( product_prod @ A @ A ),I5: multiset @ A] :
( ( J5
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ K5 ) )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ J5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa2 ) @ R3 ) ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I5 @ K5 ) @ ( plus_plus @ ( multiset @ A ) @ I5 @ J5 ) ) @ ( mult @ A @ R3 ) ) ) ) ).
% one_step_implies_mult
thf(fact_7035_multp__code__iff__mult,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > A > $o,N7: multiset @ A,M6: multiset @ A] :
( ( irrefl @ A @ R2 )
=> ( ( trans @ A @ R2 )
=> ( ! [X3: A,Y3: A] :
( ( P @ X3 @ Y3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 ) )
=> ( ( multp_code @ A @ P @ N7 @ M6 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ M6 ) @ ( mult @ A @ R2 ) ) ) ) ) ) ).
% multp_code_iff_mult
thf(fact_7036_multeqp__code__iff__reflcl__mult,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > A > $o,N7: multiset @ A,M6: multiset @ A] :
( ( irrefl @ A @ R2 )
=> ( ( trans @ A @ R2 )
=> ( ! [X3: A,Y3: A] :
( ( P @ X3 @ Y3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 ) )
=> ( ( multeqp_code @ A @ P @ N7 @ M6 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N7 @ M6 ) @ ( sup_sup @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R2 ) @ ( id2 @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% multeqp_code_iff_reflcl_mult
thf(fact_7037_multiset_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_mset @ A @ B )
= ( ^ [R6: A > B > $o,A7: multiset @ A,B6: multiset @ B] :
? [Z7: multiset @ ( product_prod @ A @ B )] :
( ( member @ ( multiset @ ( product_prod @ A @ B ) ) @ Z7
@ ( collect @ ( multiset @ ( product_prod @ A @ B ) )
@ ^ [X4: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( image_mset @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z7 )
= A7 )
& ( ( image_mset @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z7 )
= B6 ) ) ) ) ).
% multiset.in_rel
thf(fact_7038_multp__code__def,axiom,
! [A: $tType] :
( ( multp_code @ A )
= ( ^ [P4: A > A > $o,N8: multiset @ A,M10: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ M10 @ ( inter_mset @ A @ M10 @ N8 ) )
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ N8 @ ( inter_mset @ A @ M10 @ N8 ) ) ) )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M10 @ ( inter_mset @ A @ M10 @ N8 ) ) ) )
& ( P4 @ X4 @ Y4 ) ) ) ) ) ) ).
% multp_code_def
thf(fact_7039_set__mset__inter,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A] :
( ( set_mset @ A @ ( inter_mset @ A @ A5 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ ( set_mset @ A @ A5 ) @ ( set_mset @ A @ B4 ) ) ) ).
% set_mset_inter
thf(fact_7040_multiset_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: multiset @ A,G3: B > C,Y: multiset @ B] :
( ( rel_mset @ A @ C @ Sa @ X @ ( image_mset @ B @ C @ G3 @ Y ) )
= ( rel_mset @ A @ B
@ ^ [X4: A,Y4: B] : ( Sa @ X4 @ ( G3 @ Y4 ) )
@ X
@ Y ) ) ).
% multiset.rel_map(2)
thf(fact_7041_multiset_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I2: A > C,X: multiset @ A,Y: multiset @ B] :
( ( rel_mset @ C @ B @ Sb @ ( image_mset @ A @ C @ I2 @ X ) @ Y )
= ( rel_mset @ A @ B
@ ^ [X4: A] : ( Sb @ ( I2 @ X4 ) )
@ X
@ Y ) ) ).
% multiset.rel_map(1)
thf(fact_7042_multiset_Orel__mono,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,Ra2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ Ra2 )
=> ( ord_less_eq @ ( ( multiset @ A ) > ( multiset @ B ) > $o ) @ ( rel_mset @ A @ B @ R2 ) @ ( rel_mset @ A @ B @ Ra2 ) ) ) ).
% multiset.rel_mono
thf(fact_7043_mult__cancel__max0,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A ),X6: multiset @ A,Y7: multiset @ A] :
( ( trans @ A @ S2 )
=> ( ( irrefl @ A @ S2 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X6 @ Y7 ) @ ( mult @ A @ S2 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X6 @ ( inter_mset @ A @ X6 @ Y7 ) ) @ ( minus_minus @ ( multiset @ A ) @ Y7 @ ( inter_mset @ A @ X6 @ Y7 ) ) ) @ ( mult @ A @ S2 ) ) ) ) ) ).
% mult_cancel_max0
thf(fact_7044_multeqp__code__def,axiom,
! [A: $tType] :
( ( multeqp_code @ A )
= ( ^ [P4: A > A > $o,N8: multiset @ A,M10: multiset @ A] :
! [X4: A] :
( ( member @ A @ X4 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ N8 @ ( inter_mset @ A @ M10 @ N8 ) ) ) )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M10 @ ( inter_mset @ A @ M10 @ N8 ) ) ) )
& ( P4 @ X4 @ Y4 ) ) ) ) ) ).
% multeqp_code_def
thf(fact_7045_count__image__mset,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: multiset @ B,X: A] :
( ( count @ A @ ( image_mset @ B @ A @ F3 @ A5 ) @ X )
= ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ A5 ) @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( set_mset @ B @ A5 ) ) ) ) ).
% count_image_mset
thf(fact_7046_set__mset__replicate__mset__subset,axiom,
! [A: $tType,N2: nat,X: A] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N2 @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N2 @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_7047_mset__empty__count,axiom,
! [A: $tType,M6: multiset @ A] :
( ( ! [P6: A] :
( ( count @ A @ M6 @ P6 )
= ( zero_zero @ nat ) ) )
= ( M6
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% mset_empty_count
thf(fact_7048_count__greater__zero__iff,axiom,
! [A: $tType,M6: multiset @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ M6 @ X ) )
= ( member @ A @ X @ ( set_mset @ A @ M6 ) ) ) ).
% count_greater_zero_iff
thf(fact_7049_count__greater__eq__one__iff,axiom,
! [A: $tType,M6: multiset @ A,X: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( count @ A @ M6 @ X ) )
= ( member @ A @ X @ ( set_mset @ A @ M6 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_7050_in__replicate__mset,axiom,
! [A: $tType,X: A,N2: nat,Y: A] :
( ( member @ A @ X @ ( set_mset @ A @ ( replicate_mset @ A @ N2 @ Y ) ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( X = Y ) ) ) ).
% in_replicate_mset
thf(fact_7051_sum__mset__replicate__mset,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N2: nat,A3: A] :
( ( comm_m7189776963980413722m_mset @ A @ ( replicate_mset @ A @ N2 @ A3 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ A3 ) ) ) ).
% sum_mset_replicate_mset
thf(fact_7052_count__greater__eq__Suc__zero__iff,axiom,
! [A: $tType,M6: multiset @ A,X: A] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( count @ A @ M6 @ X ) )
= ( member @ A @ X @ ( set_mset @ A @ M6 ) ) ) ).
% count_greater_eq_Suc_zero_iff
thf(fact_7053_msubseteq__replicate__msetE,axiom,
! [A: $tType,A5: multiset @ A,N2: nat,A3: A] :
( ( subseteq_mset @ A @ A5 @ ( replicate_mset @ A @ N2 @ A3 ) )
=> ~ ! [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N2 )
=> ( A5
!= ( replicate_mset @ A @ M5 @ A3 ) ) ) ) ).
% msubseteq_replicate_msetE
thf(fact_7054_count__le__replicate__mset__subset__eq,axiom,
! [A: $tType,N2: nat,M6: multiset @ A,X: A] :
( ( ord_less_eq @ nat @ N2 @ ( count @ A @ M6 @ X ) )
= ( subseteq_mset @ A @ ( replicate_mset @ A @ N2 @ X ) @ M6 ) ) ).
% count_le_replicate_mset_subset_eq
thf(fact_7055_mset__subset__eqI,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A] :
( ! [A6: A] : ( ord_less_eq @ nat @ ( count @ A @ A5 @ A6 ) @ ( count @ A @ B4 @ A6 ) )
=> ( subseteq_mset @ A @ A5 @ B4 ) ) ).
% mset_subset_eqI
thf(fact_7056_subseteq__mset__def,axiom,
! [A: $tType] :
( ( subseteq_mset @ A )
= ( ^ [A8: multiset @ A,B7: multiset @ A] :
! [A7: A] : ( ord_less_eq @ nat @ ( count @ A @ A8 @ A7 ) @ ( count @ A @ B7 @ A7 ) ) ) ) ).
% subseteq_mset_def
thf(fact_7057_mset__subset__eq__count,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A,A3: A] :
( ( subseteq_mset @ A @ A5 @ B4 )
=> ( ord_less_eq @ nat @ ( count @ A @ A5 @ A3 ) @ ( count @ A @ B4 @ A3 ) ) ) ).
% mset_subset_eq_count
thf(fact_7058_count__sum,axiom,
! [A: $tType,B: $tType,F3: B > ( multiset @ A ),A5: set @ B,X: A] :
( ( count @ A @ ( groups7311177749621191930dd_sum @ B @ ( multiset @ A ) @ F3 @ A5 ) @ X )
= ( groups7311177749621191930dd_sum @ B @ nat
@ ^ [A7: B] : ( count @ A @ ( F3 @ A7 ) @ X )
@ A5 ) ) ).
% count_sum
thf(fact_7059_in__diff__count,axiom,
! [A: $tType,A3: A,M6: multiset @ A,N7: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M6 @ N7 ) ) )
= ( ord_less @ nat @ ( count @ A @ N7 @ A3 ) @ ( count @ A @ M6 @ A3 ) ) ) ).
% in_diff_count
thf(fact_7060_count__ne__remove,axiom,
! [A: $tType,X: A,T6: A,S: multiset @ A] :
( ( X != T6 )
=> ( ( count @ A @ S @ X )
= ( count @ A @ ( minus_minus @ ( multiset @ A ) @ S @ ( add_mset @ A @ T6 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ X ) ) ) ).
% count_ne_remove
thf(fact_7061_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_7062_set__mset__diff,axiom,
! [A: $tType,M6: multiset @ A,N7: multiset @ A] :
( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M6 @ N7 ) )
= ( collect @ A
@ ^ [A7: A] : ( ord_less @ nat @ ( count @ A @ N7 @ A7 ) @ ( count @ A @ M6 @ A7 ) ) ) ) ).
% set_mset_diff
thf(fact_7063_count__mset,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( count @ A @ ( mset @ A @ Xs ) @ X )
= ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X )
@ Xs ) ) ) ).
% count_mset
thf(fact_7064_count__image__mset_H,axiom,
! [A: $tType,B: $tType,F3: B > A,X6: multiset @ B,Y: A] :
( ( count @ A @ ( image_mset @ B @ A @ F3 @ X6 ) @ Y )
= ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ X6 )
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ ( set_mset @ B @ X6 ) )
& ( Y
= ( F3 @ X4 ) ) ) ) ) ) ).
% count_image_mset'
thf(fact_7065_size__multiset__overloaded__eq,axiom,
! [B: $tType] :
( ( size_size @ ( multiset @ B ) )
= ( ^ [X4: multiset @ B] : ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ X4 ) @ ( set_mset @ B @ X4 ) ) ) ) ).
% size_multiset_overloaded_eq
thf(fact_7066_count__induct,axiom,
! [A: $tType,Y: A > nat,P: ( A > nat ) > $o] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
=> ( ! [X3: multiset @ A] : ( P @ ( count @ A @ X3 ) )
=> ( P @ Y ) ) ) ).
% count_induct
thf(fact_7067_count__cases,axiom,
! [A: $tType,Y: A > nat] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
=> ~ ! [X3: multiset @ A] :
( Y
!= ( count @ A @ X3 ) ) ) ).
% count_cases
thf(fact_7068_count,axiom,
! [A: $tType,X: multiset @ A] :
( member @ ( A > nat ) @ ( count @ A @ X )
@ ( collect @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) ) ).
% count
thf(fact_7069_image__mset__const__eq,axiom,
! [B: $tType,A: $tType,C2: A,M6: multiset @ B] :
( ( image_mset @ B @ A
@ ^ [A7: B] : C2
@ M6 )
= ( replicate_mset @ A @ ( size_size @ ( multiset @ B ) @ M6 ) @ C2 ) ) ).
% image_mset_const_eq
thf(fact_7070_Inf__multiset_Orep__eq,axiom,
! [A: $tType,X: set @ ( multiset @ A )] :
( ( count @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ X ) )
= ( ^ [I4: A] :
( if @ nat
@ ( ( image2 @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ X )
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ ( image2 @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ X ) ) ) ) ) ) ).
% Inf_multiset.rep_eq
thf(fact_7071_count__Inf__multiset__nonempty,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: A] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( count @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) @ X )
= ( complete_Inf_Inf @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X11: multiset @ A] : ( count @ A @ X11 @ X )
@ A5 ) ) ) ) ).
% count_Inf_multiset_nonempty
thf(fact_7072_count__mset__gt__0,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ ( mset @ A @ Xs ) @ X ) ) ) ).
% count_mset_gt_0
thf(fact_7073_replicate__mset__msubseteq__iff,axiom,
! [A: $tType,M: nat,A3: A,N2: nat,B2: A] :
( ( subseteq_mset @ A @ ( replicate_mset @ A @ M @ A3 ) @ ( replicate_mset @ A @ N2 @ B2 ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( ( A3 = B2 )
& ( ord_less_eq @ nat @ M @ N2 ) ) ) ) ).
% replicate_mset_msubseteq_iff
thf(fact_7074_sum__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C2: A,A5: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( if @ A @ ( Y = X4 ) @ C2 @ ( zero_zero @ A ) )
@ A5 ) )
= ( times_times @ A @ C2 @ ( semiring_1_of_nat @ A @ ( count @ B @ A5 @ Y ) ) ) ) ) ).
% sum_mset_delta'
thf(fact_7075_sum__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [Y: B,C2: A,A5: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( if @ A @ ( X4 = Y ) @ C2 @ ( zero_zero @ A ) )
@ A5 ) )
= ( times_times @ A @ C2 @ ( semiring_1_of_nat @ A @ ( count @ B @ A5 @ Y ) ) ) ) ) ).
% sum_mset_delta
thf(fact_7076_bdd__above__multiset__imp__finite__support,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( finite_finite2 @ A
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( multiset @ A ) @ ( set @ A )
@ ^ [X11: multiset @ A] :
( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ X11 @ X4 ) ) )
@ A5 ) ) ) ) ) ).
% bdd_above_multiset_imp_finite_support
thf(fact_7077_size__multiset__eq,axiom,
! [A: $tType] :
( ( size_multiset @ A )
= ( ^ [F4: A > nat,M10: multiset @ A] :
( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count @ A @ M10 @ X4 ) @ ( suc @ ( F4 @ X4 ) ) )
@ ( set_mset @ A @ M10 ) ) ) ) ).
% size_multiset_eq
thf(fact_7078_subset__mset_Obdd__above__Un,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) )
= ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
& ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 ) ) ) ).
% subset_mset.bdd_above_Un
thf(fact_7079_subset__mset_Obdd__above__empty,axiom,
! [A: $tType] : ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ).
% subset_mset.bdd_above_empty
thf(fact_7080_subset__mset_Obdd__above__UN,axiom,
! [A: $tType,B: $tType,I5: set @ B,A5: B > ( set @ ( multiset @ A ) )] :
( ( finite_finite2 @ B @ I5 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( complete_Sup_Sup @ ( set @ ( multiset @ A ) ) @ ( image2 @ B @ ( set @ ( multiset @ A ) ) @ A5 @ I5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( A5 @ X4 ) ) ) ) ) ) ).
% subset_mset.bdd_above_UN
thf(fact_7081_subset__mset_Obdd__above__mono,axiom,
! [A: $tType,B4: set @ ( multiset @ A ),A5: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A5 @ B4 )
=> ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 ) ) ) ).
% subset_mset.bdd_above_mono
thf(fact_7082_subset__mset_Obdd__above__Int2,axiom,
! [A: $tType,B4: set @ ( multiset @ A ),A5: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) ) ).
% subset_mset.bdd_above_Int2
thf(fact_7083_subset__mset_Obdd__above__Int1,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) ) ).
% subset_mset.bdd_above_Int1
thf(fact_7084_subset__mset_OcSup__mono,axiom,
! [A: $tType,B4: set @ ( multiset @ A ),A5: set @ ( multiset @ A )] :
( ( B4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ! [B5: multiset @ A] :
( ( member @ ( multiset @ A ) @ B5 @ B4 )
=> ? [X7: multiset @ A] :
( ( member @ ( multiset @ A ) @ X7 @ A5 )
& ( subseteq_mset @ A @ B5 @ X7 ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ B4 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ A5 ) ) ) ) ) ).
% subset_mset.cSup_mono
thf(fact_7085_subset__mset_OcSup__le__iff,axiom,
! [A: $tType,S: set @ ( multiset @ A ),A3: multiset @ A] :
( ( S
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ S )
=> ( ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ S ) @ A3 )
= ( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ S )
=> ( subseteq_mset @ A @ X4 @ A3 ) ) ) ) ) ) ).
% subset_mset.cSup_le_iff
thf(fact_7086_size__multiset__overloaded__def,axiom,
! [B: $tType] :
( ( size_size @ ( multiset @ B ) )
= ( size_multiset @ B
@ ^ [Uu2: B] : ( zero_zero @ nat ) ) ) ).
% size_multiset_overloaded_def
thf(fact_7087_subset__mset_OcSUP__le__iff,axiom,
! [A: $tType,B: $tType,A5: set @ B,F3: B > ( multiset @ A ),U: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
=> ( ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( subseteq_mset @ A @ ( F3 @ X4 ) @ U ) ) ) ) ) ) ).
% subset_mset.cSUP_le_iff
thf(fact_7088_subset__mset_OcSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType,A5: set @ B,G3: C > ( multiset @ A ),B4: set @ C,F3: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G3 @ B4 ) )
=> ( ! [N5: B] :
( ( member @ B @ N5 @ A5 )
=> ? [X7: C] :
( ( member @ C @ X7 @ B4 )
& ( subseteq_mset @ A @ ( F3 @ N5 ) @ ( G3 @ X7 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ G3 @ B4 ) ) ) ) ) ) ).
% subset_mset.cSUP_mono
thf(fact_7089_subset__mset_OcSup__subset__mono,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A5 @ B4 )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B4 ) ) ) ) ) ).
% subset_mset.cSup_subset_mono
thf(fact_7090_subset__mset_OcSUP__lessD,axiom,
! [B: $tType,A: $tType,F3: B > ( multiset @ A ),A5: set @ B,Y: multiset @ A,I2: B] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
=> ( ( subset_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) @ Y )
=> ( ( member @ B @ I2 @ A5 )
=> ( subset_mset @ A @ ( F3 @ I2 ) @ Y ) ) ) ) ).
% subset_mset.cSUP_lessD
thf(fact_7091_subset__mset_OcSup__cInf,axiom,
! [A: $tType,S: set @ ( multiset @ A )] :
( ( S
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ S )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ S )
= ( complete_Inf_Inf @ ( multiset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [X4: multiset @ A] :
! [Y4: multiset @ A] :
( ( member @ ( multiset @ A ) @ Y4 @ S )
=> ( subseteq_mset @ A @ Y4 @ X4 ) ) ) ) ) ) ) ).
% subset_mset.cSup_cInf
thf(fact_7092_subset__mset_OcSUP__subset__mono,axiom,
! [A: $tType,B: $tType,A5: set @ B,G3: B > ( multiset @ A ),B4: set @ B,F3: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G3 @ B4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( subseteq_mset @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G3 @ B4 ) ) ) ) ) ) ) ).
% subset_mset.cSUP_subset_mono
thf(fact_7093_set__mset__Sup,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( set_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( multiset @ A ) @ ( set @ A ) @ ( set_mset @ A ) @ A5 ) ) ) ) ).
% set_mset_Sup
thf(fact_7094_count__Sup__multiset__nonempty,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: A] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( count @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A5 ) @ X )
= ( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X11: multiset @ A] : ( count @ A @ X11 @ X )
@ A5 ) ) ) ) ) ).
% count_Sup_multiset_nonempty
thf(fact_7095_Sup__multiset__in__multiset,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( 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 )
@ A5 ) ) ) ) ) ) ) ).
% Sup_multiset_in_multiset
thf(fact_7096_subset__mset_Omono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F3: ( multiset @ A ) > B,A5: C > ( multiset @ A ),I5: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A5 @ X4 ) )
@ I5 ) )
@ ( F3 @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) ) ) ) ) ) ) ) ).
% subset_mset.mono_cSUP
thf(fact_7097_subset__mset_Omono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F3: ( multiset @ A ) > B,A5: set @ ( multiset @ A )] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ ( multiset @ A ) @ B @ F3 @ A5 ) ) @ ( F3 @ ( complete_Sup_Sup @ ( multiset @ A ) @ A5 ) ) ) ) ) ) ) ).
% subset_mset.mono_cSup
thf(fact_7098_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( mono @ A @ B )
= ( mono @ A @ B ) ) ) ).
% order.mono.cong
thf(fact_7099_subset__mset_OmonoD,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: ( multiset @ A ) > B,X: multiset @ A,Y: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 )
=> ( ( subseteq_mset @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% subset_mset.monoD
thf(fact_7100_subset__mset_OmonoE,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: ( multiset @ A ) > B,X: multiset @ A,Y: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 )
=> ( ( subseteq_mset @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% subset_mset.monoE
thf(fact_7101_subset__mset_OmonoI,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: ( multiset @ A ) > B] :
( ! [X3: multiset @ A,Y3: multiset @ A] :
( ( subseteq_mset @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) )
=> ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 ) ) ) ).
% subset_mset.monoI
thf(fact_7102_subset__mset_Omono__def,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: ( multiset @ A ) > B] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 )
= ( ! [X4: multiset @ A,Y4: multiset @ A] :
( ( subseteq_mset @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) ) ) ) ) ).
% subset_mset.mono_def
thf(fact_7103_subset__mset_Omono__inf,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ! [F3: ( multiset @ A ) > B,A5: multiset @ A,B4: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( inter_mset @ A @ A5 @ B4 ) ) @ ( inf_inf @ B @ ( F3 @ A5 ) @ ( F3 @ B4 ) ) ) ) ) ).
% subset_mset.mono_inf
thf(fact_7104_Inf__multiset__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( multiset @ A ) )
= ( map_fun @ ( set @ ( multiset @ A ) ) @ ( set @ ( A > nat ) ) @ ( A > nat ) @ ( multiset @ A ) @ ( image2 @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) ) @ ( abs_multiset @ A )
@ ^ [A8: set @ ( A > nat ),I4: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ A8 ) ) ) ) ) ).
% Inf_multiset_def
thf(fact_7105_Sup__multiset__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( multiset @ A ) )
= ( ^ [A8: set @ ( multiset @ A )] :
( if @ ( multiset @ A )
@ ( ( A8
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
& ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A8 ) )
@ ( abs_multiset @ A
@ ^ [I4: A] :
( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X11: multiset @ A] : ( count @ A @ X11 @ I4 )
@ A8 ) ) )
@ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Sup_multiset_def
thf(fact_7106_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_7107_zero__multiset__def,axiom,
! [A: $tType] :
( ( zero_zero @ ( multiset @ A ) )
= ( abs_multiset @ A
@ ^ [A7: A] : ( zero_zero @ nat ) ) ) ).
% zero_multiset_def
thf(fact_7108_Abs__multiset__cases,axiom,
! [A: $tType,X: multiset @ A] :
~ ! [Y3: A > nat] :
( ( X
= ( abs_multiset @ A @ Y3 ) )
=> ~ ( member @ ( A > nat ) @ Y3
@ ( collect @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) ) ) ).
% Abs_multiset_cases
thf(fact_7109_Abs__multiset__induct,axiom,
! [A: $tType,P: ( multiset @ A ) > $o,X: multiset @ A] :
( ! [Y3: A > nat] :
( ( member @ ( A > nat ) @ Y3
@ ( collect @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
=> ( P @ ( abs_multiset @ A @ Y3 ) ) )
=> ( P @ X ) ) ).
% Abs_multiset_induct
thf(fact_7110_Abs__multiset__inject,axiom,
! [A: $tType,X: A > nat,Y: A > nat] :
( ( member @ ( A > nat ) @ X
@ ( collect @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
=> ( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
=> ( ( ( abs_multiset @ A @ X )
= ( abs_multiset @ A @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_multiset_inject
thf(fact_7111_plus__multiset__def,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( count @ A ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [M10: A > nat,N8: A > nat,A7: A] : ( plus_plus @ nat @ ( M10 @ A7 ) @ ( N8 @ A7 ) ) ) ) ).
% plus_multiset_def
thf(fact_7112_minus__multiset__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( multiset @ A ) )
= ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( count @ A ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [M10: A > nat,N8: A > nat,A7: A] : ( minus_minus @ nat @ ( M10 @ A7 ) @ ( N8 @ A7 ) ) ) ) ).
% minus_multiset_def
thf(fact_7113_Abs__multiset__inverse,axiom,
! [A: $tType,Y: A > nat] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
=> ( ( count @ A @ ( abs_multiset @ A @ Y ) )
= Y ) ) ).
% Abs_multiset_inverse
thf(fact_7114_add__mset__def,axiom,
! [A: $tType] :
( ( add_mset @ A )
= ( map_fun @ A @ A @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( id @ A ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [A7: A,M10: A > nat,B6: A] : ( if @ nat @ ( B6 = A7 ) @ ( suc @ ( M10 @ B6 ) ) @ ( M10 @ B6 ) ) ) ) ).
% add_mset_def
thf(fact_7115_subset__mset_Omono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F3: ( multiset @ A ) > B,A5: C > ( multiset @ A ),I5: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F3 @ ( A5 @ X4 ) )
@ I5 ) ) ) ) ) ) ) ).
% subset_mset.mono_cINF
thf(fact_7116_subset__mset_Omono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F3: ( multiset @ A ) > B,A5: set @ ( multiset @ A )] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ ( multiset @ A ) @ B @ F3 @ A5 ) ) ) ) ) ) ) ).
% subset_mset.mono_cInf
thf(fact_7117_subset__mset_Obdd__below__empty,axiom,
! [A: $tType] : ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ).
% subset_mset.bdd_below_empty
thf(fact_7118_subset__mset_Obdd__below__Un,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) )
= ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
& ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 ) ) ) ).
% subset_mset.bdd_below_Un
thf(fact_7119_subset__mset_Obdd__below__image__inf,axiom,
! [A: $tType,B: $tType,F3: B > ( multiset @ A ),G3: B > ( multiset @ A ),A5: set @ B] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X4: B] : ( inter_mset @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ A5 ) )
= ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
& ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G3 @ A5 ) ) ) ) ).
% subset_mset.bdd_below_image_inf
thf(fact_7120_subset__mset_Obdd__below__UN,axiom,
! [A: $tType,B: $tType,I5: set @ B,A5: B > ( set @ ( multiset @ A ) )] :
( ( finite_finite2 @ B @ I5 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( complete_Sup_Sup @ ( set @ ( multiset @ A ) ) @ ( image2 @ B @ ( set @ ( multiset @ A ) ) @ A5 @ I5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( A5 @ X4 ) ) ) ) ) ) ).
% subset_mset.bdd_below_UN
thf(fact_7121_subset__mset_Obdd__below__Int1,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) ) ).
% subset_mset.bdd_below_Int1
thf(fact_7122_subset__mset_Obdd__below__Int2,axiom,
! [A: $tType,B4: set @ ( multiset @ A ),A5: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) ) ).
% subset_mset.bdd_below_Int2
thf(fact_7123_subset__mset_Obdd__below__mono,axiom,
! [A: $tType,B4: set @ ( multiset @ A ),A5: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A5 @ B4 )
=> ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 ) ) ) ).
% subset_mset.bdd_below_mono
thf(fact_7124_subset__mset_Ole__cInf__iff,axiom,
! [A: $tType,S: set @ ( multiset @ A ),A3: multiset @ A] :
( ( S
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ S )
=> ( ( subseteq_mset @ A @ A3 @ ( complete_Inf_Inf @ ( multiset @ A ) @ S ) )
= ( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ S )
=> ( subseteq_mset @ A @ A3 @ X4 ) ) ) ) ) ) ).
% subset_mset.le_cInf_iff
thf(fact_7125_subset__mset_OcInf__mono,axiom,
! [A: $tType,B4: set @ ( multiset @ A ),A5: set @ ( multiset @ A )] :
( ( B4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ! [B5: multiset @ A] :
( ( member @ ( multiset @ A ) @ B5 @ B4 )
=> ? [X7: multiset @ A] :
( ( member @ ( multiset @ A ) @ X7 @ A5 )
& ( subseteq_mset @ A @ X7 @ B5 ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B4 ) ) ) ) ) ).
% subset_mset.cInf_mono
thf(fact_7126_subset__mset_OcINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType,B4: set @ B,F3: C > ( multiset @ A ),A5: set @ C,G3: B > ( multiset @ A )] :
( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F3 @ A5 ) )
=> ( ! [M5: B] :
( ( member @ B @ M5 @ B4 )
=> ? [X7: C] :
( ( member @ C @ X7 @ A5 )
& ( subseteq_mset @ A @ ( F3 @ X7 ) @ ( G3 @ M5 ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G3 @ B4 ) ) ) ) ) ) ).
% subset_mset.cINF_mono
thf(fact_7127_subset__mset_Ole__cINF__iff,axiom,
! [A: $tType,B: $tType,A5: set @ B,F3: B > ( multiset @ A ),U: multiset @ A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
=> ( ( subseteq_mset @ A @ U @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( subseteq_mset @ A @ U @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% subset_mset.le_cINF_iff
thf(fact_7128_subset__mset_OcInf__superset__mono,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A5 @ B4 )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ B4 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) ) ) ) ) ).
% subset_mset.cInf_superset_mono
thf(fact_7129_subset__mset_Oless__cINF__D,axiom,
! [A: $tType,B: $tType,F3: B > ( multiset @ A ),A5: set @ B,Y: multiset @ A,I2: B] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
=> ( ( subset_mset @ A @ Y @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) )
=> ( ( member @ B @ I2 @ A5 )
=> ( subset_mset @ A @ Y @ ( F3 @ I2 ) ) ) ) ) ).
% subset_mset.less_cINF_D
thf(fact_7130_subset__mset_OcINF__superset__mono,axiom,
! [A: $tType,B: $tType,A5: set @ B,G3: B > ( multiset @ A ),B4: set @ B,F3: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G3 @ B4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B4 )
=> ( subseteq_mset @ A @ ( G3 @ X3 ) @ ( F3 @ X3 ) ) )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G3 @ B4 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) ) ) ) ) ) ).
% subset_mset.cINF_superset_mono
thf(fact_7131_subset__mset_OcInf__insert__If,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A3: multiset @ A] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X6 )
=> ( ( ( X6
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert @ ( multiset @ A ) @ A3 @ X6 ) )
= A3 ) )
& ( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert @ ( multiset @ A ) @ A3 @ X6 ) )
= ( inter_mset @ A @ A3 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X6 ) ) ) ) ) ) ).
% subset_mset.cInf_insert_If
thf(fact_7132_subset__mset_OcInf__insert,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A3: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X6 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( insert @ ( multiset @ A ) @ A3 @ X6 ) )
= ( inter_mset @ A @ A3 @ ( complete_Inf_Inf @ ( multiset @ A ) @ X6 ) ) ) ) ) ).
% subset_mset.cInf_insert
thf(fact_7133_subset__mset_OcInf__le__cSup,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( subseteq_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ A5 ) ) ) ) ) ).
% subset_mset.cInf_le_cSup
thf(fact_7134_subset__mset_Oless__eq__cInf__inter,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B4 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) ) ) ) ) ).
% subset_mset.less_eq_cInf_inter
thf(fact_7135_subset__mset_OcINF__inf__distrib,axiom,
! [A: $tType,B: $tType,A5: set @ B,F3: B > ( multiset @ A ),G3: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G3 @ A5 ) )
=> ( ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G3 @ A5 ) ) )
= ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A7: B] : ( inter_mset @ A @ ( F3 @ A7 ) @ ( G3 @ A7 ) )
@ A5 ) ) ) ) ) ) ).
% subset_mset.cINF_inf_distrib
thf(fact_7136_subset__mset_OcInf__union__distrib,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) )
= ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ B4 ) ) ) ) ) ) ) ).
% subset_mset.cInf_union_distrib
thf(fact_7137_subset__mset_OcInf__cSup,axiom,
! [A: $tType,S: set @ ( multiset @ A )] :
( ( S
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ S )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ S )
= ( complete_Sup_Sup @ ( multiset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [X4: multiset @ A] :
! [Y4: multiset @ A] :
( ( member @ ( multiset @ A ) @ Y4 @ S )
=> ( subseteq_mset @ A @ X4 @ Y4 ) ) ) ) ) ) ) ).
% subset_mset.cInf_cSup
thf(fact_7138_subset__mset_OcINF__insert,axiom,
! [A: $tType,B: $tType,A5: set @ B,F3: B > ( multiset @ A ),A3: B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ ( insert @ B @ A3 @ A5 ) ) )
= ( inter_mset @ A @ ( F3 @ A3 ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) ) ) ) ) ).
% subset_mset.cINF_insert
thf(fact_7139_subset__mset_OcINF__union,axiom,
! [A: $tType,B: $tType,A5: set @ B,F3: B > ( multiset @ A ),B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ B4 ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ B4 ) ) ) ) ) ) ) ) ).
% subset_mset.cINF_union
thf(fact_7140_repeat__mset__def,axiom,
! [A: $tType] :
( ( repeat_mset @ A )
= ( map_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( id @ nat ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [N: nat,M10: A > nat,A7: A] : ( times_times @ nat @ N @ ( M10 @ A7 ) ) ) ) ).
% repeat_mset_def
thf(fact_7141_fold__mset__def,axiom,
! [B: $tType,A: $tType] :
( ( fold_mset @ A @ B )
= ( ^ [F4: A > B > B,S6: B,M10: multiset @ A] :
( finite_fold @ A @ B
@ ^ [X4: A] : ( compow @ ( B > B ) @ ( count @ A @ M10 @ X4 ) @ ( F4 @ X4 ) )
@ S6
@ ( set_mset @ A @ M10 ) ) ) ) ).
% fold_mset_def
thf(fact_7142_mset__subseteq__add__iff1,axiom,
! [A: $tType,J: nat,I2: nat,U: multiset @ A,M: multiset @ A,N2: multiset @ A] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I2 @ U ) @ M ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N2 ) )
= ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% mset_subseteq_add_iff1
thf(fact_7143_mset__subseteq__add__iff2,axiom,
! [A: $tType,I2: nat,J: nat,U: multiset @ A,M: multiset @ A,N2: multiset @ A] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I2 @ U ) @ M ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N2 ) )
= ( subseteq_mset @ A @ M @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% mset_subseteq_add_iff2
thf(fact_7144_mset__subset__add__iff2,axiom,
! [A: $tType,I2: nat,J: nat,U: multiset @ A,M: multiset @ A,N2: multiset @ A] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I2 @ U ) @ M ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N2 ) )
= ( subset_mset @ A @ M @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% mset_subset_add_iff2
thf(fact_7145_mset__subset__add__iff1,axiom,
! [A: $tType,J: nat,I2: nat,U: multiset @ A,M: multiset @ A,N2: multiset @ A] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I2 @ U ) @ M ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N2 ) )
= ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% mset_subset_add_iff1
thf(fact_7146_sorted__list__of__multiset__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord6283353356039996273ltiset @ A )
= ( fold_mset @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4 )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_multiset_def
thf(fact_7147_subset__mset_OInf__fin_Oremove,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( member @ ( multiset @ A ) @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.remove
thf(fact_7148_subset__mset_OInf__fin_Oinsert__remove,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert @ ( multiset @ A ) @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert @ ( multiset @ A ) @ X @ A5 ) )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.insert_remove
thf(fact_7149_subset__mset_OInf__fin_Osingleton,axiom,
! [A: $tType,X: multiset @ A] :
( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= X ) ).
% subset_mset.Inf_fin.singleton
thf(fact_7150_subset__mset_OInf__fin_Oinsert,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert @ ( multiset @ A ) @ X @ A5 ) )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) ) ) ) ) ).
% subset_mset.Inf_fin.insert
thf(fact_7151_subset__mset_OInf__fin_Oeq__fold_H,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 )
= ( the2 @ ( multiset @ A )
@ ( finite_fold @ ( multiset @ A ) @ ( option @ ( multiset @ A ) )
@ ^ [X4: multiset @ A,Y4: option @ ( multiset @ A )] : ( some @ ( multiset @ A ) @ ( case_option @ ( multiset @ A ) @ ( multiset @ A ) @ X4 @ ( inter_mset @ A @ X4 ) @ Y4 ) )
@ ( none @ ( multiset @ A ) )
@ A5 ) ) ) ).
% subset_mset.Inf_fin.eq_fold'
thf(fact_7152_semilattice__inf_OInf__fin_Ocong,axiom,
! [A: $tType] :
( ( lattic8678736583308907530nf_fin @ A )
= ( lattic8678736583308907530nf_fin @ A ) ) ).
% semilattice_inf.Inf_fin.cong
thf(fact_7153_subset__mset_OInf__fin_Ohom__commute,axiom,
! [A: $tType,H: ( multiset @ A ) > ( multiset @ A ),N7: set @ ( multiset @ A )] :
( ! [X3: multiset @ A,Y3: multiset @ A] :
( ( H @ ( inter_mset @ A @ X3 @ Y3 ) )
= ( inter_mset @ A @ ( H @ X3 ) @ ( H @ Y3 ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( H @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ N7 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( image2 @ ( multiset @ A ) @ ( multiset @ A ) @ H @ N7 ) ) ) ) ) ) ).
% subset_mset.Inf_fin.hom_commute
thf(fact_7154_subset__mset_OInf__fin_Obounded__iff,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) )
= ( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ A5 )
=> ( subseteq_mset @ A @ X @ X4 ) ) ) ) ) ) ).
% subset_mset.Inf_fin.bounded_iff
thf(fact_7155_subset__mset_OInf__fin_OboundedI,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [A6: multiset @ A] :
( ( member @ ( multiset @ A ) @ A6 @ A5 )
=> ( subseteq_mset @ A @ X @ A6 ) )
=> ( subseteq_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) ) ) ) ) ).
% subset_mset.Inf_fin.boundedI
thf(fact_7156_subset__mset_OInf__fin_OboundedE,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) )
=> ! [A15: multiset @ A] :
( ( member @ ( multiset @ A ) @ A15 @ A5 )
=> ( subseteq_mset @ A @ X @ A15 ) ) ) ) ) ).
% subset_mset.Inf_fin.boundedE
thf(fact_7157_subset__mset_OcInf__eq__Inf__fin,axiom,
! [A: $tType,X6: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ X6 )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ X6 ) ) ) ) ).
% subset_mset.cInf_eq_Inf_fin
thf(fact_7158_subset__mset_OInf__fin_Oclosed,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A,Y3: multiset @ A] : ( member @ ( multiset @ A ) @ ( inter_mset @ A @ X3 @ Y3 ) @ ( insert @ ( multiset @ A ) @ X3 @ ( insert @ ( multiset @ A ) @ Y3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) )
=> ( member @ ( multiset @ A ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) @ A5 ) ) ) ) ).
% subset_mset.Inf_fin.closed
thf(fact_7159_subset__mset_OInf__fin_Oinsert__not__elem,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ~ ( member @ ( multiset @ A ) @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( insert @ ( multiset @ A ) @ X @ A5 ) )
= ( inter_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) ) ) ) ) ) ).
% subset_mset.Inf_fin.insert_not_elem
thf(fact_7160_subset__mset_OInf__fin_Osubset,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ B4 @ A5 )
=> ( ( inter_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B4 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) ) ) ) ) ).
% subset_mset.Inf_fin.subset
thf(fact_7161_subset__mset_OInf__fin_Ounion,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) )
= ( inter_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B4 ) ) ) ) ) ) ) ).
% subset_mset.Inf_fin.union
thf(fact_7162_subset__mset_OInf__fin_Osubset__imp,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A5 @ B4 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B4 )
=> ( subseteq_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B4 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) ) ) ) ) ).
% subset_mset.Inf_fin.subset_imp
thf(fact_7163_repeat__mset_Oabs__eq,axiom,
! [A: $tType,X: A > nat,Xa: nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) )
@ X
@ X )
=> ( ( repeat_mset @ A @ Xa @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [A7: A] : ( times_times @ nat @ Xa @ ( X @ A7 ) ) ) ) ) ).
% repeat_mset.abs_eq
thf(fact_7164_subset__mset_OcSUP__union,axiom,
! [A: $tType,B: $tType,A5: set @ B,F3: B > ( multiset @ A ),B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ B4 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ B4 ) ) ) ) ) ) ) ) ).
% subset_mset.cSUP_union
thf(fact_7165_set__mset__sup,axiom,
! [A: $tType,A5: multiset @ A,B4: multiset @ A] :
( ( set_mset @ A @ ( union_mset @ A @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ ( set_mset @ A @ A5 ) @ ( set_mset @ A @ B4 ) ) ) ).
% set_mset_sup
thf(fact_7166_subset__mset_Obdd__above__image__sup,axiom,
! [A: $tType,B: $tType,F3: B > ( multiset @ A ),G3: B > ( multiset @ A ),A5: set @ B] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X4: B] : ( union_mset @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ A5 ) )
= ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
& ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G3 @ A5 ) ) ) ) ).
% subset_mset.bdd_above_image_sup
thf(fact_7167_eq__onp__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P ) @ ( bNF_eq_onp @ A @ Q ) )
= ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% eq_onp_mono_iff
thf(fact_7168_eq__onp__le__eq,axiom,
! [A: $tType,P: A > $o] :
( ord_less_eq @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P )
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) ).
% eq_onp_le_eq
thf(fact_7169_rel__fun__eq__onp__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R2: A > $o,S: B > C > $o] :
( ( bNF_rel_fun @ A @ A @ B @ C @ ( bNF_eq_onp @ A @ R2 ) @ S )
= ( ^ [F4: A > B,G4: A > C] :
! [X4: A] :
( ( R2 @ X4 )
=> ( S @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% rel_fun_eq_onp_rel
thf(fact_7170_rel__fun__eq__eq__onp,axiom,
! [B: $tType,A: $tType,P: B > $o] :
( ( bNF_rel_fun @ A @ A @ B @ B
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( bNF_eq_onp @ B @ P ) )
= ( bNF_eq_onp @ ( A > B )
@ ^ [F4: A > B] :
! [X4: A] : ( P @ ( F4 @ X4 ) ) ) ) ).
% rel_fun_eq_eq_onp
thf(fact_7171_eq__onp__True,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A
@ ^ [Uu2: A] : $true )
= ( ^ [Y5: A,Z4: A] : Y5 = Z4 ) ) ).
% eq_onp_True
thf(fact_7172_eq__onp__def,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A )
= ( ^ [R6: A > $o,X4: A,Y4: A] :
( ( R6 @ X4 )
& ( X4 = Y4 ) ) ) ) ).
% eq_onp_def
thf(fact_7173_subset__mset_OSup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( lattic4895041142388067077er_set @ ( multiset @ A ) @ ( union_mset @ A )
@ ^ [X4: multiset @ A,Y4: multiset @ A] : ( subseteq_mset @ A @ Y4 @ X4 )
@ ^ [X4: multiset @ A,Y4: multiset @ A] : ( subset_mset @ A @ Y4 @ X4 ) ) ).
% subset_mset.Sup_fin.semilattice_order_set_axioms
thf(fact_7174_zero__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) )
@ ^ [A7: A] : ( zero_zero @ nat )
@ ^ [A7: A] : ( zero_zero @ nat ) ) ).
% zero_multiset.rsp
thf(fact_7175_filter__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y5: A > $o,Z4: A > $o] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
@ ^ [P4: A > $o,M10: A > nat,X4: A] : ( if @ nat @ ( P4 @ X4 ) @ ( M10 @ X4 ) @ ( zero_zero @ nat ) )
@ ^ [P4: A > $o,M10: A > nat,X4: A] : ( if @ nat @ ( P4 @ X4 ) @ ( M10 @ X4 ) @ ( zero_zero @ nat ) ) ) ).
% filter_mset.rsp
thf(fact_7176_subset__mset_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( semila1105856199041335345_order @ ( multiset @ A ) @ ( union_mset @ A ) @ ( zero_zero @ ( multiset @ A ) )
@ ^ [A8: multiset @ A,B7: multiset @ A] : ( subseteq_mset @ A @ B7 @ A8 )
@ ^ [A8: multiset @ A,B7: multiset @ A] : ( subset_mset @ A @ B7 @ A8 ) ) ).
% subset_mset.semilattice_neutr_order_axioms
thf(fact_7177_add__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ A @ A @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
@ ^ [A7: A,M10: A > nat,B6: A] : ( if @ nat @ ( B6 = A7 ) @ ( suc @ ( M10 @ B6 ) ) @ ( M10 @ B6 ) )
@ ^ [A7: A,M10: A > nat,B6: A] : ( if @ nat @ ( B6 = A7 ) @ ( suc @ ( M10 @ B6 ) ) @ ( M10 @ B6 ) ) ) ).
% add_mset.rsp
thf(fact_7178_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 )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) )
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
@ ^ [M10: A > nat,N8: A > nat,A7: A] : ( plus_plus @ nat @ ( M10 @ A7 ) @ ( N8 @ A7 ) )
@ ^ [M10: A > nat,N8: A > nat,A7: A] : ( plus_plus @ nat @ ( M10 @ A7 ) @ ( N8 @ A7 ) ) ) ).
% plus_multiset.rsp
thf(fact_7179_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 )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) )
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
@ ^ [M10: A > nat,N8: A > nat,A7: A] : ( minus_minus @ nat @ ( M10 @ A7 ) @ ( N8 @ A7 ) )
@ ^ [M10: A > nat,N8: A > nat,A7: A] : ( minus_minus @ nat @ ( M10 @ A7 ) @ ( N8 @ A7 ) ) ) ).
% minus_multiset.rsp
thf(fact_7180_repeat__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
@ ^ [N: nat,M10: A > nat,A7: A] : ( times_times @ nat @ N @ ( M10 @ A7 ) )
@ ^ [N: nat,M10: A > nat,A7: A] : ( times_times @ nat @ N @ ( M10 @ A7 ) ) ) ).
% repeat_mset.rsp
thf(fact_7181_subset__mset_Omono__sup,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ! [F3: ( multiset @ A ) > B,A5: multiset @ A,B4: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F3 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F3 @ A5 ) @ ( F3 @ B4 ) ) @ ( F3 @ ( union_mset @ A @ A5 @ B4 ) ) ) ) ) ).
% subset_mset.mono_sup
thf(fact_7182_subset__mset_OcSup__insert__If,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A3: multiset @ A] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X6 )
=> ( ( ( X6
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert @ ( multiset @ A ) @ A3 @ X6 ) )
= A3 ) )
& ( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert @ ( multiset @ A ) @ A3 @ X6 ) )
= ( union_mset @ A @ A3 @ ( complete_Sup_Sup @ ( multiset @ A ) @ X6 ) ) ) ) ) ) ).
% subset_mset.cSup_insert_If
thf(fact_7183_subset__mset_OcSup__insert,axiom,
! [A: $tType,X6: set @ ( multiset @ A ),A3: multiset @ A] :
( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X6 )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( insert @ ( multiset @ A ) @ A3 @ X6 ) )
= ( union_mset @ A @ A3 @ ( complete_Sup_Sup @ ( multiset @ A ) @ X6 ) ) ) ) ) ).
% subset_mset.cSup_insert
thf(fact_7184_subset__mset_OcSup__inter__less__eq,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( ( ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( inf_inf @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) ) @ ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B4 ) ) ) ) ) ) ).
% subset_mset.cSup_inter_less_eq
thf(fact_7185_subset__mset_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType,A5: set @ B,F3: B > ( multiset @ A ),G3: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G3 @ A5 ) )
=> ( ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G3 @ A5 ) ) )
= ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A7: B] : ( union_mset @ A @ ( F3 @ A7 ) @ ( G3 @ A7 ) )
@ A5 ) ) ) ) ) ) ).
% subset_mset.SUP_sup_distrib
thf(fact_7186_subset__mset_OcSup__union__distrib,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ B4 )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) )
= ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A5 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ B4 ) ) ) ) ) ) ) ).
% subset_mset.cSup_union_distrib
thf(fact_7187_subset__mset_Osup__Inf2__distrib,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( union_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B4 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu2: multiset @ A] :
? [A7: multiset @ A,B6: multiset @ A] :
( ( Uu2
= ( union_mset @ A @ A7 @ B6 ) )
& ( member @ ( multiset @ A ) @ A7 @ A5 )
& ( member @ ( multiset @ A ) @ B6 @ B4 ) ) ) ) ) ) ) ) ) ).
% subset_mset.sup_Inf2_distrib
thf(fact_7188_subset__mset_Osup__Inf1__distrib,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( union_mset @ A @ X @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu2: multiset @ A] :
? [A7: multiset @ A] :
( ( Uu2
= ( union_mset @ A @ X @ A7 ) )
& ( member @ ( multiset @ A ) @ A7 @ A5 ) ) ) ) ) ) ) ).
% subset_mset.sup_Inf1_distrib
thf(fact_7189_add__mset_Oabs__eq,axiom,
! [A: $tType,X: A > nat,Xa: A] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) )
@ X
@ X )
=> ( ( add_mset @ A @ Xa @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [B6: A] : ( if @ nat @ ( B6 = Xa ) @ ( suc @ ( X @ B6 ) ) @ ( X @ B6 ) ) ) ) ) ).
% add_mset.abs_eq
thf(fact_7190_plus__multiset_Oabs__eq,axiom,
! [A: $tType,Xa: A > nat,X: A > nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) )
@ X
@ X )
=> ( ( plus_plus @ ( multiset @ A ) @ ( abs_multiset @ A @ Xa ) @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [A7: A] : ( plus_plus @ nat @ ( Xa @ A7 ) @ ( X @ A7 ) ) ) ) ) ) ).
% plus_multiset.abs_eq
thf(fact_7191_minus__multiset_Oabs__eq,axiom,
! [A: $tType,Xa: A > nat,X: A > nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) )
@ X
@ X )
=> ( ( minus_minus @ ( multiset @ A ) @ ( abs_multiset @ A @ Xa ) @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [A7: A] : ( minus_minus @ nat @ ( Xa @ A7 ) @ ( X @ A7 ) ) ) ) ) ) ).
% minus_multiset.abs_eq
thf(fact_7192_subset__mset_OcSUP__insert,axiom,
! [A: $tType,B: $tType,A5: set @ B,F3: B > ( multiset @ A ),A3: B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ ( insert @ B @ A3 @ A5 ) ) )
= ( union_mset @ A @ ( F3 @ A3 ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F3 @ A5 ) ) ) ) ) ) ).
% subset_mset.cSUP_insert
thf(fact_7193_subset__mset_OSup__fin_Oeq__fold_H,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 )
= ( the2 @ ( multiset @ A )
@ ( finite_fold @ ( multiset @ A ) @ ( option @ ( multiset @ A ) )
@ ^ [X4: multiset @ A,Y4: option @ ( multiset @ A )] : ( some @ ( multiset @ A ) @ ( case_option @ ( multiset @ A ) @ ( multiset @ A ) @ X4 @ ( union_mset @ A @ X4 ) @ Y4 ) )
@ ( none @ ( multiset @ A ) )
@ A5 ) ) ) ).
% subset_mset.Sup_fin.eq_fold'
thf(fact_7194_Inf__multiset_Oabs__eq,axiom,
! [A: $tType,X: set @ ( A > nat )] :
( ( bNF_rel_set @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) )
@ X
@ X )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ ( A > nat ) @ ( multiset @ A ) @ ( abs_multiset @ A ) @ X ) )
= ( abs_multiset @ A
@ ^ [I4: A] :
( if @ nat
@ ( X
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ X ) ) ) ) ) ) ).
% Inf_multiset.abs_eq
thf(fact_7195_subset__mset_OSup__fin_Osingleton,axiom,
! [A: $tType,X: multiset @ A] :
( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= X ) ).
% subset_mset.Sup_fin.singleton
thf(fact_7196_subset__mset_OSup__fin_Oinsert,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert @ ( multiset @ A ) @ X @ A5 ) )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) ) ) ) ) ).
% subset_mset.Sup_fin.insert
thf(fact_7197_semilattice__sup_OSup__fin_Ocong,axiom,
! [A: $tType] :
( ( lattic4630905495605216202up_fin @ A )
= ( lattic4630905495605216202up_fin @ A ) ) ).
% semilattice_sup.Sup_fin.cong
thf(fact_7198_rel__set__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_rel_set @ A @ B )
= ( ^ [R6: A > B > $o,A8: set @ A,B7: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ? [Y4: B] :
( ( member @ B @ Y4 @ B7 )
& ( R6 @ X4 @ Y4 ) ) )
& ! [X4: B] :
( ( member @ B @ X4 @ B7 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A8 )
& ( R6 @ Y4 @ X4 ) ) ) ) ) ) ).
% rel_set_def
thf(fact_7199_fun_Oset__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,R2: A > B > $o] :
( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ ( set @ A ) @ ( set @ B )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R2 )
@ ( bNF_rel_set @ A @ B @ R2 )
@ ^ [F4: D > A] : ( image2 @ D @ A @ F4 @ ( top_top @ ( set @ D ) ) )
@ ^ [F4: D > B] : ( image2 @ D @ B @ F4 @ ( top_top @ ( set @ D ) ) ) ) ).
% fun.set_transfer
thf(fact_7200_subset__mset_OSup__fin_OboundedE,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) @ X )
=> ! [A15: multiset @ A] :
( ( member @ ( multiset @ A ) @ A15 @ A5 )
=> ( subseteq_mset @ A @ A15 @ X ) ) ) ) ) ).
% subset_mset.Sup_fin.boundedE
thf(fact_7201_subset__mset_OSup__fin_OboundedI,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [A6: multiset @ A] :
( ( member @ ( multiset @ A ) @ A6 @ A5 )
=> ( subseteq_mset @ A @ A6 @ X ) )
=> ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) @ X ) ) ) ) ).
% subset_mset.Sup_fin.boundedI
thf(fact_7202_subset__mset_OSup__fin_Obounded__iff,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) @ X )
= ( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ A5 )
=> ( subseteq_mset @ A @ X4 @ X ) ) ) ) ) ) ).
% subset_mset.Sup_fin.bounded_iff
thf(fact_7203_subset__mset_OcSup__eq__Sup__fin,axiom,
! [A: $tType,X6: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ X6 )
=> ( ( X6
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ X6 )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ X6 ) ) ) ) ).
% subset_mset.cSup_eq_Sup_fin
thf(fact_7204_subset__mset_OSup__fin_Oinsert__not__elem,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ~ ( member @ ( multiset @ A ) @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert @ ( multiset @ A ) @ X @ A5 ) )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) ) ) ) ) ) ).
% subset_mset.Sup_fin.insert_not_elem
thf(fact_7205_subset__mset_OSup__fin_Oclosed,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ! [X3: multiset @ A,Y3: multiset @ A] : ( member @ ( multiset @ A ) @ ( union_mset @ A @ X3 @ Y3 ) @ ( insert @ ( multiset @ A ) @ X3 @ ( insert @ ( multiset @ A ) @ Y3 @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) )
=> ( member @ ( multiset @ A ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) @ A5 ) ) ) ) ).
% subset_mset.Sup_fin.closed
thf(fact_7206_subset__mset_OSup__fin_Osubset,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ B4 @ A5 )
=> ( ( union_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B4 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) ) ) ) ) ).
% subset_mset.Sup_fin.subset
thf(fact_7207_subset__mset_OSup__fin_Ounion,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( sup_sup @ ( set @ ( multiset @ A ) ) @ A5 @ B4 ) )
= ( union_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B4 ) ) ) ) ) ) ) ).
% subset_mset.Sup_fin.union
thf(fact_7208_subset__mset_OSup__fin_Ohom__commute,axiom,
! [A: $tType,H: ( multiset @ A ) > ( multiset @ A ),N7: set @ ( multiset @ A )] :
( ! [X3: multiset @ A,Y3: multiset @ A] :
( ( H @ ( union_mset @ A @ X3 @ Y3 ) )
= ( union_mset @ A @ ( H @ X3 ) @ ( H @ Y3 ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ N7 )
=> ( ( N7
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( H @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ N7 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( image2 @ ( multiset @ A ) @ ( multiset @ A ) @ H @ N7 ) ) ) ) ) ) ).
% subset_mset.Sup_fin.hom_commute
thf(fact_7209_subset__mset_OSup__fin_Osubset__imp,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( ord_less_eq @ ( set @ ( multiset @ A ) ) @ A5 @ B4 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B4 )
=> ( subseteq_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B4 ) ) ) ) ) ).
% subset_mset.Sup_fin.subset_imp
thf(fact_7210_subset__mset_Oinf__Sup1__distrib,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( inter_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu2: multiset @ A] :
? [A7: multiset @ A] :
( ( Uu2
= ( inter_mset @ A @ X @ A7 ) )
& ( member @ ( multiset @ A ) @ A7 @ A5 ) ) ) ) ) ) ) ).
% subset_mset.inf_Sup1_distrib
thf(fact_7211_subset__mset_Oinf__Sup2__distrib,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B4: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( inter_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B4 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu2: multiset @ A] :
? [A7: multiset @ A,B6: multiset @ A] :
( ( Uu2
= ( inter_mset @ A @ A7 @ B6 ) )
& ( member @ ( multiset @ A ) @ A7 @ A5 )
& ( member @ ( multiset @ A ) @ B6 @ B4 ) ) ) ) ) ) ) ) ) ).
% subset_mset.inf_Sup2_distrib
thf(fact_7212_subset__mset_OSup__fin_Oremove,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( member @ ( multiset @ A ) @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Sup_fin.remove
thf(fact_7213_subset__mset_OSup__fin_Oinsert__remove,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert @ ( multiset @ A ) @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) )
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( insert @ ( multiset @ A ) @ X @ A5 ) )
= ( union_mset @ A @ X @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ ( minus_minus @ ( set @ ( multiset @ A ) ) @ A5 @ ( insert @ ( multiset @ A ) @ X @ ( bot_bot @ ( set @ ( multiset @ A ) ) ) ) ) ) ) ) ) ) ) ).
% subset_mset.Sup_fin.insert_remove
thf(fact_7214_subset__mset_OInf__fin__le__Sup__fin,axiom,
! [A: $tType,A5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) ) ) ) ).
% subset_mset.Inf_fin_le_Sup_fin
thf(fact_7215_Inf__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( set @ ( A > nat ) ) @ ( set @ ( A > nat ) ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_rel_set @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) )
@ ^ [A8: set @ ( A > nat ),I4: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ A8 ) ) )
@ ^ [A8: set @ ( A > nat ),I4: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ A8 ) ) ) ) ).
% Inf_multiset.rsp
thf(fact_7216_INF__parametric,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( complete_Inf @ C )
=> ! [A5: A > B > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > C ) > C ) @ ( ( B > C ) > C ) @ ( bNF_rel_set @ A @ B @ A5 )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > C ) @ C @ C
@ ( bNF_rel_fun @ A @ B @ C @ C @ A5
@ ^ [Y5: C,Z4: C] : Y5 = Z4 )
@ ^ [Y5: C,Z4: C] : Y5 = Z4 )
@ ^ [A8: set @ A,F4: A > C] : ( complete_Inf_Inf @ C @ ( image2 @ A @ C @ F4 @ A8 ) )
@ ^ [A8: set @ B,F4: B > C] : ( complete_Inf_Inf @ C @ ( image2 @ B @ C @ F4 @ A8 ) ) ) ) ).
% INF_parametric
thf(fact_7217_SUP__parametric,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( complete_Sup @ C )
=> ! [R2: A > B > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > C ) > C ) @ ( ( B > C ) > C ) @ ( bNF_rel_set @ A @ B @ R2 )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > C ) @ C @ C
@ ( bNF_rel_fun @ A @ B @ C @ C @ R2
@ ^ [Y5: C,Z4: C] : Y5 = Z4 )
@ ^ [Y5: C,Z4: C] : Y5 = Z4 )
@ ^ [A8: set @ A,F4: A > C] : ( complete_Sup_Sup @ C @ ( image2 @ A @ C @ F4 @ A8 ) )
@ ^ [A8: set @ B,F4: B > C] : ( complete_Sup_Sup @ C @ ( image2 @ B @ C @ F4 @ A8 ) ) ) ) ).
% SUP_parametric
thf(fact_7218_empty__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_set @ A @ B @ A5 @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ B ) ) ) ).
% empty_transfer
thf(fact_7219_union__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( set @ A ) > ( set @ A ) ) @ ( ( set @ B ) > ( set @ B ) ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_set @ A @ B @ A5 ) ) @ ( sup_sup @ ( set @ A ) ) @ ( sup_sup @ ( set @ B ) ) ) ).
% union_transfer
thf(fact_7220_rel__set__mono,axiom,
! [B: $tType,A: $tType,A5: A > B > $o,B4: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A5 @ B4 )
=> ( ord_less_eq @ ( ( set @ A ) > ( set @ B ) > $o ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_set @ A @ B @ B4 ) ) ) ).
% rel_set_mono
thf(fact_7221_Union__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ ( set @ ( set @ A ) ) @ ( set @ ( set @ B ) ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A5 ) ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ B ) ) ) ).
% Union_transfer
thf(fact_7222_set__relator__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( bNF_rel_set @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( set @ A )
@ ^ [A8: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( P @ X4 ) ) ) ) ).
% set_relator_eq_onp
thf(fact_7223_UNION__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B4: C > D > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > ( set @ C ) ) > ( set @ C ) ) @ ( ( B > ( set @ D ) ) > ( set @ D ) ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( A > ( set @ C ) ) @ ( B > ( set @ D ) ) @ ( set @ C ) @ ( set @ D ) @ ( bNF_rel_fun @ A @ B @ ( set @ C ) @ ( set @ D ) @ A5 @ ( bNF_rel_set @ C @ D @ B4 ) ) @ ( bNF_rel_set @ C @ D @ B4 ) )
@ ^ [A8: set @ A,F4: A > ( set @ C )] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ F4 @ A8 ) )
@ ^ [A8: set @ B,F4: B > ( set @ D )] : ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ B @ ( set @ D ) @ F4 @ A8 ) ) ) ).
% UNION_transfer
thf(fact_7224_Inf__multiset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( set @ ( A > nat ) ) @ ( set @ ( multiset @ A ) ) @ ( A > nat ) @ ( multiset @ A )
@ ( bNF_rel_set @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ^ [A8: set @ ( A > nat ),I4: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ I4 )
@ A8 ) ) )
@ ( complete_Inf_Inf @ ( multiset @ A ) ) ) ).
% Inf_multiset.transfer
thf(fact_7225_filter__mset_Oabs__eq,axiom,
! [A: $tType,X: A > nat,Xa: A > $o] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) )
@ X
@ X )
=> ( ( filter_mset @ A @ Xa @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [X4: A] : ( if @ nat @ ( Xa @ X4 ) @ ( X @ X4 ) @ ( zero_zero @ nat ) ) ) ) ) ).
% filter_mset.abs_eq
thf(fact_7226_filter__mset__True,axiom,
! [A: $tType,M6: multiset @ A] :
( ( filter_mset @ A
@ ^ [Y4: A] : $true
@ M6 )
= M6 ) ).
% filter_mset_True
thf(fact_7227_filter__mset__False,axiom,
! [A: $tType,M6: multiset @ A] :
( ( filter_mset @ A
@ ^ [Y4: A] : $false
@ M6 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% filter_mset_False
thf(fact_7228_set__mset__filter,axiom,
! [A: $tType,P: A > $o,M6: multiset @ A] :
( ( set_mset @ A @ ( filter_mset @ A @ P @ M6 ) )
= ( collect @ A
@ ^ [A7: A] :
( ( member @ A @ A7 @ ( set_mset @ A @ M6 ) )
& ( P @ A7 ) ) ) ) ).
% set_mset_filter
thf(fact_7229_mset__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( mset @ A @ ( filter2 @ A @ P @ Xs ) )
= ( filter_mset @ A @ P @ ( mset @ A @ Xs ) ) ) ).
% mset_filter
thf(fact_7230_filter__mset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ^ [Y5: A > $o,Z4: A > $o] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ^ [P4: A > $o,M10: A > nat,X4: A] : ( if @ nat @ ( P4 @ X4 ) @ ( M10 @ X4 ) @ ( zero_zero @ nat ) )
@ ( filter_mset @ A ) ) ).
% filter_mset.transfer
thf(fact_7231_filter__filter__mset,axiom,
! [A: $tType,P: A > $o,Q: A > $o,M6: multiset @ A] :
( ( filter_mset @ A @ P @ ( filter_mset @ A @ Q @ M6 ) )
= ( filter_mset @ A
@ ^ [X4: A] :
( ( Q @ X4 )
& ( P @ X4 ) )
@ M6 ) ) ).
% filter_filter_mset
thf(fact_7232_multiset__partition,axiom,
! [A: $tType,M6: multiset @ A,P: A > $o] :
( M6
= ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M6 )
@ ( filter_mset @ A
@ ^ [X4: A] :
~ ( P @ X4 )
@ M6 ) ) ) ).
% multiset_partition
thf(fact_7233_image__mset__If,axiom,
! [A: $tType,B: $tType,P: B > $o,F3: B > A,G3: B > A,A5: multiset @ B] :
( ( image_mset @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P @ X4 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ A5 )
= ( plus_plus @ ( multiset @ A ) @ ( image_mset @ B @ A @ F3 @ ( filter_mset @ B @ P @ A5 ) )
@ ( image_mset @ B @ A @ G3
@ ( filter_mset @ B
@ ^ [X4: B] :
~ ( P @ X4 )
@ A5 ) ) ) ) ).
% image_mset_If
thf(fact_7234_size__filter__mset__lesseq,axiom,
! [A: $tType,F3: A > $o,M6: multiset @ A] : ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ ( filter_mset @ A @ F3 @ M6 ) ) @ ( size_size @ ( multiset @ A ) @ M6 ) ) ).
% size_filter_mset_lesseq
thf(fact_7235_filter__eq__replicate__mset,axiom,
! [A: $tType,X: A,D4: multiset @ A] :
( ( filter_mset @ A
@ ^ [Y4: A] : Y4 = X
@ D4 )
= ( replicate_mset @ A @ ( count @ A @ D4 @ X ) @ X ) ) ).
% filter_eq_replicate_mset
thf(fact_7236_zero__multiset_Otransfer,axiom,
! [A: $tType] :
( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ^ [A7: A] : ( zero_zero @ nat )
@ ( zero_zero @ ( multiset @ A ) ) ) ).
% zero_multiset.transfer
thf(fact_7237_multiset_Orep__transfer,axiom,
! [D: $tType,E: $tType,T2: D > E > $o] :
( bNF_rel_fun @ ( D > nat ) @ ( multiset @ E ) @ ( D > nat ) @ ( E > nat ) @ ( pcr_multiset @ D @ E @ T2 )
@ ( bNF_rel_fun @ D @ E @ nat @ nat @ T2
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ^ [X4: D > nat] : X4
@ ( count @ E ) ) ).
% multiset.rep_transfer
thf(fact_7238_add__mset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ A @ A @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ^ [A7: A,M10: A > nat,B6: A] : ( if @ nat @ ( B6 = A7 ) @ ( suc @ ( M10 @ B6 ) ) @ ( M10 @ B6 ) )
@ ( add_mset @ A ) ) ).
% add_mset.transfer
thf(fact_7239_plus__multiset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ^ [M10: A > nat,N8: A > nat,A7: A] : ( plus_plus @ nat @ ( M10 @ A7 ) @ ( N8 @ A7 ) )
@ ( plus_plus @ ( multiset @ A ) ) ) ).
% plus_multiset.transfer
thf(fact_7240_minus__multiset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ^ [M10: A > nat,N8: A > nat,A7: A] : ( minus_minus @ nat @ ( M10 @ A7 ) @ ( N8 @ A7 ) )
@ ( minus_minus @ ( multiset @ A ) ) ) ).
% minus_multiset.transfer
thf(fact_7241_repeat__mset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ^ [N: nat,M10: A > nat,A7: A] : ( times_times @ nat @ N @ ( M10 @ A7 ) )
@ ( repeat_mset @ A ) ) ).
% repeat_mset.transfer
thf(fact_7242_filter__mset__def,axiom,
! [A: $tType] :
( ( filter_mset @ A )
= ( map_fun @ ( A > $o ) @ ( A > $o ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( id @ ( A > $o ) ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [P4: A > $o,M10: A > nat,X4: A] : ( if @ nat @ ( P4 @ X4 ) @ ( M10 @ X4 ) @ ( zero_zero @ nat ) ) ) ) ).
% filter_mset_def
thf(fact_7243_type__definition__multiset,axiom,
! [A: $tType] :
( type_definition @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ ( abs_multiset @ A )
@ ( collect @ ( A > nat )
@ ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) ) ).
% type_definition_multiset
thf(fact_7244_sum__multiset__singleton,axiom,
! [A: $tType,A5: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ ( multiset @ A )
@ ^ [N: A] : ( add_mset @ A @ N @ ( zero_zero @ ( multiset @ A ) ) )
@ A5 )
= ( mset_set @ A @ A5 ) ) ).
% sum_multiset_singleton
thf(fact_7245_mset__set_Oempty,axiom,
! [A: $tType] :
( ( mset_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% mset_set.empty
thf(fact_7246_filter__mset__mset__set,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( filter_mset @ A @ P @ ( mset_set @ A @ A5 ) )
= ( mset_set @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) ) ) ) ) ) ).
% filter_mset_mset_set
thf(fact_7247_msubset__mset__set__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( subseteq_mset @ A @ ( mset_set @ A @ A5 ) @ ( mset_set @ A @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% msubset_mset_set_iff
thf(fact_7248_typedef__rep__transfer,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T2: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T2
= ( ^ [X4: A,Y4: B] :
( X4
= ( Rep @ Y4 ) ) ) )
=> ( bNF_rel_fun @ A @ B @ A @ A @ T2
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ^ [X4: A] : X4
@ Rep ) ) ) ).
% typedef_rep_transfer
thf(fact_7249_subset__imp__msubset__mset__set,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( subseteq_mset @ A @ ( mset_set @ A @ A5 ) @ ( mset_set @ A @ B4 ) ) ) ) ).
% subset_imp_msubset_mset_set
thf(fact_7250_mset__set__empty__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( mset_set @ A @ A5 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% mset_set_empty_iff
thf(fact_7251_infinite__set__mset__mset__set,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( set_mset @ A @ ( mset_set @ A @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_set_mset_mset_set
thf(fact_7252_mset__set__Diff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ ( multiset @ A ) @ ( mset_set @ A @ A5 ) @ ( mset_set @ A @ B4 ) ) ) ) ) ).
% mset_set_Diff
thf(fact_7253_count__mset__set__finite__iff,axiom,
! [A: $tType,S: set @ A,A3: A] :
( ( finite_finite2 @ A @ S )
=> ( ( ( member @ A @ A3 @ S )
=> ( ( count @ A @ ( mset_set @ A @ S ) @ A3 )
= ( one_one @ nat ) ) )
& ( ~ ( member @ A @ A3 @ S )
=> ( ( count @ A @ ( mset_set @ A @ S ) @ A3 )
= ( zero_zero @ nat ) ) ) ) ) ).
% count_mset_set_finite_iff
thf(fact_7254_mset__set_Oremove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( mset_set @ A @ A5 )
= ( add_mset @ A @ X @ ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% mset_set.remove
thf(fact_7255_mset__set_Oinsert__remove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( mset_set @ A @ ( insert @ A @ X @ A5 ) )
= ( add_mset @ A @ X @ ( mset_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% mset_set.insert_remove
thf(fact_7256_mset__set__Union,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( mset_set @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset_set @ A @ A5 ) @ ( mset_set @ A @ B4 ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_7257_stable__sort__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord3483353639454293061rt_key @ B @ A )
= ( ^ [Sk: ( B > A ) > ( list @ B ) > ( list @ B )] :
! [F4: B > A,Xs3: list @ B,K3: A] :
( ( filter2 @ B
@ ^ [Y4: B] :
( ( F4 @ Y4 )
= K3 )
@ ( Sk @ F4 @ Xs3 ) )
= ( filter2 @ B
@ ^ [Y4: B] :
( ( F4 @ Y4 )
= K3 )
@ Xs3 ) ) ) ) ) ).
% stable_sort_key_def
thf(fact_7258_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I5: set @ A,F5: A > ( set @ B ),F3: B > C,G5: D > ( set @ C ),J5: set @ D] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F5 @ I3 ) @ ( F5 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F5 @ J2 ) @ ( F5 @ I3 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F3
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J3: D] : ( principal @ C @ ( G5 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I4: A] : ( principal @ B @ ( F5 @ I4 ) )
@ I5 ) ) )
= ( ! [X4: D] :
( ( member @ D @ X4 @ J5 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ I5 )
& ! [Z7: B] :
( ( member @ B @ Z7 @ ( F5 @ Y4 ) )
=> ( member @ C @ ( F3 @ Z7 ) @ ( G5 @ X4 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_7259_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G3: A > B,F33: filter @ B,F24: filter @ A,F3: C > A,F15: filter @ C] :
( ( filterlim @ A @ B @ G3 @ F33 @ F24 )
=> ( ( filterlim @ C @ A @ F3 @ F24 @ F15 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( G3 @ ( F3 @ X4 ) )
@ F33
@ F15 ) ) ) ).
% filterlim_compose
thf(fact_7260_filterlim__ident,axiom,
! [A: $tType,F5: filter @ A] :
( filterlim @ A @ A
@ ^ [X4: A] : X4
@ F5
@ F5 ) ).
% filterlim_ident
thf(fact_7261_filterlim__top,axiom,
! [B: $tType,A: $tType,F3: A > B,F5: filter @ A] : ( filterlim @ A @ B @ F3 @ ( top_top @ ( filter @ B ) ) @ F5 ) ).
% filterlim_top
thf(fact_7262_filterlim__inf,axiom,
! [B: $tType,A: $tType,F3: A > B,F24: filter @ B,F33: filter @ B,F15: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( inf_inf @ ( filter @ B ) @ F24 @ F33 ) @ F15 )
= ( ( filterlim @ A @ B @ F3 @ F24 @ F15 )
& ( filterlim @ A @ B @ F3 @ F33 @ F15 ) ) ) ).
% filterlim_inf
thf(fact_7263_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_7264_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_7265_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_7266_filterlim__sequentially__Suc,axiom,
! [A: $tType,F3: nat > A,F5: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X4: nat] : ( F3 @ ( suc @ X4 ) )
@ F5
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F3 @ F5 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_7267_filterlim__sup,axiom,
! [B: $tType,A: $tType,F3: A > B,F5: filter @ B,F15: filter @ A,F24: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F5 @ F15 )
=> ( ( filterlim @ A @ B @ F3 @ F5 @ F24 )
=> ( filterlim @ A @ B @ F3 @ F5 @ ( sup_sup @ ( filter @ A ) @ F15 @ F24 ) ) ) ) ).
% filterlim_sup
thf(fact_7268_filterlim__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,G5: C > ( filter @ B ),B4: set @ C,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ G5 @ B4 ) ) @ F5 )
= ( ! [X4: C] :
( ( member @ C @ X4 @ B4 )
=> ( filterlim @ A @ B @ F3 @ ( G5 @ X4 ) @ F5 ) ) ) ) ).
% filterlim_INF
thf(fact_7269_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X: A,A5: set @ A,F3: B > C,F5: filter @ C,G5: A > ( filter @ B )] :
( ( member @ A @ X @ A5 )
=> ( ( filterlim @ B @ C @ F3 @ F5 @ ( G5 @ X ) )
=> ( filterlim @ B @ C @ F3 @ F5 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ G5 @ A5 ) ) ) ) ) ).
% filterlim_INF'
thf(fact_7270_filterlim__mono,axiom,
! [B: $tType,A: $tType,F3: A > B,F24: filter @ B,F15: filter @ A,F25: filter @ B,F16: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F24 @ F15 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F24 @ F25 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F16 @ F15 )
=> ( filterlim @ A @ B @ F3 @ F25 @ F16 ) ) ) ) ).
% filterlim_mono
thf(fact_7271_filterlim__If,axiom,
! [B: $tType,A: $tType,F3: A > B,G5: filter @ B,F5: filter @ A,P: A > $o,G3: A > B] :
( ( filterlim @ A @ B @ F3 @ G5 @ ( inf_inf @ ( filter @ A ) @ F5 @ ( principal @ A @ ( collect @ A @ P ) ) ) )
=> ( ( filterlim @ A @ B @ G3 @ G5
@ ( inf_inf @ ( filter @ A ) @ F5
@ ( principal @ A
@ ( collect @ A
@ ^ [X4: A] :
~ ( P @ X4 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( if @ B @ ( P @ X4 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ G5
@ F5 ) ) ) ).
% filterlim_If
thf(fact_7272_filterlim__base,axiom,
! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J5: set @ A,I2: A > C,I5: set @ C,F5: C > ( set @ D ),F3: D > E,G5: A > ( set @ E )] :
( ! [M5: A,X3: B] :
( ( member @ A @ M5 @ J5 )
=> ( member @ C @ ( I2 @ M5 ) @ I5 ) )
=> ( ! [M5: A,X3: D] :
( ( member @ A @ M5 @ J5 )
=> ( ( member @ D @ X3 @ ( F5 @ ( I2 @ M5 ) ) )
=> ( member @ E @ ( F3 @ X3 ) @ ( G5 @ M5 ) ) ) )
=> ( filterlim @ D @ E @ F3
@ ( complete_Inf_Inf @ ( filter @ E )
@ ( image2 @ A @ ( filter @ E )
@ ^ [J3: A] : ( principal @ E @ ( G5 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ D )
@ ( image2 @ C @ ( filter @ D )
@ ^ [I4: C] : ( principal @ D @ ( F5 @ I4 ) )
@ I5 ) ) ) ) ) ).
% filterlim_base
thf(fact_7273_map__rec,axiom,
! [A: $tType,B: $tType] :
( ( map @ B @ A )
= ( ^ [F4: B > A] :
( rec_list @ ( list @ A ) @ B @ ( nil @ A )
@ ^ [X4: B,Uu2: list @ B] : ( cons @ A @ ( F4 @ X4 ) ) ) ) ) ).
% map_rec
thf(fact_7274_zipf__zip,axiom,
! [A: $tType,B: $tType,L1: list @ A,L22: list @ B] :
( ( ( size_size @ ( list @ A ) @ L1 )
= ( size_size @ ( list @ B ) @ L22 ) )
=> ( ( zipf @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ L1 @ L22 )
= ( zip @ A @ B @ L1 @ L22 ) ) ) ).
% zipf_zip
thf(fact_7275_trivial__limit__sequentially,axiom,
( ( at_top @ nat )
!= ( bot_bot @ ( filter @ nat ) ) ) ).
% trivial_limit_sequentially
thf(fact_7276_rec__list__Cons__imp,axiom,
! [B: $tType,A: $tType,F3: ( list @ A ) > B,F1: B,F22: A > ( list @ A ) > B > B,X: A,Xs: list @ A] :
( ( F3
= ( rec_list @ B @ A @ F1 @ F22 ) )
=> ( ( F3 @ ( cons @ A @ X @ Xs ) )
= ( F22 @ X @ Xs @ ( F3 @ Xs ) ) ) ) ).
% rec_list_Cons_imp
thf(fact_7277_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_7278_zipf_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B > C,A3: A,As: list @ A,B2: B,Bs: list @ B] :
( ( zipf @ A @ B @ C @ F3 @ ( cons @ A @ A3 @ As ) @ ( cons @ B @ B2 @ Bs ) )
= ( cons @ C @ ( F3 @ A3 @ B2 ) @ ( zipf @ A @ B @ C @ F3 @ As @ Bs ) ) ) ).
% zipf.simps(2)
thf(fact_7279_rec__list__Nil__imp,axiom,
! [A: $tType,B: $tType,F3: ( list @ A ) > B,F1: B,F22: A > ( list @ A ) > B > B] :
( ( F3
= ( rec_list @ B @ A @ F1 @ F22 ) )
=> ( ( F3 @ ( nil @ A ) )
= F1 ) ) ).
% rec_list_Nil_imp
thf(fact_7280_zipf_Oelims,axiom,
! [B: $tType,A: $tType,C: $tType,X: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
( ( ( zipf @ A @ B @ C @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
!= ( nil @ C ) ) ) )
=> ( ! [A6: A,As4: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As4 ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ( Y
!= ( cons @ C @ ( X @ A6 @ B5 ) @ ( zipf @ A @ B @ C @ X @ As4 @ Bs2 ) ) ) ) )
=> ( ( ? [V4: A,Va: list @ A] :
( Xa
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V4: B,Va: list @ B] :
( Xb
= ( cons @ B @ V4 @ Va ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) ) ) ) ) ) ).
% zipf.elims
thf(fact_7281_list_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G3: C,Ga: B > ( list @ B ) > C > C,F3: A > B] :
( ( comp @ ( list @ B ) @ C @ ( list @ A ) @ ( rec_list @ C @ B @ G3 @ Ga ) @ ( map @ A @ B @ F3 ) )
= ( rec_list @ C @ A @ G3
@ ^ [X4: A,Xa4: list @ A] : ( Ga @ ( F3 @ X4 ) @ ( map @ A @ B @ F3 @ Xa4 ) ) ) ) ).
% list.rec_o_map
thf(fact_7282_zipf_Opelims,axiom,
! [C: $tType,A: $tType,B: $tType,X: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
( ( ( zipf @ A @ B @ C @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( ( Y
= ( nil @ C ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
=> ( ! [A6: A,As4: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As4 ) )
=> ! [B5: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B5 @ Bs2 ) )
=> ( ( Y
= ( cons @ C @ ( X @ A6 @ B5 ) @ ( zipf @ A @ B @ C @ X @ As4 @ Bs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As4 ) @ ( cons @ B @ B5 @ Bs2 ) ) ) ) ) ) )
=> ( ! [V4: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( ( Y
= ( undefined @ ( list @ C ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V4: B,Va: list @ B] :
( ( Xb
= ( cons @ B @ V4 @ Va ) )
=> ( ( Y
= ( undefined @ ( list @ C ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% zipf.pelims
thf(fact_7283_set__rec,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( rec_list @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X4: A,Uu2: list @ A] : ( insert @ A @ X4 ) ) ) ).
% set_rec
thf(fact_7284_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite2 @ A @ A5 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_7285_map__tailrec__rev_Opelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y = Xb )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xb ) ) ) ) )
=> ~ ! [A6: A,As4: list @ A] :
( ( Xa
= ( cons @ A @ A6 @ As4 ) )
=> ( ( Y
= ( map_tailrec_rev @ A @ B @ X @ As4 @ ( cons @ B @ ( X @ A6 ) @ Xb ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A6 @ As4 ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_7286_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_7287_Gcd__fin_Osubset,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B4: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( gcd_gcd @ A @ ( semiring_gcd_Gcd_fin @ A @ B4 ) @ ( semiring_gcd_Gcd_fin @ A @ A5 ) )
= ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ).
% Gcd_fin.subset
thf(fact_7288_Gcd__fin_Ounion,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( gcd_gcd @ A @ ( semiring_gcd_Gcd_fin @ A @ A5 ) @ ( semiring_gcd_Gcd_fin @ A @ B4 ) ) ) ) ).
% Gcd_fin.union
thf(fact_7289_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert @ A @ A3 @ A5 ) )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Gcd_fin.insert_remove
thf(fact_7290_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Gcd_fin.remove
thf(fact_7291_curr__in,axiom,
! [A: $tType,B: $tType,C: $tType,F3: ( product_prod @ A @ B ) > C,A5: set @ A,B4: set @ B,C4: set @ C] :
( ( member @ ( ( product_prod @ A @ B ) > C ) @ F3
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 )
@ C4 ) )
=> ( member @ ( A > B > C ) @ ( bNF_Wellorder_curr @ A @ B @ C @ A5 @ F3 ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A5 @ ( bNF_Wellorder_Func @ B @ C @ B4 @ C4 ) ) ) ) ).
% curr_in
thf(fact_7292_curr__surj,axiom,
! [C: $tType,B: $tType,A: $tType,G3: A > B > C,A5: set @ A,B4: set @ B,C4: set @ C] :
( ( member @ ( A > B > C ) @ G3 @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A5 @ ( bNF_Wellorder_Func @ B @ C @ B4 @ C4 ) ) )
=> ? [X3: ( product_prod @ A @ B ) > C] :
( ( member @ ( ( product_prod @ A @ B ) > C ) @ X3
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 )
@ C4 ) )
& ( ( bNF_Wellorder_curr @ A @ B @ C @ A5 @ X3 )
= G3 ) ) ) ).
% curr_surj
thf(fact_7293_curr__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( bNF_Wellorder_curr @ A @ B @ C )
= ( ^ [A8: set @ A,F4: ( product_prod @ A @ B ) > C,A7: A] :
( if @ ( B > C ) @ ( member @ A @ A7 @ A8 )
@ ^ [B6: B] : ( F4 @ ( product_Pair @ A @ B @ A7 @ B6 ) )
@ ( undefined @ ( B > C ) ) ) ) ) ).
% curr_def
thf(fact_7294_curr__inj,axiom,
! [C: $tType,B: $tType,A: $tType,F1: ( product_prod @ A @ B ) > C,A5: set @ A,B4: set @ B,C4: set @ C,F22: ( product_prod @ A @ B ) > C] :
( ( member @ ( ( product_prod @ A @ B ) > C ) @ F1
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 )
@ C4 ) )
=> ( ( member @ ( ( product_prod @ A @ B ) > C ) @ F22
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 )
@ C4 ) )
=> ( ( ( bNF_Wellorder_curr @ A @ B @ C @ A5 @ F1 )
= ( bNF_Wellorder_curr @ A @ B @ C @ A5 @ F22 ) )
= ( F1 = F22 ) ) ) ) ).
% curr_inj
thf(fact_7295_bij__betw__curr,axiom,
! [A: $tType,B: $tType,C: $tType,A5: set @ A,B4: set @ B,C4: set @ C] :
( bij_betw @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) @ ( bNF_Wellorder_curr @ A @ B @ C @ A5 )
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 )
@ C4 )
@ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A5 @ ( bNF_Wellorder_Func @ B @ C @ B4 @ C4 ) ) ) ).
% bij_betw_curr
thf(fact_7296_disjnt__equiv__class,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( disjnt @ A @ ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ).
% disjnt_equiv_class
thf(fact_7297_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_7298_disjnt__insert2,axiom,
! [A: $tType,Y7: set @ A,A3: A,X6: set @ A] :
( ( disjnt @ A @ Y7 @ ( insert @ A @ A3 @ X6 ) )
= ( ~ ( member @ A @ A3 @ Y7 )
& ( disjnt @ A @ Y7 @ X6 ) ) ) ).
% disjnt_insert2
thf(fact_7299_disjnt__insert1,axiom,
! [A: $tType,A3: A,X6: set @ A,Y7: set @ A] :
( ( disjnt @ A @ ( insert @ A @ A3 @ X6 ) @ Y7 )
= ( ~ ( member @ A @ A3 @ Y7 )
& ( disjnt @ A @ X6 @ Y7 ) ) ) ).
% disjnt_insert1
thf(fact_7300_disjnt__Un2,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,B4: set @ A] :
( ( disjnt @ A @ C4 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ( disjnt @ A @ C4 @ A5 )
& ( disjnt @ A @ C4 @ B4 ) ) ) ).
% disjnt_Un2
thf(fact_7301_disjnt__Un1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C4: set @ A] :
( ( disjnt @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C4 )
= ( ( disjnt @ A @ A5 @ C4 )
& ( disjnt @ A @ B4 @ C4 ) ) ) ).
% disjnt_Un1
thf(fact_7302_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C4: set @ A,A5: set @ B,B4: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C4
@ ^ [Uu2: A] : A5 )
@ ( product_Sigma @ A @ B @ C4
@ ^ [Uu2: A] : B4 ) )
= ( ( C4
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A5 @ B4 ) ) ) ).
% disjnt_Times1_iff
thf(fact_7303_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A5: set @ A,C4: set @ B,B4: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : C4 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu2: A] : C4 ) )
= ( ( C4
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A5 @ B4 ) ) ) ).
% disjnt_Times2_iff
thf(fact_7304_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A5: set @ A,C4: A > ( set @ B ),B4: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ C4 ) @ ( product_Sigma @ A @ B @ B4 @ C4 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
=> ( ( C4 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A5 @ B4 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_7305_disjnt__def,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A8: set @ A,B7: set @ A] :
( ( inf_inf @ ( set @ A ) @ A8 @ B7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjnt_def
thf(fact_7306_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A] :
( ( bij_betw @ A @ B @ F3 @ A5 @ ( bot_bot @ ( set @ B ) ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_7307_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) @ A5 )
=> ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_7308_disjnt__empty1,axiom,
! [A: $tType,A5: set @ A] : ( disjnt @ A @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% disjnt_empty1
thf(fact_7309_disjnt__empty2,axiom,
! [A: $tType,A5: set @ A] : ( disjnt @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ).
% disjnt_empty2
thf(fact_7310_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A5: set @ A,F3: A > B,A17: set @ B] :
( ~ ( member @ A @ B2 @ A5 )
=> ( ~ ( member @ B @ ( F3 @ B2 ) @ A17 )
=> ( ( bij_betw @ A @ B @ F3 @ A5 @ A17 )
=> ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A17 @ ( insert @ B @ ( F3 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_7311_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A5: set @ A,F3: A > B,A17: set @ B] :
( ~ ( member @ A @ B2 @ A5 )
=> ( ~ ( member @ B @ ( F3 @ B2 ) @ A17 )
=> ( ( bij_betw @ A @ B @ F3 @ A5 @ A17 )
= ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A17 @ ( insert @ B @ ( F3 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_7312_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,C4: set @ A,B4: set @ B,D4: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ C4 ) @ ( sup_sup @ ( set @ B ) @ B4 @ D4 ) )
=> ( ( bij_betw @ A @ B @ F3 @ C4 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ C4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B4 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F3 @ A5 @ B4 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_7313_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B4: set @ B,C4: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A5 @ B4 )
=> ( ( bij_betw @ A @ B @ F3 @ C4 @ D4 )
=> ( ( ( inf_inf @ ( set @ B ) @ B4 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ C4 ) @ ( sup_sup @ ( set @ B ) @ B4 @ D4 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_7314_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F12: A > B,A17: set @ A,A19: set @ B,F3: C > A,A5: set @ C] :
( ( bij_betw @ A @ B @ F12 @ A17 @ A19 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F3 @ A5 ) @ A17 )
=> ( ( bij_betw @ C @ A @ F3 @ A5 @ A17 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F12 @ F3 ) @ A5 @ A19 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_7315_disjnt__subset1,axiom,
! [A: $tType,X6: set @ A,Y7: set @ A,Z5: set @ A] :
( ( disjnt @ A @ X6 @ Y7 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5 @ X6 )
=> ( disjnt @ A @ Z5 @ Y7 ) ) ) ).
% disjnt_subset1
thf(fact_7316_disjnt__subset2,axiom,
! [A: $tType,X6: set @ A,Y7: set @ A,Z5: set @ A] :
( ( disjnt @ A @ X6 @ Y7 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5 @ Y7 )
=> ( disjnt @ A @ X6 @ Z5 ) ) ) ).
% disjnt_subset2
thf(fact_7317_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,A17: set @ B,B4: set @ A,B15: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A5 @ A17 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( ( image2 @ A @ B @ F3 @ B4 )
= B15 )
=> ( bij_betw @ A @ B @ F3 @ B4 @ B15 ) ) ) ) ).
% bij_betw_subset
thf(fact_7318_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A5: set @ A,F12: B > A,F3: A > B,A17: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( F12 @ ( F3 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A17 )
=> ( ( F3 @ ( F12 @ X3 ) )
= X3 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ A17 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F12 @ A17 ) @ A5 )
=> ( bij_betw @ A @ B @ F3 @ A5 @ A17 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_7319_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B4: set @ B,G3: B > A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ B4 )
=> ( ( inj_on @ B @ A @ G3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G3 @ B4 ) @ A5 )
=> ? [H4: A > B] : ( bij_betw @ A @ B @ H4 @ A5 @ B4 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_7320_disjoint__UN__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: B > ( set @ A ),I5: set @ B] :
( ( disjnt @ A @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ I5 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( disjnt @ A @ A5 @ ( B4 @ X4 ) ) ) ) ) ).
% disjoint_UN_iff
thf(fact_7321_disjnt__insert,axiom,
! [A: $tType,X: A,N7: set @ A,M6: set @ A] :
( ~ ( member @ A @ X @ N7 )
=> ( ( disjnt @ A @ M6 @ N7 )
=> ( disjnt @ A @ ( insert @ A @ X @ M6 ) @ N7 ) ) ) ).
% disjnt_insert
thf(fact_7322_bij__fn,axiom,
! [A: $tType,F3: A > A,N2: nat] :
( ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N2 @ F3 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_fn
thf(fact_7323_disjnt__sym,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( disjnt @ A @ A5 @ B4 )
=> ( disjnt @ A @ B4 @ A5 ) ) ).
% disjnt_sym
thf(fact_7324_disjnt__iff,axiom,
! [A: $tType] :
( ( disjnt @ A )
= ( ^ [A8: set @ A,B7: set @ A] :
! [X4: A] :
~ ( ( member @ A @ X4 @ A8 )
& ( member @ A @ X4 @ B7 ) ) ) ) ).
% disjnt_iff
thf(fact_7325_bij__betwI_H,axiom,
! [A: $tType,B: $tType,X6: set @ A,F3: A > B,Y7: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ X6 )
=> ( ( ( F3 @ X3 )
= ( F3 @ Y3 ) )
= ( X3 = Y3 ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( member @ B @ ( F3 @ X3 ) @ Y7 ) )
=> ( ! [Y3: B] :
( ( member @ B @ Y3 @ Y7 )
=> ? [X7: A] :
( ( member @ A @ X7 @ X6 )
& ( Y3
= ( F3 @ X7 ) ) ) )
=> ( bij_betw @ A @ B @ F3 @ X6 @ Y7 ) ) ) ) ).
% bij_betwI'
thf(fact_7326_bij__betw__funpow,axiom,
! [A: $tType,F3: A > A,S: set @ A,N2: nat] :
( ( bij_betw @ A @ A @ F3 @ S @ S )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N2 @ F3 ) @ S @ S ) ) ).
% bij_betw_funpow
thf(fact_7327_prod_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H: B > C,S: set @ B,T2: set @ C,G3: C > A] :
( ( bij_betw @ B @ C @ H @ S @ T2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( G3 @ ( H @ X4 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G3 @ T2 ) ) ) ) ).
% prod.reindex_bij_betw
thf(fact_7328_sum_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [H: B > C,S: set @ B,T2: set @ C,G3: C > A] :
( ( bij_betw @ B @ C @ H @ S @ T2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( G3 @ ( H @ X4 ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G3 @ T2 ) ) ) ) ).
% sum.reindex_bij_betw
thf(fact_7329_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,C4: set @ B,G3: A > B,B4: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A5 @ C4 )
=> ( ( bij_betw @ A @ B @ G3 @ B4 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C4 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ A5 ) @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ ( sup_sup @ ( set @ A ) @ A5 @ B4 )
@ ( sup_sup @ ( set @ B ) @ C4 @ D4 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_7330_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),F3: B > C,A17: A > ( set @ C )] :
( ! [I3: A,J2: A] :
( ( member @ A @ I3 @ I5 )
=> ( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A5 @ I3 ) @ ( A5 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A5 @ J2 ) @ ( A5 @ I3 ) ) ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( bij_betw @ B @ C @ F3 @ ( A5 @ I3 ) @ ( A17 @ I3 ) ) )
=> ( bij_betw @ B @ C @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A17 @ I5 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_7331_infinite__imp__bij__betw2,axiom,
! [A: $tType,A5: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [H4: A > A] : ( bij_betw @ A @ A @ H4 @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_7332_infinite__imp__bij__betw,axiom,
! [A: $tType,A5: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [H4: A > A] : ( bij_betw @ A @ A @ H4 @ A5 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_7333_disjnt__ge__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y7: set @ A,X6: set @ A] :
( ( finite_finite2 @ A @ Y7 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X6 )
=> ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ Y7 ) @ X3 ) )
=> ( disjnt @ A @ X6 @ Y7 ) ) ) ) ).
% disjnt_ge_max
thf(fact_7334_vimage__subset__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,B4: set @ B,A5: set @ A] :
( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ B4 ) @ A5 )
= ( ord_less_eq @ ( set @ B ) @ B4 @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ) ).
% vimage_subset_eq
thf(fact_7335_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T9: set @ C,H: B > C,S: set @ B,T2: set @ C,G3: C > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T9 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S @ S3 ) @ ( minus_minus @ ( set @ C ) @ T2 @ T9 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S3 )
=> ( ( G3 @ ( H @ A6 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T9 )
=> ( ( G3 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( G3 @ ( H @ X4 ) )
@ S )
= ( groups7311177749621191930dd_sum @ C @ A @ G3 @ T2 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_7336_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T9: set @ C,H: B > C,S: set @ B,T2: set @ C,G3: C > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T9 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S @ S3 ) @ ( minus_minus @ ( set @ C ) @ T2 @ T9 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S3 )
=> ( ( G3 @ ( H @ A6 ) )
= ( one_one @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T9 )
=> ( ( G3 @ B5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( G3 @ ( H @ X4 ) )
@ S )
= ( groups7121269368397514597t_prod @ C @ A @ G3 @ T2 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_7337_bij__image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B,B4: C > ( set @ A ),A5: set @ C] :
( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ A @ B @ F3 @ ( B4 @ X4 ) )
@ A5 ) ) ) ) ).
% bij_image_INT
thf(fact_7338_card__Un__disjnt,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( disjnt @ A @ A5 @ B4 )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) ) ) ) ) ) ).
% card_Un_disjnt
thf(fact_7339_sum__card__image,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( pairwise @ A
@ ^ [S6: A,T4: A] : ( disjnt @ B @ ( F3 @ S6 ) @ ( F3 @ T4 ) )
@ A5 )
=> ( ( groups7311177749621191930dd_sum @ ( set @ B ) @ nat @ ( finite_card @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ A5 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A7: A] : ( finite_card @ B @ ( F3 @ A7 ) )
@ A5 ) ) ) ) ).
% sum_card_image
thf(fact_7340_init__seg__of__def,axiom,
! [A: $tType] :
( ( init_seg_of @ A )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),S6: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ S6 )
& ! [A7: A,B6: A,C5: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B6 ) @ S6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ C5 ) @ R4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B6 ) @ R4 ) ) ) ) ) ) ).
% init_seg_of_def
thf(fact_7341_bij__betw__Suc,axiom,
! [M6: set @ nat,N7: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M6 @ N7 )
= ( ( image2 @ nat @ nat @ suc @ M6 )
= N7 ) ) ).
% bij_betw_Suc
thf(fact_7342_bij__swap,axiom,
! [A: $tType,B: $tType] : ( bij_betw @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) ) ).
% bij_swap
thf(fact_7343_refl__on__init__seg__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( init_seg_of @ A ) ) ).
% refl_on_init_seg_of
thf(fact_7344_antisym__init__seg__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 ) @ ( init_seg_of @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ R3 ) @ ( init_seg_of @ A ) )
=> ( R3 = S2 ) ) ) ).
% antisym_init_seg_of
thf(fact_7345_trans__init__seg__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A ),T6: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 ) @ ( init_seg_of @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ T6 ) @ ( init_seg_of @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ T6 ) @ ( init_seg_of @ A ) ) ) ) ).
% trans_init_seg_of
thf(fact_7346_pairwise__trivial,axiom,
! [A: $tType,I5: set @ A] :
( pairwise @ A
@ ^ [I4: A,J3: A] : J3 != I4
@ I5 ) ).
% pairwise_trivial
thf(fact_7347_pairwiseD,axiom,
! [A: $tType,R2: A > A > $o,S: set @ A,X: A,Y: A] :
( ( pairwise @ A @ R2 @ S )
=> ( ( member @ A @ X @ S )
=> ( ( member @ A @ Y @ S )
=> ( ( X != Y )
=> ( R2 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_7348_pairwiseI,axiom,
! [A: $tType,S: set @ A,R2: A > A > $o] :
( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ S )
=> ( ( member @ A @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R2 @ X3 @ Y3 ) ) ) )
=> ( pairwise @ A @ R2 @ S ) ) ).
% pairwiseI
thf(fact_7349_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R6: A > A > $o,S5: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S5 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ S5 )
=> ( ( X4 != Y4 )
=> ( R6 @ X4 @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_7350_pairwise__insert,axiom,
! [A: $tType,R3: A > A > $o,X: A,S2: set @ A] :
( ( pairwise @ A @ R3 @ ( insert @ A @ X @ S2 ) )
= ( ! [Y4: A] :
( ( ( member @ A @ Y4 @ S2 )
& ( Y4 != X ) )
=> ( ( R3 @ X @ Y4 )
& ( R3 @ Y4 @ X ) ) )
& ( pairwise @ A @ R3 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_7351_pairwise__imageI,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B,P: B > B > $o] :
( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( X3 != Y3 )
=> ( ( ( F3 @ X3 )
!= ( F3 @ Y3 ) )
=> ( P @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) ) )
=> ( pairwise @ B @ P @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ).
% pairwise_imageI
thf(fact_7352_pairwise__image,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,F3: B > A,S2: set @ B] :
( ( pairwise @ A @ R3 @ ( image2 @ B @ A @ F3 @ S2 ) )
= ( pairwise @ B
@ ^ [X4: B,Y4: B] :
( ( ( F3 @ X4 )
!= ( F3 @ Y4 ) )
=> ( R3 @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) )
@ S2 ) ) ).
% pairwise_image
thf(fact_7353_pairwise__mono,axiom,
! [A: $tType,P: A > A > $o,A5: set @ A,Q: A > A > $o,B4: set @ A] :
( ( pairwise @ A @ P @ A5 )
=> ( ! [X3: A,Y3: A] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( pairwise @ A @ Q @ B4 ) ) ) ) ).
% pairwise_mono
thf(fact_7354_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S: set @ A,T2: set @ A] :
( ( pairwise @ A @ P @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( pairwise @ A @ P @ T2 ) ) ) ).
% pairwise_subset
thf(fact_7355_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A5: A] : ( pairwise @ A @ P @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_7356_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_7357_initial__segment__of__Diff,axiom,
! [A: $tType,P3: set @ ( product_prod @ A @ A ),Q4: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P3 @ 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 ) ) @ P3 @ S2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ Q4 @ S2 ) ) @ ( init_seg_of @ A ) ) ) ).
% initial_segment_of_Diff
thf(fact_7358_pairwise__alt,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R6: A > A > $o,S5: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S5 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ ( minus_minus @ ( set @ A ) @ S5 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( R6 @ X4 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_7359_disjoint__image__subset,axiom,
! [A: $tType,A20: set @ ( set @ A ),F3: ( set @ A ) > ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A20 )
=> ( ! [X14: set @ A] :
( ( member @ ( set @ A ) @ X14 @ A20 )
=> ( ord_less_eq @ ( set @ A ) @ ( F3 @ X14 ) @ X14 ) )
=> ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ F3 @ A20 ) ) ) ) ).
% disjoint_image_subset
thf(fact_7360_Chains__init__seg__of__Union,axiom,
! [A: $tType,R2: set @ ( set @ ( product_prod @ A @ A ) ),R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) @ R2 @ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) )
=> ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ 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 ) ) @ R3 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 ) ) @ ( init_seg_of @ A ) ) ) ) ).
% Chains_init_seg_of_Union
thf(fact_7361_arg__min__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F3: A > B,P: A > $o,A3: A] :
( ( inj_on @ A @ B @ F3 @ ( collect @ A @ P ) )
=> ( ( P @ A3 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ A3 ) @ ( F3 @ Y3 ) ) )
=> ( ( lattices_ord_arg_min @ A @ B @ F3 @ P )
= A3 ) ) ) ) ) ).
% arg_min_inj_eq
thf(fact_7362_arg__min__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,M: A > nat] :
( ( P @ X )
=> ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ X ) ) ) ).
% arg_min_nat_le
thf(fact_7363_arg__min__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) )
& ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ Y6 ) ) ) ) ) ).
% arg_min_nat_lemma
thf(fact_7364_arg__min__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P: C > $o,K: C,F3: C > A] :
( ( P @ K )
=> ( ! [X3: C] :
( ( P @ X3 )
=> ( ord_less_eq @ A @ ( F3 @ K ) @ ( F3 @ X3 ) ) )
=> ( ( F3 @ ( lattices_ord_arg_min @ C @ A @ F3 @ P ) )
= ( F3 @ K ) ) ) ) ) ).
% arg_min_equality
thf(fact_7365_arg__minI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X: A,F3: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ~ ( ord_less @ B @ ( F3 @ Y3 ) @ ( F3 @ X ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ~ ( ord_less @ B @ ( F3 @ Y6 ) @ ( F3 @ X3 ) ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( lattices_ord_arg_min @ A @ B @ F3 @ P ) ) ) ) ) ) ).
% arg_minI
thf(fact_7366_mono__Chains,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R3 ) @ ( chains @ A @ S2 ) ) ) ).
% mono_Chains
thf(fact_7367_Chains__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( set @ A )
@ ^ [C8: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ C8 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ C8 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R4 ) ) ) ) ) ) ) ).
% Chains_def
thf(fact_7368_Chains__relation__of,axiom,
! [A: $tType,C4: set @ A,P: A > A > $o,A5: set @ A] :
( ( member @ ( set @ A ) @ C4 @ ( chains @ A @ ( order_relation_of @ A @ P @ A5 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ C4 @ A5 ) ) ).
% Chains_relation_of
thf(fact_7369_Chains__inits__DiffI,axiom,
! [A: $tType,R2: set @ ( set @ ( product_prod @ A @ A ) ),S2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) @ R2 @ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) )
=> ( member @ ( set @ ( set @ ( product_prod @ A @ A ) ) )
@ ( collect @ ( set @ ( product_prod @ A @ A ) )
@ ^ [Uu2: set @ ( product_prod @ A @ A )] :
? [R4: set @ ( product_prod @ A @ A )] :
( ( Uu2
= ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ S2 ) )
& ( member @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R2 ) ) )
@ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) ) ) ).
% Chains_inits_DiffI
thf(fact_7370_arg__min__on__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic7623131987881927897min_on @ B @ A )
= ( ^ [F4: B > A,S5: set @ B] :
( lattices_ord_arg_min @ B @ A @ F4
@ ^ [X4: B] : ( member @ B @ X4 @ S5 ) ) ) ) ) ).
% arg_min_on_def
thf(fact_7371_Chains__subset_H,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( ord_less_eq @ ( set @ ( set @ A ) )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) ) )
@ ( chains @ A @ R3 ) ) ) ).
% Chains_subset'
thf(fact_7372_Chains__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R3 )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) ) ) ) ).
% Chains_subset
thf(fact_7373_arg__min__natI,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% arg_min_natI
thf(fact_7374_chains__alt__def,axiom,
! [A: $tType] :
( ( chains2 @ A )
= ( ^ [A8: set @ ( set @ A )] : ( collect @ ( set @ ( set @ A ) ) @ ( pred_chain @ ( set @ A ) @ A8 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% chains_alt_def
thf(fact_7375_pred__on_Ochain__total,axiom,
! [A: $tType,A5: set @ A,P: A > A > $o,C4: set @ A,X: A,Y: A] :
( ( pred_chain @ A @ A5 @ P @ C4 )
=> ( ( member @ A @ X @ C4 )
=> ( ( member @ A @ Y @ C4 )
=> ( ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X
@ Y )
| ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ Y
@ X ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_7376_subset_Ochain__total,axiom,
! [A: $tType,A5: set @ ( set @ A ),C4: set @ ( set @ A ),X: set @ A,Y: set @ A] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C4 )
=> ( ( member @ ( set @ A ) @ X @ C4 )
=> ( ( member @ ( set @ A ) @ Y @ C4 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ X
@ Y )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ Y
@ X ) ) ) ) ) ).
% subset.chain_total
thf(fact_7377_subset_OchainI,axiom,
! [A: $tType,C4: set @ ( set @ A ),A5: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C4 @ A5 )
=> ( ! [X3: set @ A,Y3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C4 )
=> ( ( member @ ( set @ A ) @ Y3 @ C4 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ X3
@ Y3 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ Y3
@ X3 ) ) ) )
=> ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C4 ) ) ) ).
% subset.chainI
thf(fact_7378_subset_Ochain__def,axiom,
! [A: $tType,A5: set @ ( set @ A ),C4: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C4 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C4 @ A5 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C4 )
=> ! [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ C4 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ X4
@ Y4 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ Y4
@ X4 ) ) ) ) ) ) ).
% subset.chain_def
thf(fact_7379_pred__on_Ochain__def,axiom,
! [A: $tType] :
( ( pred_chain @ A )
= ( ^ [A8: set @ A,P4: A > A > $o,C8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C8 @ A8 )
& ! [X4: A] :
( ( member @ A @ X4 @ C8 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ C8 )
=> ( ( sup_sup @ ( A > A > $o ) @ P4
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X4
@ Y4 )
| ( sup_sup @ ( A > A > $o ) @ P4
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ Y4
@ X4 ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_7380_pred__on_OchainI,axiom,
! [A: $tType,C4: set @ A,A5: set @ A,P: A > A > $o] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ A5 )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ C4 )
=> ( ( member @ A @ Y3 @ C4 )
=> ( ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X3
@ Y3 )
| ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ Y3
@ X3 ) ) ) )
=> ( pred_chain @ A @ A5 @ P @ C4 ) ) ) ).
% pred_on.chainI
thf(fact_7381_subset__Zorn,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ! [C6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ? [X7: set @ A] :
( ( member @ ( set @ A ) @ X7 @ A5 )
& ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C6 )
=> ( ord_less_eq @ ( set @ A ) @ Xa3 @ X7 ) ) ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A5 )
& ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa2 )
=> ( Xa2 = X3 ) ) ) ) ) ).
% subset_Zorn
thf(fact_7382_subset_Ochain__empty,axiom,
! [A: $tType,A5: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% subset.chain_empty
thf(fact_7383_pred__on_Ochain__empty,axiom,
! [A: $tType,A5: set @ A,P: A > A > $o] : ( pred_chain @ A @ A5 @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pred_on.chain_empty
thf(fact_7384_subset__Zorn_H,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ! [C6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) @ A5 ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A5 )
& ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa2 )
=> ( Xa2 = X3 ) ) ) ) ) ).
% subset_Zorn'
thf(fact_7385_subset__chain__def,axiom,
! [A: $tType,A20: set @ ( set @ A ),C10: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ C10 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C10 @ A20 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C10 )
=> ! [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ C10 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y4 )
| ( ord_less_eq @ ( set @ A ) @ Y4 @ X4 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_7386_subset__chain__insert,axiom,
! [A: $tType,A20: set @ ( set @ A ),B4: set @ A,B13: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ B4 @ B13 ) )
= ( ( member @ ( set @ A ) @ B4 @ A20 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ B13 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ B4 )
| ( ord_less_eq @ ( set @ A ) @ B4 @ X4 ) ) )
& ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B13 ) ) ) ).
% subset_chain_insert
thf(fact_7387_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_7388_subset__Zorn__nonempty,axiom,
! [A: $tType,A20: set @ ( set @ A )] :
( ( A20
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [C11: set @ ( set @ A )] :
( ( C11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ C11 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C11 ) @ A20 ) ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A20 )
& ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ A20 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa2 )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_7389_Union__in__chain,axiom,
! [A: $tType,B13: set @ ( set @ A ),A20: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B13 )
=> ( ( B13
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B13 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B13 ) @ B13 ) ) ) ) ).
% Union_in_chain
thf(fact_7390_Inter__in__chain,axiom,
! [A: $tType,B13: set @ ( set @ A ),A20: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B13 )
=> ( ( B13
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B13 )
=> ( member @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B13 ) @ B13 ) ) ) ) ).
% Inter_in_chain
thf(fact_7391_subset_Ochain__extend,axiom,
! [A: $tType,A5: set @ ( set @ A ),C4: set @ ( set @ A ),Z2: set @ A] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C4 )
=> ( ( member @ ( set @ A ) @ Z2 @ A5 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C4 )
=> ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z4: set @ A] : Y5 = Z4
@ X3
@ Z2 ) )
=> ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z2 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C4 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_7392_pred__on_Ochain__extend,axiom,
! [A: $tType,A5: set @ A,P: A > A > $o,C4: set @ A,Z2: A] :
( ( pred_chain @ A @ A5 @ P @ C4 )
=> ( ( member @ A @ Z2 @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ C4 )
=> ( sup_sup @ ( A > A > $o ) @ P
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ X3
@ Z2 ) )
=> ( pred_chain @ A @ A5 @ P @ ( sup_sup @ ( set @ A ) @ ( insert @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) @ C4 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_7393_finite__subset__Union__chain,axiom,
! [A: $tType,A5: set @ A,B13: set @ ( set @ A ),A20: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ B13 ) )
=> ( ( B13
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A20 @ ( ord_less @ ( set @ A ) ) @ B13 )
=> ~ ! [B11: set @ A] :
( ( member @ ( set @ A ) @ B11 @ B13 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A5 @ B11 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_7394_Chains__alt__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R3 )
=> ( ( chains @ A @ R3 )
= ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) ) ) ) ) ).
% Chains_alt_def
thf(fact_7395_multiset_Odomain,axiom,
! [B: $tType,C: $tType,T2: C > B > $o] :
( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T2 ) )
= ( ^ [X4: C > nat] :
? [Y4: B > nat] :
( ( bNF_rel_fun @ C @ B @ nat @ nat @ T2
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ X4
@ Y4 )
& ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Z7: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( Y4 @ Z7 ) ) ) ) ) ) ) ).
% multiset.domain
thf(fact_7396_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_7397_DomainpE,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,A3: A] :
( ( domainp @ A @ B @ R3 @ A3 )
=> ~ ! [B5: B] :
~ ( R3 @ A3 @ B5 ) ) ).
% DomainpE
thf(fact_7398_DomainPI,axiom,
! [B: $tType,A: $tType,R3: A > B > $o,A3: A,B2: B] :
( ( R3 @ A3 @ B2 )
=> ( domainp @ A @ B @ R3 @ A3 ) ) ).
% DomainPI
thf(fact_7399_Domainp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [R4: A > B > $o,A7: A] :
? [B6: A,C5: B] :
( ( A7 = B6 )
& ( R4 @ B6 @ C5 ) ) ) ) ).
% Domainp.simps
thf(fact_7400_Domainp_Ocases,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,A3: A] :
( ( domainp @ A @ B @ R3 @ A3 )
=> ~ ! [B5: B] :
~ ( R3 @ A3 @ B5 ) ) ).
% Domainp.cases
thf(fact_7401_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Y: A,Z2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ ( Conj @ Y @ Z2 ) @ X )
= ( Conj @ ( Disj @ Y @ X ) @ ( Disj @ Z2 @ X ) ) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib2
thf(fact_7402_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,Y: A,Z2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ ( Disj @ Y @ Z2 ) @ X )
= ( Disj @ ( Conj @ Y @ X ) @ ( Conj @ Z2 @ X ) ) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib2
thf(fact_7403_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( ( Compl @ X )
= ( Compl @ Y ) )
= ( X = Y ) ) ) ).
% abstract_boolean_algebra.compl_eq_compl_iff
thf(fact_7404_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A,Z2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ ( Conj @ Y @ Z2 ) )
= ( Conj @ ( Disj @ X @ Y ) @ ( Disj @ X @ Z2 ) ) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib
thf(fact_7405_abstract__boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ ( Compl @ X ) )
= One ) ) ).
% abstract_boolean_algebra.disj_cancel_right
thf(fact_7406_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A,Z2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ ( Disj @ Y @ Z2 ) )
= ( Disj @ ( Conj @ X @ Y ) @ ( Conj @ X @ Z2 ) ) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib
thf(fact_7407_abstract__boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ ( Compl @ X ) )
= Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_right
thf(fact_7408_abstract__boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,A3: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( ( Conj @ A3 @ X )
= Zero )
=> ( ( ( Disj @ A3 @ X )
= One )
=> ( ( ( Conj @ A3 @ Y )
= Zero )
=> ( ( ( Disj @ A3 @ Y )
= One )
=> ( X = Y ) ) ) ) ) ) ).
% abstract_boolean_algebra.complement_unique
thf(fact_7409_abstract__boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ ( Compl @ X ) @ X )
= One ) ) ).
% abstract_boolean_algebra.disj_cancel_left
thf(fact_7410_abstract__boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ ( Compl @ X ) @ X )
= Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_left
thf(fact_7411_abstract__boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ Zero )
= X ) ) ).
% abstract_boolean_algebra.disj_zero_right
thf(fact_7412_abstract__boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ Zero )
= Zero ) ) ).
% abstract_boolean_algebra.conj_zero_right
thf(fact_7413_abstract__boolean__algebra_Odisj__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ X @ One )
= One ) ) ).
% abstract_boolean_algebra.disj_one_right
thf(fact_7414_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ ( Disj @ X @ Y ) )
= ( Conj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ).
% abstract_boolean_algebra.de_Morgan_disj
thf(fact_7415_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ ( Conj @ X @ Y ) )
= ( Disj @ ( Compl @ X ) @ ( Compl @ Y ) ) ) ) ).
% abstract_boolean_algebra.de_Morgan_conj
thf(fact_7416_abstract__boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ Zero @ X )
= Zero ) ) ).
% abstract_boolean_algebra.conj_zero_left
thf(fact_7417_abstract__boolean__algebra_Oconj__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Conj @ X @ One )
= X ) ) ).
% abstract_boolean_algebra.conj_one_right
thf(fact_7418_abstract__boolean__algebra_Odisj__one__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Disj @ One @ X )
= One ) ) ).
% abstract_boolean_algebra.disj_one_left
thf(fact_7419_abstract__boolean__algebra_Odouble__compl,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ ( Compl @ X ) )
= X ) ) ).
% abstract_boolean_algebra.double_compl
thf(fact_7420_abstract__boolean__algebra_Ocompl__unique,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A,X: A,Y: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( ( Conj @ X @ Y )
= Zero )
=> ( ( ( Disj @ X @ Y )
= One )
=> ( ( Compl @ X )
= Y ) ) ) ) ).
% abstract_boolean_algebra.compl_unique
thf(fact_7421_abstract__boolean__algebra_Ocompl__zero,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ Zero )
= One ) ) ).
% abstract_boolean_algebra.compl_zero
thf(fact_7422_abstract__boolean__algebra_Ocompl__one,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One )
=> ( ( Compl @ One )
= Zero ) ) ).
% abstract_boolean_algebra.compl_one
thf(fact_7423_multiset_Odomain__eq,axiom,
! [A: $tType] :
( ( domainp @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
= ( ^ [F4: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) ) ) ).
% multiset.domain_eq
thf(fact_7424_multiset_Odomain__par__left__total,axiom,
! [B: $tType,C: $tType,T2: C > B > $o,P8: ( C > nat ) > $o] :
( ( left_total @ ( C > nat ) @ ( B > nat )
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T2
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 ) )
=> ( ( bNF_rel_fun @ ( C > nat ) @ ( B > nat ) @ $o @ $o
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T2
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P8
@ ^ [F4: B > nat] :
( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) )
=> ( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T2 ) )
= P8 ) ) ) ).
% multiset.domain_par_left_total
thf(fact_7425_Rep__unit,axiom,
! [X: product_unit] : ( member @ $o @ ( product_Rep_unit @ X ) @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ).
% Rep_unit
thf(fact_7426_Rep__unit__inject,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( product_Rep_unit @ X )
= ( product_Rep_unit @ Y ) )
= ( X = Y ) ) ).
% Rep_unit_inject
thf(fact_7427_Rep__unit__induct,axiom,
! [Y: $o,P: $o > $o] :
( ( member @ $o @ Y @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ! [X3: product_unit] : ( P @ ( product_Rep_unit @ X3 ) )
=> ( P @ Y ) ) ) ).
% Rep_unit_induct
thf(fact_7428_Rep__unit__cases,axiom,
! [Y: $o] :
( ( member @ $o @ Y @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ~ ! [X3: product_unit] :
( Y
= ( ~ ( product_Rep_unit @ X3 ) ) ) ) ).
% Rep_unit_cases
thf(fact_7429_type__definition__unit,axiom,
type_definition @ product_unit @ $o @ product_Rep_unit @ product_Abs_unit @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ).
% type_definition_unit
thf(fact_7430_Abs__unit__inverse,axiom,
! [Y: $o] :
( ( member @ $o @ Y @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
= Y ) ) ).
% Abs_unit_inverse
thf(fact_7431_Rep__unit__inverse,axiom,
! [X: product_unit] :
( ( product_Abs_unit @ ( product_Rep_unit @ X ) )
= X ) ).
% Rep_unit_inverse
thf(fact_7432_Abs__unit__cases,axiom,
! [X: product_unit] :
~ ! [Y3: $o] :
( ( X
= ( product_Abs_unit @ Y3 ) )
=> ~ ( member @ $o @ Y3 @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% Abs_unit_cases
thf(fact_7433_Abs__unit__induct,axiom,
! [P: product_unit > $o,X: product_unit] :
( ! [Y3: $o] :
( ( member @ $o @ Y3 @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( P @ ( product_Abs_unit @ Y3 ) ) )
=> ( P @ X ) ) ).
% Abs_unit_induct
thf(fact_7434_Abs__unit__inject,axiom,
! [X: $o,Y: $o] :
( ( member @ $o @ X @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( member @ $o @ Y @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( ( product_Abs_unit @ X )
= ( product_Abs_unit @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_unit_inject
thf(fact_7435_multiset_Odomain__par,axiom,
! [B: $tType,C: $tType,T2: C > B > $o,DT: C > $o,DS: nat > $o,P23: ( C > nat ) > $o] :
( ( ( domainp @ C @ B @ T2 )
= DT )
=> ( ( ( domainp @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
= DS )
=> ( ( left_unique @ C @ B @ T2 )
=> ( ( bNF_rel_fun @ ( C > nat ) @ ( B > nat ) @ $o @ $o
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T2
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P23
@ ^ [F4: B > nat] :
( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F4 @ X4 ) ) ) ) )
=> ( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T2 ) )
= ( inf_inf @ ( ( C > nat ) > $o ) @ ( basic_pred_fun @ C @ nat @ DT @ DS ) @ P23 ) ) ) ) ) ) ).
% multiset.domain_par
thf(fact_7436_mono__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ D )
& ( order @ C )
& ( order @ A ) )
=> ! [A5: A > B > $o,B4: C > D > $o] :
( ( bi_total @ A @ B @ A5 )
=> ( ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A5
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less_eq @ A )
@ ( ord_less_eq @ B ) )
=> ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B4
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less_eq @ C )
@ ( ord_less_eq @ D ) )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B4 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ( order_mono @ A @ C )
@ ( order_mono @ B @ D ) ) ) ) ) ) ).
% mono_transfer
thf(fact_7437_typedef__left__unique,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T2: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T2
= ( ^ [X4: A,Y4: B] :
( X4
= ( Rep @ Y4 ) ) ) )
=> ( left_unique @ A @ B @ T2 ) ) ) ).
% typedef_left_unique
thf(fact_7438_pred__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_fun @ A @ B )
= ( ^ [A8: A > $o,B7: B > $o,F4: A > B] :
! [X4: A] :
( ( A8 @ X4 )
=> ( B7 @ ( F4 @ X4 ) ) ) ) ) ).
% pred_fun_def
thf(fact_7439_fun_Opred__True,axiom,
! [A: $tType,D: $tType] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ ^ [Uu2: A] : $true )
= ( ^ [Uu2: D > A] : $true ) ) ).
% fun.pred_True
thf(fact_7440_fun_Opred__mono,axiom,
! [D: $tType,A: $tType,P: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
=> ( ord_less_eq @ ( ( D > A ) > $o )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ P )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ Pa ) ) ) ).
% fun.pred_mono
thf(fact_7441_fun_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F3: A > B,G3: A > B] :
( ( X = Ya )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ ^ [Z7: A] :
( ( F3 @ Z7 )
= ( G3 @ Z7 ) )
@ Ya )
=> ( ( comp @ A @ B @ D @ F3 @ X )
= ( comp @ A @ B @ D @ G3 @ Ya ) ) ) ) ).
% fun.map_cong_pred
thf(fact_7442_pred__fun__True__id,axiom,
! [A: $tType,B: $tType,C: $tType,P3: B > $o,F3: C > B] :
( ( nO_MATCH @ ( A > A ) @ ( B > $o ) @ ( id @ A ) @ P3 )
=> ( ( basic_pred_fun @ C @ B
@ ^ [X4: C] : $true
@ P3
@ F3 )
= ( basic_pred_fun @ C @ $o
@ ^ [X4: C] : $true
@ ( id @ $o )
@ ( comp @ B @ $o @ C @ P3 @ F3 ) ) ) ) ).
% pred_fun_True_id
thf(fact_7443_fun_Opred__mono__strong,axiom,
! [A: $tType,D: $tType,P: A > $o,X: D > A,Pa: A > $o] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ P
@ X )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ Pa
@ X ) ) ) ).
% fun.pred_mono_strong
thf(fact_7444_fun_Opred__cong,axiom,
! [A: $tType,D: $tType,X: D > A,Ya: D > A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ P
@ X )
= ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ Pa
@ Ya ) ) ) ) ).
% fun.pred_cong
thf(fact_7445_fun_Opred__rel,axiom,
! [A: $tType,D: $tType,P: A > $o,X: D > A] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ P
@ X )
= ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ( bNF_eq_onp @ A @ P )
@ X
@ X ) ) ).
% fun.pred_rel
thf(fact_7446_fun_ODomainp__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R2: A > B > $o] :
( ( domainp @ ( C > A ) @ ( C > B )
@ ( bNF_rel_fun @ C @ C @ A @ B
@ ^ [Y5: C,Z4: C] : Y5 = Z4
@ R2 ) )
= ( basic_pred_fun @ C @ A
@ ^ [Uu2: C] : $true
@ ( domainp @ A @ B @ R2 ) ) ) ).
% fun.Domainp_rel
thf(fact_7447_fun_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q: B > $o,F3: A > B,X: D > A] :
( ( basic_pred_fun @ D @ B
@ ^ [Uu2: D] : $true
@ Q
@ ( comp @ A @ B @ D @ F3 @ X ) )
= ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ ( comp @ B @ $o @ A @ Q @ F3 )
@ X ) ) ).
% fun.pred_map
thf(fact_7448_fun_Opred__set,axiom,
! [A: $tType,D: $tType,P: A > $o] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ P )
= ( ^ [X4: D > A] :
! [Y4: A] :
( ( member @ A @ Y4 @ ( image2 @ D @ A @ X4 @ ( top_top @ ( set @ D ) ) ) )
=> ( P @ Y4 ) ) ) ) ).
% fun.pred_set
thf(fact_7449_fun_Opred__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,R2: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( D > A ) > $o ) @ ( ( D > B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R2
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ $o @ $o
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R2 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true )
@ ( basic_pred_fun @ D @ B
@ ^ [Uu2: D] : $true ) ) ).
% fun.pred_transfer
thf(fact_7450_fun_Orel__eq__onp,axiom,
! [D: $tType,A: $tType,P: A > $o] :
( ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( D > A )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu2: D] : $true
@ P ) ) ) ).
% fun.rel_eq_onp
thf(fact_7451_Inter__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( ( bi_total @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( set @ ( set @ A ) ) @ ( set @ ( set @ B ) ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A5 ) ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( complete_Inf_Inf @ ( set @ A ) ) @ ( complete_Inf_Inf @ ( set @ B ) ) ) ) ) ).
% Inter_transfer
thf(fact_7452_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) )
= ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_7453_in__measures_I1_J,axiom,
! [A: $tType,X: A,Y: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_7454_subset__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( set @ A ) > $o ) @ ( ( set @ B ) > $o ) @ ( bNF_rel_set @ A @ B @ A5 )
@ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ $o @ $o @ ( bNF_rel_set @ A @ B @ A5 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less_eq @ ( set @ A ) )
@ ( ord_less_eq @ ( set @ B ) ) ) ) ).
% subset_transfer
thf(fact_7455_strict__subset__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( set @ A ) > $o ) @ ( ( set @ B ) > $o ) @ ( bNF_rel_set @ A @ B @ A5 )
@ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ $o @ $o @ ( bNF_rel_set @ A @ B @ A5 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less @ ( set @ A ) )
@ ( ord_less @ ( set @ B ) ) ) ) ).
% strict_subset_transfer
thf(fact_7456_typedef__bi__unique,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T2: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T2
= ( ^ [X4: A,Y4: B] :
( X4
= ( Rep @ Y4 ) ) ) )
=> ( bi_unique @ A @ B @ T2 ) ) ) ).
% typedef_bi_unique
thf(fact_7457_measures__less,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ).
% measures_less
thf(fact_7458_measures__lesseq,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_7459_measures__def,axiom,
! [A: $tType] :
( ( measures @ A )
= ( ^ [Fs2: list @ ( A > nat )] :
( inv_image @ ( list @ nat ) @ A @ ( lex @ nat @ less_than )
@ ^ [A7: A] :
( map @ ( A > nat ) @ nat
@ ^ [F4: A > nat] : ( F4 @ A7 )
@ Fs2 ) ) ) ) ).
% measures_def
thf(fact_7460_inter__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( set @ A ) > ( set @ A ) ) @ ( ( set @ B ) > ( set @ B ) ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_set @ A @ B @ A5 ) ) @ ( inf_inf @ ( set @ A ) ) @ ( inf_inf @ ( set @ B ) ) ) ) ).
% inter_transfer
thf(fact_7461_right__total__Inter__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( ( right_total @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( set @ ( set @ A ) ) @ ( set @ ( set @ B ) ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A5 ) ) @ ( bNF_rel_set @ A @ B @ A5 )
@ ^ [S5: set @ ( set @ A )] : ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ S5 ) @ ( collect @ A @ ( domainp @ A @ B @ A5 ) ) )
@ ( complete_Inf_Inf @ ( set @ B ) ) ) ) ) ).
% right_total_Inter_transfer
thf(fact_7462_vimage__right__total__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B4: A > B > $o,A5: C > D > $o] :
( ( bi_unique @ A @ B @ B4 )
=> ( ( right_total @ C @ D @ A5 )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( set @ A ) > ( set @ C ) ) @ ( ( set @ B ) > ( set @ D ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ A5 @ B4 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ C ) @ ( set @ D ) @ ( bNF_rel_set @ A @ B @ B4 ) @ ( bNF_rel_set @ C @ D @ A5 ) )
@ ^ [F4: C > A,X11: set @ A] : ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F4 @ X11 ) @ ( collect @ C @ ( domainp @ C @ D @ A5 ) ) )
@ ( vimage @ D @ B ) ) ) ) ).
% vimage_right_total_transfer
thf(fact_7463_typedef__right__total,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T2: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T2
= ( ^ [X4: A,Y4: B] :
( X4
= ( Rep @ Y4 ) ) ) )
=> ( right_total @ A @ B @ T2 ) ) ) ).
% typedef_right_total
thf(fact_7464_right__total__Collect__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( right_total @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( set @ A ) @ ( set @ B )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( bNF_rel_set @ A @ B @ A5 )
@ ^ [P4: A > $o] :
( collect @ A
@ ^ [X4: A] :
( ( P4 @ X4 )
& ( domainp @ A @ B @ A5 @ X4 ) ) )
@ ( collect @ B ) ) ) ).
% right_total_Collect_transfer
thf(fact_7465_right__total__Domainp__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B4: A > B > $o,A5: C > D > $o] :
( ( right_total @ A @ B @ B4 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A5
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ A5
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ^ [T8: C > A > $o,X4: C] :
? [Y4: A] :
( ( member @ A @ Y4 @ ( collect @ A @ ( domainp @ A @ B @ B4 ) ) )
& ( T8 @ X4 @ Y4 ) )
@ ( domainp @ D @ B ) ) ) ).
% right_total_Domainp_transfer
thf(fact_7466_right__total__Compl__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( ( bi_unique @ A @ B @ A5 )
=> ( ( right_total @ A @ B @ A5 )
=> ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_set @ A @ B @ A5 )
@ ^ [S5: set @ A] : ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ S5 ) @ ( collect @ A @ ( domainp @ A @ B @ A5 ) ) )
@ ( uminus_uminus @ ( set @ B ) ) ) ) ) ).
% right_total_Compl_transfer
thf(fact_7467_right__total__fun__eq__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B4: C > D > $o] :
( ( right_total @ A @ B @ A5 )
=> ( ( bi_unique @ C @ D @ B4 )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B4 )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B4 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ^ [F4: A > C,G4: A > C] :
! [X4: A] :
( ( member @ A @ X4 @ ( collect @ A @ ( domainp @ A @ B @ A5 ) ) )
=> ( ( F4 @ X4 )
= ( G4 @ X4 ) ) )
@ ^ [Y5: B > D,Z4: B > D] : Y5 = Z4 ) ) ) ).
% right_total_fun_eq_transfer
thf(fact_7468_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X4: A,N: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N ) @ ( power_power @ A @ X4 @ ( nat2 @ N ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X4 ) @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_7469_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( image @ A @ A @ R3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A3 = B2 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_7470_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W2: num,Y: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) @ M )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_7471_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,W2: num,M: int] :
( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W2 ) ) @ M )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_7472_power__int__minus__one__mult__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) )
= ( one_one @ A ) ) ) ).
% power_int_minus_one_mult_self
thf(fact_7473_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int,B2: A] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ B2 ) )
= B2 ) ) ).
% power_int_minus_one_mult_self'
thf(fact_7474_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: num,N2: num] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N2 ) ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N2 ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_7475_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: num,N2: num,B2: A] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N2 ) ) @ B2 ) )
= ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N2 ) ) ) @ B2 ) ) ) ).
% power_int_add_numeral2
thf(fact_7476_power__int__mono__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N2 ) @ ( power_int @ A @ B2 @ N2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_7477_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_7478_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N7: int,A3: A] :
( ( ord_less @ int @ N2 @ N7 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N2 ) @ ( power_int @ A @ A3 @ N7 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_7479_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N2: int] :
( ( times_times @ A @ ( power_int @ A @ X @ N2 ) @ X )
= ( times_times @ A @ X @ ( power_int @ A @ X @ N2 ) ) ) ) ).
% power_int_commutes
thf(fact_7480_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ X @ Y ) @ M )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ).
% power_int_mult_distrib
thf(fact_7481_zero__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N2: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N2 ) ) ) ) ).
% zero_less_power_int
thf(fact_7482_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N7: int,A3: A] :
( ( ord_less_eq @ int @ N2 @ N7 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N2 ) @ ( power_int @ A @ A3 @ N7 ) ) ) ) ) ).
% power_int_increasing
thf(fact_7483_zero__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N2 ) ) ) ) ).
% zero_le_power_int
thf(fact_7484_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N7: int,A3: A] :
( ( ord_less @ int @ N2 @ N7 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N7 ) @ ( power_int @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_7485_power__int__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,N2: int] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N2 ) @ ( power_int @ A @ Y @ N2 ) ) ) ) ) ) ).
% power_int_mono
thf(fact_7486_power__int__strict__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N2: int] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N2 @ ( zero_zero @ int ) )
=> ( ord_less @ A @ ( power_int @ A @ B2 @ N2 ) @ ( power_int @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_7487_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N2: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N2 ) ) ) ) ) ).
% one_le_power_int
thf(fact_7488_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N2 ) ) ) ) ) ).
% one_less_power_int
thf(fact_7489_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int,N2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M @ N2 )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ N2 ) )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N2 ) ) ) ) ) ).
% power_int_add
thf(fact_7490_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( division_ring @ A )
& ( one @ B )
& ( uminus @ B ) )
=> ! [X: C,A3: A,N2: int] :
( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N2 )
= ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N2 ) @ ( power_int @ A @ A3 @ N2 ) ) ) ) ) ).
% power_int_minus_left_distrib
thf(fact_7491_power__int__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N2: int] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ B2 @ N2 ) @ ( power_int @ A @ A3 @ N2 ) ) ) ) ) ) ).
% power_int_antimono
thf(fact_7492_power__int__strict__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N2: int] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N2 ) @ ( power_int @ A @ B2 @ N2 ) ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_7493_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_7494_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N2: int,N7: int,A3: A] :
( ( ord_less_eq @ int @ N2 @ N7 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( A3
!= ( zero_zero @ A ) )
| ( N7
!= ( zero_zero @ int ) )
| ( N2
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N7 ) @ ( power_int @ A @ A3 @ N2 ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_7495_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M: int,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less_eq @ int @ M @ N2 ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_7496_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M: int,N2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 )
=> ( ord_less @ int @ M @ N2 ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_7497_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N2
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X @ ( minus_minus @ int @ N2 @ ( one_one @ int ) ) ) @ X )
= ( power_int @ A @ X @ N2 ) ) ) ) ).
% power_int_minus_mult
thf(fact_7498_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ X ) ) ) ) ).
% power_int_add_1
thf(fact_7499_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ X @ ( power_int @ A @ X @ M ) ) ) ) ) ).
% power_int_add_1'
thf(fact_7500_Zorns__po__lemma,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ! [C6: set @ A] :
( ( member @ ( set @ A ) @ C6 @ ( chains @ A @ R3 ) )
=> ? [X7: A] :
( ( member @ A @ X7 @ ( field2 @ A @ R3 ) )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ C6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X7 ) @ R3 ) ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa2 ) @ R3 )
=> ( Xa2 = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_7501_Partial__order__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_7125193373082350890der_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ) ).
% Partial_order_Restr
thf(fact_7502_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( A3 = B2 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_7503_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B4: set @ A,F5: A > ( filter @ B ),P: B > $o] :
( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ B4 )
=> ! [B5: A] :
( ( member @ A @ B5 @ B4 )
=> ? [X7: A] :
( ( member @ A @ X7 @ B4 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X7 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ A6 ) @ ( F5 @ B5 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ B4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ B4 )
& ( eventually @ B @ P @ ( F5 @ X4 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_7504_eventually__top,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( top_top @ ( filter @ A ) ) )
= ( ! [X11: A] : ( P @ X11 ) ) ) ).
% eventually_top
thf(fact_7505_eventually__const,axiom,
! [A: $tType,F5: filter @ A,P: $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X4: A] : P
@ F5 )
= P ) ) ).
% eventually_const
thf(fact_7506_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_7507_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N: nat] : ( P @ ( plus_plus @ nat @ N @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_7508_eventually__finite__subsets__at__top__weakI,axiom,
! [A: $tType,A5: set @ A,P: ( set @ A ) > $o] :
( ! [X14: set @ A] :
( ( finite_finite2 @ A @ X14 )
=> ( ( ord_less_eq @ ( set @ A ) @ X14 @ A5 )
=> ( P @ X14 ) ) )
=> ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A5 ) ) ) ).
% eventually_finite_subsets_at_top_weakI
thf(fact_7509_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P: B > $o,G3: B > A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F3 @ ( G3 @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G3 @ X3 ) ) )
=> ( ( 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_7510_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_7511_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N8: A] :
! [N: A] :
( ( ord_less_eq @ A @ N @ N8 )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_7512_le__filter__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F8: filter @ A] :
! [P4: A > $o] :
( ( eventually @ A @ P4 @ F8 )
=> ( eventually @ A @ P4 @ F7 ) ) ) ) ).
% le_filter_def
thf(fact_7513_filter__leI,axiom,
! [A: $tType,F6: filter @ A,F5: filter @ A] :
( ! [P2: A > $o] :
( ( eventually @ A @ P2 @ F6 )
=> ( eventually @ A @ P2 @ F5 ) )
=> ( ord_less_eq @ ( filter @ A ) @ F5 @ F6 ) ) ).
% filter_leI
thf(fact_7514_filter__leD,axiom,
! [A: $tType,F5: filter @ A,F6: filter @ A,P: A > $o] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ F6 )
=> ( ( eventually @ A @ P @ F6 )
=> ( eventually @ A @ P @ F5 ) ) ) ).
% filter_leD
thf(fact_7515_le__principal,axiom,
! [A: $tType,F5: filter @ A,A5: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ ( principal @ A @ A5 ) )
= ( eventually @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 )
@ F5 ) ) ).
% le_principal
thf(fact_7516_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_7517_le__sequentially,axiom,
! [F5: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F5 @ ( at_top @ nat ) )
= ( ! [N8: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N8 ) @ F5 ) ) ) ).
% le_sequentially
thf(fact_7518_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,P: A > $o] :
( ! [X3: A] :
( ( ord_less_eq @ A @ C2 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ A @ P @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_7519_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N8: A] :
! [N: A] :
( ( ord_less_eq @ A @ N8 @ N )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_7520_eventually__sequentiallyI,axiom,
! [C2: nat,P: nat > $o] :
( ! [X3: nat] :
( ( ord_less_eq @ nat @ C2 @ X3 )
=> ( P @ X3 ) )
=> ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_7521_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N8: nat] :
! [N: nat] :
( ( ord_less_eq @ nat @ N8 @ N )
=> ( P @ N ) ) ) ) ).
% eventually_sequentially
thf(fact_7522_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F3: A > B,F5: filter @ B,G5: filter @ A,F6: filter @ B,G8: filter @ A,F12: A > B] :
( ( filterlim @ A @ B @ F3 @ F5 @ G5 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F5 @ F6 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G8 @ G5 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= ( F12 @ X4 ) )
@ G8 )
=> ( filterlim @ A @ B @ F12 @ F6 @ G8 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_7523_eventually__finite__subsets__at__top,axiom,
! [A: $tType,P: ( set @ A ) > $o,A5: set @ A] :
( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A5 ) )
= ( ? [X11: set @ A] :
( ( finite_finite2 @ A @ X11 )
& ( ord_less_eq @ ( set @ A ) @ X11 @ A5 )
& ! [Y10: set @ A] :
( ( ( finite_finite2 @ A @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ X11 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A5 ) )
=> ( P @ Y10 ) ) ) ) ) ).
% eventually_finite_subsets_at_top
thf(fact_7524_eventually__ex,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
? [X11: B] : ( P @ X4 @ X11 )
@ F5 )
= ( ? [Y10: A > B] :
( eventually @ A
@ ^ [X4: A] : ( P @ X4 @ ( Y10 @ X4 ) )
@ F5 ) ) ) ).
% eventually_ex
thf(fact_7525_eventually__sup,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,F6: filter @ A] :
( ( eventually @ A @ P @ ( sup_sup @ ( filter @ A ) @ F5 @ F6 ) )
= ( ( eventually @ A @ P @ F5 )
& ( eventually @ A @ P @ F6 ) ) ) ).
% eventually_sup
thf(fact_7526_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F5: filter @ A,F3: B > A,G5: filter @ B] :
( ( eventually @ A @ P @ F5 )
=> ( ( filterlim @ B @ A @ F3 @ F5 @ G5 )
=> ( eventually @ B
@ ^ [X4: B] : ( P @ ( F3 @ X4 ) )
@ G5 ) ) ) ).
% eventually_compose_filterlim
thf(fact_7527_filterlim__cong,axiom,
! [A: $tType,B: $tType,F15: filter @ A,F16: filter @ A,F24: filter @ B,F25: filter @ B,F3: B > A,G3: B > A] :
( ( F15 = F16 )
=> ( ( F24 = F25 )
=> ( ( eventually @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= ( G3 @ X4 ) )
@ F24 )
=> ( ( filterlim @ B @ A @ F3 @ F15 @ F24 )
= ( filterlim @ B @ A @ G3 @ F16 @ F25 ) ) ) ) ) ).
% filterlim_cong
thf(fact_7528_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F4: A > B,F26: filter @ B,F17: filter @ A] :
! [P4: B > $o] :
( ( eventually @ B @ P4 @ F26 )
=> ( eventually @ A
@ ^ [X4: A] : ( P4 @ ( F4 @ X4 ) )
@ F17 ) ) ) ) ).
% filterlim_iff
thf(fact_7529_filterlim__principal,axiom,
! [B: $tType,A: $tType,F3: A > B,S: set @ B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( principal @ B @ S ) @ F5 )
= ( eventually @ A
@ ^ [X4: A] : ( member @ B @ ( F3 @ X4 ) @ S )
@ F5 ) ) ).
% filterlim_principal
thf(fact_7530_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N8: A] :
! [N: A] :
( ( ord_less @ A @ N8 @ N )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_7531_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_7532_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_7533_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_7534_eventually__finite__subsets__at__top__finite,axiom,
! [A: $tType,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( eventually @ ( set @ A ) @ P @ ( finite5375528669736107172at_top @ A @ A5 ) )
= ( P @ A5 ) ) ) ).
% eventually_finite_subsets_at_top_finite
thf(fact_7535_wo__rel_Omax2__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= B2 ) )
& ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= A3 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_7536_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( ( Y6 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R3 ) )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_7537_wo__rel_OTOTALS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ! [X7: A] :
( ( member @ A @ X7 @ ( field2 @ A @ R3 ) )
=> ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa2 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X7 ) @ R3 ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_7538_well__order__induct__imp,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ! [X3: A] :
( ! [Y6: A] :
( ( ( Y6 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X3 ) @ R3 ) )
=> ( ( member @ A @ Y6 @ ( field2 @ A @ R3 ) )
=> ( P @ Y6 ) ) )
=> ( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
=> ( P @ X3 ) ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( P @ A3 ) ) ) ) ).
% well_order_induct_imp
thf(fact_7539_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= A3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_7540_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 )
= B2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_7541_wo__rel_Omax2__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_7542_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y3: B] :
( eventually @ A
@ ^ [X4: A] : ( P @ X4 @ Y3 )
@ Net )
=> ( eventually @ A
@ ^ [X4: A] :
! [X11: B] : ( P @ X4 @ X11 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_7543_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_7544_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N8: A] :
! [N: A] :
( ( ord_less @ A @ N @ N8 )
=> ( P @ N ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_7545_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ~ ( eventually @ A @ Q @ F5 )
=> ~ ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F5 ) ) ) ).
% not_eventually_impI
thf(fact_7546_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) )
@ F5 )
= ( ( eventually @ A @ P @ F5 )
& ( eventually @ A @ Q @ F5 ) ) ) ).
% eventually_conj_iff
thf(fact_7547_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F5 )
=> ( eventually @ A @ Q @ F5 ) ) ) ).
% eventually_rev_mp
thf(fact_7548_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [N: A] :
( ( P @ N )
= ( Q @ N ) )
@ F5 )
=> ( ( eventually @ A @ P @ F5 )
= ( eventually @ A @ Q @ F5 ) ) ) ).
% eventually_subst
thf(fact_7549_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q: A > $o,R2: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A @ Q @ F5 )
=> ( ! [I3: A] :
( ( P @ I3 )
=> ( ( Q @ I3 )
=> ( R2 @ I3 ) ) )
=> ( eventually @ A @ R2 @ F5 ) ) ) ) ).
% eventually_elim2
thf(fact_7550_eventually__conj,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ( eventually @ A @ Q @ F5 )
=> ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) )
@ F5 ) ) ) ).
% eventually_conj
thf(fact_7551_eventually__True,axiom,
! [A: $tType,F5: filter @ A] :
( eventually @ A
@ ^ [X4: A] : $true
@ F5 ) ).
% eventually_True
thf(fact_7552_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
@ F5 )
=> ( ( eventually @ A @ P @ F5 )
=> ( eventually @ A @ Q @ F5 ) ) ) ).
% eventually_mp
thf(fact_7553_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C4: $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P @ X4 )
| C4 )
@ F5 )
= ( ( eventually @ A @ P @ F5 )
| C4 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_7554_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C4: $o,P: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( C4
| ( P @ X4 ) )
@ F5 )
= ( C4
| ( eventually @ A @ P @ F5 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_7555_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C4: $o,P: A > $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( C4
=> ( P @ X4 ) )
@ F5 )
= ( C4
=> ( eventually @ A @ P @ F5 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_7556_eventuallyI,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ F5 ) ) ).
% eventuallyI
thf(fact_7557_filter__eq__iff,axiom,
! [A: $tType] :
( ( ^ [Y5: filter @ A,Z4: filter @ A] : Y5 = Z4 )
= ( ^ [F7: filter @ A,F8: filter @ A] :
! [P4: A > $o] :
( ( eventually @ A @ P4 @ F7 )
= ( eventually @ A @ P4 @ F8 ) ) ) ) ).
% filter_eq_iff
thf(fact_7558_eventually__mono,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F5 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( eventually @ A @ Q @ F5 ) ) ) ).
% eventually_mono
thf(fact_7559_not__eventuallyD,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ~ ( eventually @ A @ P @ F5 )
=> ? [X3: A] :
~ ( P @ X3 ) ) ).
% not_eventuallyD
thf(fact_7560_always__eventually,axiom,
! [A: $tType,P: A > $o,F5: filter @ A] :
( ! [X_1: A] : ( P @ X_1 )
=> ( eventually @ A @ P @ F5 ) ) ).
% always_eventually
thf(fact_7561_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_7562_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_7563_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_7564_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A5: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( eventually @ B
@ ^ [X4: B] :
! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( P @ X4 @ Y4 ) )
@ Net )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( eventually @ B
@ ^ [Y4: B] : ( P @ Y4 @ X4 )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_7565_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A5: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( eventually @ B
@ ^ [Y4: B] : ( P @ Y4 @ X3 )
@ Net ) )
=> ( eventually @ B
@ ^ [X4: B] :
! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( P @ X4 @ Y4 ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_7566_eventually__INF1,axiom,
! [B: $tType,A: $tType,I2: A,I5: set @ A,P: B > $o,F5: A > ( filter @ B )] :
( ( member @ A @ I2 @ I5 )
=> ( ( eventually @ B @ P @ ( F5 @ I2 ) )
=> ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ I5 ) ) ) ) ) ).
% eventually_INF1
thf(fact_7567_trivial__limit__def,axiom,
! [A: $tType,F5: filter @ A] :
( ( F5
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X4: A] : $false
@ F5 ) ) ).
% trivial_limit_def
thf(fact_7568_eventually__const__iff,axiom,
! [A: $tType,P: $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] : P
@ F5 )
= ( P
| ( F5
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_7569_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X3: A] :
~ ( P @ X3 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_7570_eventually__happens_H,axiom,
! [A: $tType,F5: filter @ A,P: A > $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P @ F5 )
=> ? [X_1: A] : ( P @ X_1 ) ) ) ).
% eventually_happens'
thf(fact_7571_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_7572_eventually__bot,axiom,
! [A: $tType,P: A > $o] : ( eventually @ A @ P @ ( bot_bot @ ( filter @ A ) ) ) ).
% eventually_bot
thf(fact_7573_eventually__inf__principal,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,S2: set @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F5 @ ( principal @ A @ S2 ) ) )
= ( eventually @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( P @ X4 ) )
@ F5 ) ) ).
% eventually_inf_principal
thf(fact_7574_eventually__inf,axiom,
! [A: $tType,P: A > $o,F5: filter @ A,F6: filter @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F5 @ F6 ) )
= ( ? [Q3: A > $o,R6: A > $o] :
( ( eventually @ A @ Q3 @ F5 )
& ( eventually @ A @ R6 @ F6 )
& ! [X4: A] :
( ( ( Q3 @ X4 )
& ( R6 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_inf
thf(fact_7575_wo__rel_Omax2__among,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% wo_rel.max2_among
thf(fact_7576_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X4: A] :
! [Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( P @ Y4 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_7577_eventually__Inf__base,axiom,
! [A: $tType,B4: set @ ( filter @ A ),P: A > $o] :
( ( B4
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F9: filter @ A] :
( ( member @ ( filter @ A ) @ F9 @ B4 )
=> ! [G6: filter @ A] :
( ( member @ ( filter @ A ) @ G6 @ B4 )
=> ? [X7: filter @ A] :
( ( member @ ( filter @ A ) @ X7 @ B4 )
& ( ord_less_eq @ ( filter @ A ) @ X7 @ ( inf_inf @ ( filter @ A ) @ F9 @ G6 ) ) ) ) )
=> ( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B4 ) )
= ( ? [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ B4 )
& ( eventually @ A @ P @ X4 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_7578_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z11: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z11 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top
thf(fact_7579_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z11: B] :
( ( ord_less_eq @ B @ C2 @ Z11 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z11 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_7580_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,F5: filter @ B,G3: B > A] :
( ( filterlim @ B @ A @ F3 @ ( at_top @ A ) @ F5 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
@ F5 )
=> ( filterlim @ B @ A @ G3 @ ( at_top @ A ) @ F5 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_7581_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z11: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ Z11 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_7582_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A5: set @ A,P: B > $o,F5: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ A5 ) ) )
= ( ? [Q3: A > B > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( eventually @ B @ ( Q3 @ X4 ) @ ( F5 @ X4 ) ) )
& ! [Y4: B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( Q3 @ X4 @ Y4 ) )
=> ( P @ Y4 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_7583_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z11: B] :
( ( ord_less_eq @ B @ Z11 @ C2 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z11 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_7584_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z11: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z11 )
@ F5 ) ) ) ) ).
% filterlim_at_bot
thf(fact_7585_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z11: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F3 @ X4 ) @ Z11 )
@ F5 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_7586_wo__rel_Ocases__Total,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R3 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R3 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_7587_natLeq__on__wo__rel,axiom,
! [N2: nat] :
( bNF_Wellorder_wo_rel @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y4: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y4 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_7588_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) ) @ R3 )
& ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A3 @ B2 ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_7589_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z11: B] :
( ( ord_less @ B @ C2 @ Z11 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z11 @ ( F3 @ X4 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_7590_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F5: B > ( filter @ A ),B4: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F5 @ B4 ) ) )
= ( ? [X11: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X11 @ B4 )
& ( finite_finite2 @ B @ X11 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F5 @ X11 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_7591_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z11: B] :
( ( ord_less @ B @ Z11 @ C2 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z11 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_7592_eventually__Inf,axiom,
! [A: $tType,P: A > $o,B4: set @ ( filter @ A )] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ B4 ) )
= ( ? [X11: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X11 @ B4 )
& ( finite_finite2 @ ( filter @ A ) @ X11 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ X11 ) ) ) ) ) ).
% eventually_Inf
thf(fact_7593_filterlim__finite__subsets__at__top,axiom,
! [A: $tType,B: $tType,F3: A > ( set @ B ),A5: set @ B,F5: filter @ A] :
( ( filterlim @ A @ ( set @ B ) @ F3 @ ( finite5375528669736107172at_top @ B @ A5 ) @ F5 )
= ( ! [X11: set @ B] :
( ( ( finite_finite2 @ B @ X11 )
& ( ord_less_eq @ ( set @ B ) @ X11 @ A5 ) )
=> ( eventually @ A
@ ^ [Y4: A] :
( ( finite_finite2 @ B @ ( F3 @ Y4 ) )
& ( ord_less_eq @ ( set @ B ) @ X11 @ ( F3 @ Y4 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F3 @ Y4 ) @ A5 ) )
@ F5 ) ) ) ) ).
% filterlim_finite_subsets_at_top
thf(fact_7594_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G3: B > A,Y7: set @ B,X6: set @ A,F5: filter @ B,F3: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G3 @ Y7 ) @ X6 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ B @ X4 @ Y7 )
@ F5 )
=> ( ( map_filter_on @ A @ C @ X6 @ F3 @ ( map_filter_on @ B @ A @ Y7 @ G3 @ F5 ) )
= ( map_filter_on @ B @ C @ Y7 @ ( comp @ A @ C @ B @ F3 @ G3 ) @ F5 ) ) ) ) ).
% map_filter_on_comp
thf(fact_7595_wo__rel_OWell__order__isMinim__exists,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_1: A] : ( bNF_We4791949203932849705sMinim @ A @ R3 @ B4 @ X_1 ) ) ) ) ).
% wo_rel.Well_order_isMinim_exists
thf(fact_7596_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X6: set @ A,F5: filter @ A,P: B > $o,F3: A > B] :
( ( eventually @ A
@ ^ [X4: A] : ( member @ A @ X4 @ X6 )
@ F5 )
=> ( ( eventually @ B @ P @ ( map_filter_on @ A @ B @ X6 @ F3 @ F5 ) )
= ( eventually @ A
@ ^ [X4: A] :
( ( P @ ( F3 @ X4 ) )
& ( member @ A @ X4 @ X6 ) )
@ F5 ) ) ) ).
% eventually_map_filter_on
thf(fact_7597_wo__rel_OisMinim__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( bNF_We4791949203932849705sMinim @ A @ R3 @ A5 @ B2 )
= ( ( member @ A @ B2 @ A5 )
& ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ X4 ) @ R3 ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_7598_wo__rel_Ominim__isMinim,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( bNF_We4791949203932849705sMinim @ A @ R3 @ B4 @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B4 ) ) ) ) ) ).
% wo_rel.minim_isMinim
thf(fact_7599_wo__rel_Ominim__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( bNF_We6954850376910717587_minim @ A @ R3 @ A5 )
= ( the @ A @ ( bNF_We4791949203932849705sMinim @ A @ R3 @ A5 ) ) ) ) ).
% wo_rel.minim_def
thf(fact_7600_wo__rel_Oequals__minim,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A3 @ B4 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B5 ) @ R3 ) )
=> ( A3
= ( bNF_We6954850376910717587_minim @ A @ R3 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_7601_wo__rel_Ominim__least,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ B4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B4 ) @ B2 ) @ R3 ) ) ) ) ).
% wo_rel.minim_least
thf(fact_7602_wo__rel_Ominim__inField,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B4 ) @ ( field2 @ A @ R3 ) ) ) ) ) ).
% wo_rel.minim_inField
thf(fact_7603_wo__rel_Ominim__in,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_We6954850376910717587_minim @ A @ R3 @ B4 ) @ B4 ) ) ) ) ).
% wo_rel.minim_in
thf(fact_7604_Sup__filter__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( filter @ A ) )
= ( ^ [S5: set @ ( filter @ A )] :
( abs_filter @ A
@ ^ [P4: A > $o] :
! [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ S5 )
=> ( eventually @ A @ P4 @ X4 ) ) ) ) ) ).
% Sup_filter_def
thf(fact_7605_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,X: 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 @ X @ Xs ) ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_7606_insort__insert__triv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs )
= Xs ) ) ) ).
% insort_insert_triv
thf(fact_7607_bot__filter__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P4: A > $o] : $true ) ) ).
% bot_filter_def
thf(fact_7608_sup__filter__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F8: filter @ A] :
( abs_filter @ A
@ ^ [P4: A > $o] :
( ( eventually @ A @ P4 @ F7 )
& ( eventually @ A @ P4 @ F8 ) ) ) ) ) ).
% sup_filter_def
thf(fact_7609_principal__def,axiom,
! [A: $tType] :
( ( principal @ A )
= ( ^ [S5: set @ A] : ( abs_filter @ A @ ( ball @ A @ S5 ) ) ) ) ).
% principal_def
thf(fact_7610_map__filter__on__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B )
= ( ^ [X11: set @ A,F4: A > B,F7: filter @ A] :
( abs_filter @ B
@ ^ [P4: B > $o] :
( eventually @ A
@ ^ [X4: A] :
( ( P4 @ ( F4 @ X4 ) )
& ( member @ A @ X4 @ X11 ) )
@ F7 ) ) ) ) ).
% map_filter_on_def
thf(fact_7611_top__filter__def,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P5: A > $o] :
! [X5: A] : ( P5 @ X5 ) ) ) ).
% top_filter_def
thf(fact_7612_set__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( set2 @ A
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) )
= ( insert @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ).
% set_insort_insert
thf(fact_7613_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,X: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) ) ) ) ).
% sorted_insort_insert
thf(fact_7614_insort__insert__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) ) ) ) ).
% insort_insert_insort
thf(fact_7615_inf__filter__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F8: filter @ A] :
( abs_filter @ A
@ ^ [P4: A > $o] :
? [Q3: A > $o,R6: A > $o] :
( ( eventually @ A @ Q3 @ F7 )
& ( eventually @ A @ R6 @ F8 )
& ! [X4: A] :
( ( ( Q3 @ X4 )
& ( R6 @ X4 ) )
=> ( P4 @ X4 ) ) ) ) ) ) ).
% inf_filter_def
thf(fact_7616_Ref__Time_Oupdate__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_update @ A )
= ( ^ [R4: ref @ A,V3: A] :
( heap_Time_heap @ product_unit
@ ^ [H6: 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 @ R4 @ V3 @ H6 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% Ref_Time.update_def
thf(fact_7617_prod__mset_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: multiset @ B,B4: multiset @ B,G3: B > A] :
( ( ( inter_mset @ B @ A5 @ B4 )
= ( zero_zero @ ( multiset @ B ) ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G3 @ ( union_mset @ B @ A5 @ B4 ) ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G3 @ A5 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G3 @ B4 ) ) ) ) ) ) ).
% prod_mset.union_disjoint
thf(fact_7618_unit__abs__eta__conv,axiom,
! [A: $tType,F3: product_unit > A] :
( ( ^ [U2: product_unit] : ( F3 @ product_Unity ) )
= F3 ) ).
% unit_abs_eta_conv
thf(fact_7619_prod__mset_Oadd__mset,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [X: A,N7: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( add_mset @ A @ X @ N7 ) )
= ( times_times @ A @ X @ ( comm_m9189036328036947845d_mset @ A @ N7 ) ) ) ) ).
% prod_mset.add_mset
thf(fact_7620_prod__mset_Ounion,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M6: multiset @ A,N7: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M6 @ N7 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ M6 ) @ ( comm_m9189036328036947845d_mset @ A @ N7 ) ) ) ) ).
% prod_mset.union
thf(fact_7621_prod__mset__Un,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: multiset @ A,B4: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B4 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ A5 ) @ ( comm_m9189036328036947845d_mset @ A @ B4 ) ) ) ) ).
% prod_mset_Un
thf(fact_7622_prod__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [Uu2: B] : ( one_one @ A )
@ A5 ) )
= ( one_one @ A ) ) ) ).
% prod_mset.neutral_const
thf(fact_7623_prod__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A,X: B,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G3 @ ( add_mset @ B @ X @ A5 ) ) )
= ( times_times @ A @ ( G3 @ X ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G3 @ A5 ) ) ) ) ) ).
% prod_mset.insert
thf(fact_7624_prod__mset__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: A,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [Uu2: B] : C2
@ A5 ) )
= ( power_power @ A @ C2 @ ( size_size @ ( multiset @ B ) @ A5 ) ) ) ) ).
% prod_mset_constant
thf(fact_7625_UNIV__unit,axiom,
( ( top_top @ ( set @ product_unit ) )
= ( insert @ product_unit @ product_Unity @ ( bot_bot @ ( set @ product_unit ) ) ) ) ).
% UNIV_unit
thf(fact_7626_bot__unit__def,axiom,
( ( bot_bot @ product_unit )
= product_Unity ) ).
% bot_unit_def
thf(fact_7627_Unity__def,axiom,
( product_Unity
= ( product_Abs_unit @ $true ) ) ).
% Unity_def
thf(fact_7628_sup__unit__def,axiom,
( ( sup_sup @ product_unit )
= ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_7629_Inf__unit__def,axiom,
( ( complete_Inf_Inf @ product_unit )
= ( ^ [Uu3: set @ product_unit] : product_Unity ) ) ).
% Inf_unit_def
thf(fact_7630_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : Y = product_Unity ).
% old.unit.exhaust
thf(fact_7631_top__unit__def,axiom,
( ( top_top @ product_unit )
= product_Unity ) ).
% top_unit_def
thf(fact_7632_uminus__unit__def,axiom,
( ( uminus_uminus @ product_unit )
= ( ^ [Uu3: product_unit] : product_Unity ) ) ).
% uminus_unit_def
thf(fact_7633_prod__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A,H: B > A,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( G3 @ X4 ) @ ( H @ X4 ) )
@ A5 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G3 @ A5 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ H @ A5 ) ) ) ) ) ).
% prod_mset.distrib
thf(fact_7634_prod__mset_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > C > A,B4: multiset @ C,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [I4: B] : ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ C @ A @ ( G3 @ I4 ) @ B4 ) )
@ A5 ) )
= ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ C @ A
@ ^ [J3: C] :
( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [I4: B] : ( G3 @ I4 @ J3 )
@ A5 ) )
@ B4 ) ) ) ) ).
% prod_mset.swap
thf(fact_7635_Sup__unit__def,axiom,
( ( complete_Sup_Sup @ product_unit )
= ( ^ [Uu3: set @ product_unit] : product_Unity ) ) ).
% Sup_unit_def
thf(fact_7636_inf__unit__def,axiom,
( ( inf_inf @ product_unit )
= ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_7637_wait__def,axiom,
( heap_Time_wait
= ( ^ [N: nat] :
( heap_Time_Heap2 @ product_unit
@ ^ [H6: 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 @ H6 @ N ) ) ) ) ) ) ).
% wait_def
thf(fact_7638_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_7639_prod__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C2: A,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( if @ A @ ( Y = X4 ) @ C2 @ ( one_one @ A ) )
@ A5 ) )
= ( power_power @ A @ C2 @ ( count @ B @ A5 @ Y ) ) ) ) ).
% prod_mset_delta'
thf(fact_7640_prod__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C2: A,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X4: B] : ( if @ A @ ( X4 = Y ) @ C2 @ ( one_one @ A ) )
@ A5 ) )
= ( power_power @ A @ C2 @ ( count @ B @ A5 @ Y ) ) ) ) ).
% prod_mset_delta
thf(fact_7641_prod__mset__multiplicity,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( comm_m9189036328036947845d_mset @ A )
= ( ^ [M10: multiset @ A] :
( groups7121269368397514597t_prod @ A @ A
@ ^ [X4: A] : ( power_power @ A @ X4 @ ( count @ A @ M10 @ X4 ) )
@ ( set_mset @ A @ M10 ) ) ) ) ) ).
% prod_mset_multiplicity
thf(fact_7642_execute__update,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R3: ref @ A,V2: A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ product_unit @ ( ref_update @ A @ R3 @ V2 ) @ H )
= ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R3 @ V2 @ H ) @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_update
thf(fact_7643_prod__mset_Oremove,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [X: A,A5: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ A5 ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ A5 )
= ( times_times @ A @ X @ ( comm_m9189036328036947845d_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% prod_mset.remove
thf(fact_7644_old_Ounit_Orec,axiom,
! [T: $tType,F1: T] :
( ( product_rec_unit @ T @ F1 @ product_Unity )
= F1 ) ).
% old.unit.rec
thf(fact_7645_default__unit__def,axiom,
( ( default_default @ product_unit )
= product_Unity ) ).
% default_unit_def
thf(fact_7646_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F13: T,X4: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F13 @ X4 ) ) ) ) ).
% old.rec_unit_def
thf(fact_7647_CODE__ABORT__def,axiom,
! [A: $tType] :
( ( cODE_ABORT @ A )
= ( ^ [F4: product_unit > A] : ( F4 @ product_Unity ) ) ) ).
% CODE_ABORT_def
thf(fact_7648_log_Osimps,axiom,
( log
= ( ^ [B6: code_natural,I4: code_natural] :
( if @ code_natural
@ ( ( ord_less_eq @ code_natural @ B6 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ I4 @ B6 ) )
@ ( one_one @ code_natural )
@ ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ B6 @ ( divide_divide @ code_natural @ I4 @ B6 ) ) ) ) ) ) ).
% log.simps
thf(fact_7649_log_Oelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( ( log @ X @ Xa )
= Y )
=> ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) ) ) ).
% log.elims
thf(fact_7650_log_Opelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( ( log @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) )
=> ~ ( ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) ) ) ) ) ).
% log.pelims
thf(fact_7651_minus__shift__def,axiom,
( minus_shift
= ( ^ [R4: code_natural,K3: code_natural,L2: code_natural] : ( if @ code_natural @ ( ord_less @ code_natural @ K3 @ L2 ) @ ( minus_minus @ code_natural @ ( plus_plus @ code_natural @ R4 @ K3 ) @ L2 ) @ ( minus_minus @ code_natural @ K3 @ L2 ) ) ) ) ).
% minus_shift_def
thf(fact_7652_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
! [A: $tType,X: product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F2: code_natural > A,N5: code_natural,M5: code_natural] :
( X
!= ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F2 @ ( product_Pair @ code_natural @ code_natural @ N5 @ M5 ) ) ) ).
% Lazy_Sequence.iterate_upto.cases
thf(fact_7653_exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( ( quickc658316121487927005ustive @ B )
& ( cl_HOL_Oequal @ A )
& ( quickc658316121487927005ustive @ A ) )
=> ! [X: product_prod @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F2: ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I3: code_natural,D2: code_natural] :
( X
!= ( product_Pair @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F2 @ ( product_Pair @ code_natural @ code_natural @ I3 @ D2 ) ) ) ) ).
% exhaustive_fun'.cases
thf(fact_7654_exhaustive__natural_H_Ocases,axiom,
! [X: product_prod @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F2: code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I3: code_natural] :
( X
!= ( product_Pair @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F2 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I3 ) ) ) ).
% exhaustive_natural'.cases
thf(fact_7655_full__exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( ( quickc3360725361186068524ustive @ B )
& ( cl_HOL_Oequal @ A )
& ( quickc3360725361186068524ustive @ A ) )
=> ! [X: product_prod @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F2: ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I3: code_natural,D2: code_natural] :
( X
!= ( product_Pair @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F2 @ ( product_Pair @ code_natural @ code_natural @ I3 @ D2 ) ) ) ) ).
% full_exhaustive_fun'.cases
thf(fact_7656_full__exhaustive__natural_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F2: ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I3: code_natural] :
( X
!= ( product_Pair @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F2 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I3 ) ) ) ).
% full_exhaustive_natural'.cases
thf(fact_7657_log_Ocases,axiom,
! [X: product_prod @ code_natural @ code_natural] :
~ ! [B5: code_natural,I3: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ B5 @ I3 ) ) ).
% log.cases
thf(fact_7658_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_7659_abstract__filter__def,axiom,
! [A: $tType] :
( ( abstract_filter @ A )
= ( ^ [F4: product_unit > ( filter @ A )] : ( F4 @ product_Unity ) ) ) ).
% abstract_filter_def
thf(fact_7660_iter_Ocases,axiom,
! [A: $tType,X: product_prod @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )] :
~ ! [Random: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ),Nrandom: code_natural,Seed: product_prod @ code_natural @ code_natural] :
( X
!= ( product_Pair @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ Random @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ Nrandom @ Seed ) ) ) ).
% iter.cases
thf(fact_7661_split__seed__def,axiom,
( split_seed
= ( ^ [S6: 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 ) )
@ ^ [V3: code_natural,W3: code_natural] :
( product_case_prod @ code_natural @ code_natural @ ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [V6: code_natural,W4: code_natural] : ( product_Pair @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_Pair @ code_natural @ code_natural @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V3 ) @ W4 ) @ ( product_Pair @ code_natural @ code_natural @ V6 @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W3 ) ) )
@ ( product_snd @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( next @ S6 ) ) )
@ S6 ) ) ) ).
% split_seed_def
thf(fact_7662_Random_Orange__def,axiom,
( range
= ( ^ [K3: 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K3 )
@ ^ [L2: code_natural] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ next
@ ^ [V3: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ V3 @ ( times_times @ code_natural @ L2 @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ code_natural ) )
@ ^ [V3: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( modulo_modulo @ code_natural @ V3 @ K3 ) ) ) ) ) ).
% Random.range_def
thf(fact_7663_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F4: B > ( product_prod @ C @ D ),G4: C > D > A,X4: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F4 @ X4 ) ) ) ) ).
% scomp_apply
thf(fact_7664_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F3: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F3 )
= ( F3 @ X ) ) ).
% Pair_scomp
thf(fact_7665_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
= X ) ).
% scomp_Pair
thf(fact_7666_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: $tType,E: $tType,F3: A > ( product_prod @ E @ F ),G3: E > F > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F @ ( product_prod @ C @ D ) @ F3 @ G3 ) @ H )
= ( product_scomp @ A @ E @ F @ B @ F3
@ ^ [X4: E] : ( product_scomp @ F @ C @ D @ B @ ( G3 @ X4 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_7667_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F4 @ X4 ) ) ) ) ).
% scomp_def
thf(fact_7668_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa @ Xb )
= Y )
=> ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) ) ) ).
% iterate.elims
thf(fact_7669_iterate_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( iterate @ B @ A )
= ( ^ [K3: code_natural,F4: B > A > ( product_prod @ B @ A ),X4: B] :
( if @ ( A > ( product_prod @ B @ A ) )
@ ( K3
= ( zero_zero @ code_natural ) )
@ ( product_Pair @ B @ A @ X4 )
@ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F4 @ X4 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K3 @ ( one_one @ code_natural ) ) @ F4 ) ) ) ) ) ).
% iterate.simps
thf(fact_7670_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( G4 @ ( product_fst @ B @ C @ ( F4 @ X4 ) ) @ ( product_snd @ B @ C @ ( F4 @ X4 ) ) ) ) ) ).
% scomp_unfold
thf(fact_7671_range,axiom,
! [K: code_natural,S2: product_prod @ code_natural @ code_natural] :
( ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K )
=> ( ord_less @ code_natural @ ( product_fst @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( range @ K @ S2 ) ) @ K ) ) ).
% range
thf(fact_7672_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) )
=> ~ ( ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) ) ) ) ) ).
% iterate.pelims
thf(fact_7673_select__weight__member,axiom,
! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A ),S2: 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 @ S2 ) ) @ ( set2 @ A @ ( map @ ( product_prod @ code_natural @ A ) @ A @ ( product_snd @ code_natural @ A ) @ Xs ) ) ) ) ).
% select_weight_member
thf(fact_7674_iterate_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B )] :
~ ! [K2: code_natural,F2: B > A > ( product_prod @ B @ A ),X3: B] :
( X
!= ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ K2 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ F2 @ X3 ) ) ) ).
% iterate.cases
thf(fact_7675_select__weight__cons__zero,axiom,
! [A: $tType,X: A,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( select_weight @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( zero_zero @ code_natural ) @ X ) @ Xs ) )
= ( select_weight @ A @ Xs ) ) ).
% select_weight_cons_zero
thf(fact_7676_select__weight__drop__zero,axiom,
! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( select_weight @ A
@ ( filter2 @ ( product_prod @ code_natural @ A )
@ ( product_case_prod @ code_natural @ A @ $o
@ ^ [K3: code_natural,Uu2: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K3 ) )
@ Xs ) )
= ( select_weight @ A @ Xs ) ) ).
% select_weight_drop_zero
thf(fact_7677_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 ) ) )
@ ^ [K3: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( pick @ A @ Xs3 @ K3 ) ) ) ) ) ).
% select_weight_def
thf(fact_7678_pick__member,axiom,
! [A: $tType,I2: code_natural,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( ord_less @ code_natural @ I2 @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs ) ) )
=> ( member @ A @ ( pick @ A @ Xs @ I2 ) @ ( set2 @ A @ ( map @ ( product_prod @ code_natural @ A ) @ A @ ( product_snd @ code_natural @ A ) @ Xs ) ) ) ) ).
% pick_member
thf(fact_7679_pick__drop__zero,axiom,
! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( pick @ A
@ ( filter2 @ ( product_prod @ code_natural @ A )
@ ( product_case_prod @ code_natural @ A @ $o
@ ^ [K3: code_natural,Uu2: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K3 ) )
@ Xs ) )
= ( pick @ A @ Xs ) ) ).
% pick_drop_zero
thf(fact_7680_pick_Osimps,axiom,
! [A: $tType,I2: code_natural,X: product_prod @ code_natural @ A,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( ( ord_less @ code_natural @ I2 @ ( product_fst @ code_natural @ A @ X ) )
=> ( ( pick @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ X @ Xs ) @ I2 )
= ( product_snd @ code_natural @ A @ X ) ) )
& ( ~ ( ord_less @ code_natural @ I2 @ ( product_fst @ code_natural @ A @ X ) )
=> ( ( pick @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ X @ Xs ) @ I2 )
= ( pick @ A @ Xs @ ( minus_minus @ code_natural @ I2 @ ( product_fst @ code_natural @ A @ X ) ) ) ) ) ) ).
% pick.simps
thf(fact_7681_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_7682_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 ) ) )
=> ( ! [F2: code_natural > A,N5: code_natural,M5: code_natural] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F2 @ ( product_Pair @ code_natural @ code_natural @ N5 @ M5 ) ) )
=> ( ! [X7: product_unit] :
( ~ ( ord_less @ code_natural @ M5 @ N5 )
=> ( P @ F2 @ ( plus_plus @ code_natural @ N5 @ ( one_one @ code_natural ) ) @ M5 ) )
=> ( P @ F2 @ N5 @ M5 ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% Predicate.iterate_upto.pinduct
thf(fact_7683_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_7684_filtercomap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtercomap @ A @ B )
= ( ^ [F4: A > B,F7: filter @ B] :
( abs_filter @ A
@ ^ [P4: A > $o] :
? [Q3: B > $o] :
( ( eventually @ B @ Q3 @ F7 )
& ! [X4: A] :
( ( Q3 @ ( F4 @ X4 ) )
=> ( P4 @ X4 ) ) ) ) ) ) ).
% filtercomap_def
thf(fact_7685_filterlim__filtercomap,axiom,
! [A: $tType,B: $tType,F3: A > B,F5: filter @ B] : ( filterlim @ A @ B @ F3 @ F5 @ ( filtercomap @ A @ B @ F3 @ F5 ) ) ).
% filterlim_filtercomap
thf(fact_7686_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_7687_eventually__filtercomapI,axiom,
! [B: $tType,A: $tType,P: A > $o,F5: filter @ A,F3: B > A] :
( ( eventually @ A @ P @ F5 )
=> ( eventually @ B
@ ^ [X4: B] : ( P @ ( F3 @ X4 ) )
@ ( filtercomap @ B @ A @ F3 @ F5 ) ) ) ).
% eventually_filtercomapI
thf(fact_7688_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_7689_filtercomap__principal,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ B] :
( ( filtercomap @ A @ B @ F3 @ ( principal @ B @ A5 ) )
= ( principal @ A @ ( vimage @ A @ B @ F3 @ A5 ) ) ) ).
% filtercomap_principal
thf(fact_7690_less__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( ord_less @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( ord_less @ nat @ Xa @ X ) ) ).
% less_natural.abs_eq
thf(fact_7691_filtercomap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,F5: C > ( filter @ B ),B4: set @ C] :
( ( filtercomap @ A @ B @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F5 @ B4 ) ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B6: C] : ( filtercomap @ A @ B @ F3 @ ( F5 @ B6 ) )
@ B4 ) ) ) ).
% filtercomap_INF
thf(fact_7692_filtercomap__ident,axiom,
! [A: $tType,F5: filter @ A] :
( ( filtercomap @ A @ A
@ ^ [X4: A] : X4
@ F5 )
= F5 ) ).
% filtercomap_ident
thf(fact_7693_filtercomap__filtercomap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,G3: B > C,F5: filter @ C] :
( ( filtercomap @ A @ B @ F3 @ ( filtercomap @ B @ C @ G3 @ F5 ) )
= ( filtercomap @ A @ C
@ ^ [X4: A] : ( G3 @ ( F3 @ X4 ) )
@ F5 ) ) ).
% filtercomap_filtercomap
thf(fact_7694_filterlim__filtercomap__iff,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B,G3: B > C,G5: filter @ C,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( filtercomap @ B @ C @ G3 @ G5 ) @ F5 )
= ( filterlim @ A @ C @ ( comp @ B @ C @ A @ G3 @ F3 ) @ G5 @ F5 ) ) ).
% filterlim_filtercomap_iff
thf(fact_7695_eventually__filtercomap,axiom,
! [A: $tType,B: $tType,P: A > $o,F3: A > B,F5: filter @ B] :
( ( eventually @ A @ P @ ( filtercomap @ A @ B @ F3 @ F5 ) )
= ( ? [Q3: B > $o] :
( ( eventually @ B @ Q3 @ F5 )
& ! [X4: A] :
( ( Q3 @ ( F3 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap
thf(fact_7696_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F3: B > A] :
( ! [P2: A > $o] :
( ( eventually @ A @ P2 @ F5 )
=> ? [X7: B] : ( P2 @ ( F3 @ X7 ) ) )
=> ( ( filtercomap @ B @ A @ F3 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtercomap_neq_bot
thf(fact_7697_filtercomap__inf,axiom,
! [A: $tType,B: $tType,F3: A > B,F15: filter @ B,F24: filter @ B] :
( ( filtercomap @ A @ B @ F3 @ ( inf_inf @ ( filter @ B ) @ F15 @ F24 ) )
= ( inf_inf @ ( filter @ A ) @ ( filtercomap @ A @ B @ F3 @ F15 ) @ ( filtercomap @ A @ B @ F3 @ F24 ) ) ) ).
% filtercomap_inf
thf(fact_7698_filtercomap__sup,axiom,
! [A: $tType,B: $tType,F3: A > B,F15: filter @ B,F24: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( sup_sup @ ( filter @ A ) @ ( filtercomap @ A @ B @ F3 @ F15 ) @ ( filtercomap @ A @ B @ F3 @ F24 ) ) @ ( filtercomap @ A @ B @ F3 @ ( sup_sup @ ( filter @ B ) @ F15 @ F24 ) ) ) ).
% filtercomap_sup
thf(fact_7699_filterlim__iff__le__filtercomap,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F4: A > B,F7: filter @ B,G9: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ G9 @ ( filtercomap @ A @ B @ F4 @ F7 ) ) ) ) ).
% filterlim_iff_le_filtercomap
thf(fact_7700_filtercomap__mono,axiom,
! [B: $tType,A: $tType,F5: filter @ A,F6: filter @ A,F3: B > A] :
( ( ord_less_eq @ ( filter @ A ) @ F5 @ F6 )
=> ( ord_less_eq @ ( filter @ B ) @ ( filtercomap @ B @ A @ F3 @ F5 ) @ ( filtercomap @ B @ A @ F3 @ F6 ) ) ) ).
% filtercomap_mono
thf(fact_7701_less__eq__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( ord_less_eq @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( ord_less_eq @ nat @ Xa @ X ) ) ).
% less_eq_natural.abs_eq
thf(fact_7702_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F3: B > A] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( filtercomap @ B @ A @ F3 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% filtercomap_neq_bot_surj
thf(fact_7703_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 ) ) )
= ( ? [N8: A] :
! [X4: B] :
( ( ord_less_eq @ A @ N8 @ ( F3 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_top_linorder
thf(fact_7704_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 ) ) )
= ( ? [N8: A] :
! [X4: B] :
( ( ord_less @ A @ N8 @ ( F3 @ X4 ) )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_top_dense
thf(fact_7705_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 ) ) )
= ( ? [N8: A] :
! [X4: B] :
( ( ord_less_eq @ A @ ( F3 @ X4 ) @ N8 )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
thf(fact_7706_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 ) ) )
= ( ? [N8: A] :
! [X4: B] :
( ( ord_less @ A @ ( F3 @ X4 ) @ N8 )
=> ( P @ X4 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_dense
thf(fact_7707_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > C,F5: B > ( filter @ C ),B4: set @ B] :
( ord_less_eq @ ( filter @ A )
@ ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [B6: B] : ( filtercomap @ A @ C @ F3 @ ( F5 @ B6 ) )
@ B4 ) )
@ ( filtercomap @ A @ C @ F3 @ ( complete_Sup_Sup @ ( filter @ C ) @ ( image2 @ B @ ( filter @ C ) @ F5 @ B4 ) ) ) ) ).
% filtercomap_SUP
thf(fact_7708_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 ) ) )
@ ^ [K3: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth @ A @ Xs3 @ ( code_nat_of_natural @ K3 ) ) ) ) ) ) ).
% select_def
thf(fact_7709_iter_H_Ocases,axiom,
! [A: $tType] :
( ( quickcheck_random @ A )
=> ! [X: product_prod @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) )] :
~ ! [T7: itself @ A,Nrandom: code_natural,Sz: code_natural,Seed: product_prod @ code_natural @ code_natural] :
( X
!= ( product_Pair @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ) @ T7 @ ( 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_7710_less__eq__natural_Orep__eq,axiom,
( ( ord_less_eq @ code_natural )
= ( ^ [X4: code_natural,Xa4: code_natural] : ( ord_less_eq @ nat @ ( code_nat_of_natural @ X4 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% less_eq_natural.rep_eq
thf(fact_7711_less__natural_Orep__eq,axiom,
( ( ord_less @ code_natural )
= ( ^ [X4: code_natural,Xa4: code_natural] : ( ord_less @ nat @ ( code_nat_of_natural @ X4 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% less_natural.rep_eq
thf(fact_7712_less__natural__def,axiom,
( ( ord_less @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( code_natural > $o ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ $o @ $o @ code_nat_of_natural @ ( id @ $o ) ) @ ( ord_less @ nat ) ) ) ).
% less_natural_def
thf(fact_7713_less__eq__natural__def,axiom,
( ( ord_less_eq @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( code_natural > $o ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ $o @ $o @ code_nat_of_natural @ ( id @ $o ) ) @ ( ord_less_eq @ nat ) ) ) ).
% less_eq_natural_def
thf(fact_7714_natural__decr,axiom,
! [N2: code_natural] :
( ( N2
!= ( zero_zero @ code_natural ) )
=> ( ord_less @ nat @ ( minus_minus @ nat @ ( code_nat_of_natural @ N2 ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( code_nat_of_natural @ N2 ) ) ) ).
% natural_decr
thf(fact_7715_admissible__chfin,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o] :
( ! [S9: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ S9 )
=> ( finite_finite2 @ A @ S9 ) )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P ) ) ) ).
% admissible_chfin
thf(fact_7716_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N2: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
= ( ( ( ord_less @ nat @ N2 @ M )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) )
| ( ( ord_less_eq @ nat @ M @ N2 )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N2 @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_7717_concat__bit__nonnegative__iff,axiom,
! [N2: nat,K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N2 @ K @ L ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_7718_concat__bit__negative__iff,axiom,
! [N2: nat,K: int,L: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N2 @ K @ L ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_7719_admissible__ball,axiom,
! [A: $tType,B: $tType,A5: set @ A,Lub: ( set @ B ) > B,Ord: B > B > $o,P: B > A > $o] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X4: B] : ( P @ X4 @ Y3 ) ) )
=> ( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X4: B] :
! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( P @ X4 @ Y4 ) ) ) ) ).
% admissible_ball
thf(fact_7720_admissible__const,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,T6: $o] :
( comple1908693960933563346ssible @ A @ Lub @ Ord
@ ^ [X4: A] : T6 ) ).
% admissible_const
thf(fact_7721_admissible__conj,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,P: A > $o,Q: A > $o] :
( ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P )
=> ( ( comple1908693960933563346ssible @ A @ Lub @ Ord @ Q )
=> ( comple1908693960933563346ssible @ A @ Lub @ Ord
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% admissible_conj
thf(fact_7722_admissible__True,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o] :
( comple1908693960933563346ssible @ A @ Lub @ Ord
@ ^ [X4: A] : $true ) ).
% admissible_True
thf(fact_7723_admissible__all,axiom,
! [A: $tType,B: $tType,Lub: ( set @ B ) > B,Ord: B > B > $o,P: B > A > $o] :
( ! [Y3: A] :
( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X4: B] : ( P @ X4 @ Y3 ) )
=> ( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X4: B] :
! [X11: A] : ( P @ X4 @ X11 ) ) ) ).
% admissible_all
thf(fact_7724_ccpo_Oadmissible__def,axiom,
! [A: $tType] :
( ( comple1908693960933563346ssible @ A )
= ( ^ [Lub2: ( set @ A ) > A,Ord2: A > A > $o,P4: A > $o] :
! [A8: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord2 @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( P4 @ X4 ) )
=> ( P4 @ ( Lub2 @ A8 ) ) ) ) ) ) ) ).
% ccpo.admissible_def
thf(fact_7725_ccpo_OadmissibleI,axiom,
! [A: $tType,Ord: A > A > $o,P: A > $o,Lub: ( set @ A ) > A] :
( ! [A13: set @ A] :
( ( comple1602240252501008431_chain @ A @ Ord @ A13 )
=> ( ( A13
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ A13 )
=> ( P @ X7 ) )
=> ( P @ ( Lub @ A13 ) ) ) ) )
=> ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P ) ) ).
% ccpo.admissibleI
thf(fact_7726_ccpo_OadmissibleD,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,P: A > $o,A5: set @ A] :
( ( comple1908693960933563346ssible @ A @ Lub @ Ord @ P )
=> ( ( comple1602240252501008431_chain @ A @ Ord @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( P @ X3 ) )
=> ( P @ ( Lub @ A5 ) ) ) ) ) ) ).
% ccpo.admissibleD
thf(fact_7727_admissible__disj,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o,Q: A > $o] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
=> ( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ Q )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A )
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) ) ) ) ).
% admissible_disj
thf(fact_7728_multp__def,axiom,
! [A: $tType] :
( ( multp @ A )
= ( ^ [R4: A > A > $o,M10: multiset @ A,N8: multiset @ A] : ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M10 @ N8 ) @ ( mult @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% multp_def
thf(fact_7729_pair__lessI2,axiom,
! [A3: nat,B2: nat,S2: nat,T6: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less @ nat @ S2 @ T6 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T6 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_7730_total__pair__less,axiom,
! [A5: set @ ( product_prod @ nat @ nat )] : ( total_on @ ( product_prod @ nat @ nat ) @ A5 @ fun_pair_less ) ).
% total_pair_less
thf(fact_7731_trans__pair__less,axiom,
trans @ ( product_prod @ nat @ nat ) @ fun_pair_less ).
% trans_pair_less
thf(fact_7732_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ X @ Z2 ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y @ Z2 ) ) ).
% pair_less_iff1
thf(fact_7733_pair__less__def,axiom,
( fun_pair_less
= ( lex_prod @ nat @ nat @ less_than @ less_than ) ) ).
% pair_less_def
thf(fact_7734_wf__pair__less,axiom,
wf @ ( product_prod @ nat @ nat ) @ fun_pair_less ).
% wf_pair_less
thf(fact_7735_less__multiset__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ ( multiset @ A ) )
= ( multp @ A @ ( ord_less @ A ) ) ) ) ).
% less_multiset_def
thf(fact_7736_pair__lessI1,axiom,
! [A3: nat,B2: nat,S2: nat,T6: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T6 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_7737_mono__multp,axiom,
! [A: $tType,R3: A > A > $o,R7: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R3 @ R7 )
=> ( ord_less_eq @ ( ( multiset @ A ) > ( multiset @ A ) > $o ) @ ( multp @ A @ R3 ) @ ( multp @ A @ R7 ) ) ) ).
% mono_multp
thf(fact_7738_smin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS2 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_min_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ ( insert @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_7739_pair__leqI2,axiom,
! [A3: nat,B2: nat,S2: nat,T6: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ S2 @ T6 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T6 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_7740_pair__leq__def,axiom,
( fun_pair_leq
= ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ) @ fun_pair_less @ ( id2 @ ( product_prod @ nat @ nat ) ) ) ) ).
% pair_leq_def
thf(fact_7741_smin__emptyI,axiom,
! [X6: set @ ( product_prod @ nat @ nat )] :
( ( X6
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X6 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_7742_min__strict__def,axiom,
( fun_min_strict
= ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% min_strict_def
thf(fact_7743_pair__leqI1,axiom,
! [A3: nat,B2: nat,S2: nat,T6: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T6 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_7744_wmax__insertI,axiom,
! [Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ YS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_max_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X @ XS2 ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_7745_wmin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS2 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_min_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ ( insert @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_7746_wmax__emptyI,axiom,
! [X6: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X6 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ X6 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_7747_wmin__emptyI,axiom,
! [X6: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X6 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_7748_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_7749_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 ) @ ( insert @ ( 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_7750_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 ) @ ( insert @ ( 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_7751_smax__insertI,axiom,
! [Y: product_prod @ nat @ nat,Y7: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,X6: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y7 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X6 @ Y7 ) @ 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 ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X @ X6 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_7752_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_7753_euclidean__size__greater__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
= ( B2
!= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_greater_0_iff
thf(fact_7754_max__strict__def,axiom,
( fun_max_strict
= ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% max_strict_def
thf(fact_7755_euclidean__size__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_mult
thf(fact_7756_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% size_mult_mono
thf(fact_7757_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B2 @ A3 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_7758_euclidean__size__times__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_times_unit
thf(fact_7759_dvd__proper__imp__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ~ ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_7760_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_7761_dvd__imp__size__le,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ).
% dvd_imp_size_le
thf(fact_7762_mod__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% mod_size_less
thf(fact_7763_smax__emptyI,axiom,
! [Y7: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y7 )
=> ( ( Y7
!= ( 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 ) ) ) @ Y7 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_7764_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A3: A] :
( ( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( A3
!= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) )
=> ( ( ( B2
!= ( zero_zero @ A ) )
=> ! [Q6: A,R: A] :
( ( ( euclid7384307370059645450egment @ A @ R )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( R
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= Q6 )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= R )
=> ( A3
!= ( plus_plus @ A @ ( times_times @ A @ Q6 @ B2 ) @ R ) ) ) ) ) ) ) )
=> ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_7765_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q4: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R3 )
= A3 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= R3 ) ) ) ) ) ) ).
% mod_eqI
thf(fact_7766_division__segment__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) ) )
= A3 ) ) ).
% division_segment_euclidean_size
thf(fact_7767_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_7768_division__segment__int__def,axiom,
( ( euclid7384307370059645450egment @ int )
= ( ^ [K3: int] : ( if @ int @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ K3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% division_segment_int_def
thf(fact_7769_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( ( euclid7384307370059645450egment @ A @ A3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_7770_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q4: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R3 )
= A3 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= Q4 ) ) ) ) ) ) ).
% div_eqI
thf(fact_7771_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R3: A,Q4: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R3 ) @ B2 )
= Q4 ) ) ) ) ) ).
% div_bounded
thf(fact_7772_wmsI,axiom,
! [A5: multiset @ ( product_prod @ nat @ nat ),B4: multiset @ ( product_prod @ nat @ nat ),Z5: 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 ) @ A5 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B4 ) ) @ fun_max_strict )
| ( ( A5
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( B4
= ( 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 ) ) @ Z5 @ A5 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z5 @ B4 ) ) @ ms_weak ) ) ).
% wmsI
thf(fact_7773_smsI,axiom,
! [A5: multiset @ ( product_prod @ nat @ nat ),B4: multiset @ ( product_prod @ nat @ nat ),Z5: 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 ) @ A5 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B4 ) ) @ 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 ) ) @ Z5 @ A5 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z5 @ B4 ) ) @ ms_strict ) ) ).
% smsI
thf(fact_7774_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_7775_ms__weak__def,axiom,
( ms_weak
= ( sup_sup @ ( set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ms_strict @ ( id2 @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ).
% ms_weak_def
thf(fact_7776_ms__strictI,axiom,
! [Z5: multiset @ ( product_prod @ nat @ nat ),Z14: multiset @ ( product_prod @ nat @ nat ),A5: multiset @ ( product_prod @ nat @ nat ),B4: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z5 @ Z14 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A5 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B4 ) ) @ 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 ) ) @ Z5 @ A5 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z14 @ B4 ) ) @ ms_strict ) ) ) ).
% ms_strictI
thf(fact_7777_ms__weakI1,axiom,
! [Z5: multiset @ ( product_prod @ nat @ nat ),Z14: multiset @ ( product_prod @ nat @ nat ),A5: multiset @ ( product_prod @ nat @ nat ),B4: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z5 @ Z14 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A5 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B4 ) ) @ 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 ) ) @ Z5 @ A5 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z14 @ B4 ) ) @ ms_weak ) ) ) ).
% ms_weakI1
thf(fact_7778_pw__leq__lstep,axiom,
! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
=> ( pw_leq @ ( add_mset @ ( product_prod @ nat @ nat ) @ X @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ).
% pw_leq_lstep
thf(fact_7779_pw__leq__step,axiom,
! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat,X6: multiset @ ( product_prod @ nat @ nat ),Y7: multiset @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
=> ( ( pw_leq @ X6 @ Y7 )
=> ( pw_leq @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X6 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y7 ) ) ) ) ).
% pw_leq_step
thf(fact_7780_pw__leq_Osimps,axiom,
( pw_leq
= ( ^ [A12: multiset @ ( product_prod @ nat @ nat ),A23: multiset @ ( product_prod @ nat @ nat )] :
( ( ( A12
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( A23
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
| ? [X4: product_prod @ nat @ nat,Y4: product_prod @ nat @ nat,X11: multiset @ ( product_prod @ nat @ nat ),Y10: multiset @ ( product_prod @ nat @ nat )] :
( ( A12
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X4 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X11 ) )
& ( A23
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y4 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y10 ) )
& ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X4 @ Y4 ) @ fun_pair_leq )
& ( pw_leq @ X11 @ Y10 ) ) ) ) ) ).
% pw_leq.simps
thf(fact_7781_pw__leq_Ocases,axiom,
! [A1: multiset @ ( product_prod @ nat @ nat ),A22: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ A1 @ A22 )
=> ( ( ( A1
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
=> ( A22
!= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
=> ~ ! [X3: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat,X14: multiset @ ( product_prod @ nat @ nat )] :
( ( A1
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X3 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X14 ) )
=> ! [Y14: multiset @ ( product_prod @ nat @ nat )] :
( ( A22
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y3 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y14 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y3 ) @ fun_pair_leq )
=> ~ ( pw_leq @ X14 @ Y14 ) ) ) ) ) ) ).
% pw_leq.cases
thf(fact_7782_ms__weakI2,axiom,
! [Z5: multiset @ ( product_prod @ nat @ nat ),Z14: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z5 @ Z14 )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z5 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z14 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) @ ms_weak ) ) ).
% ms_weakI2
thf(fact_7783_pw__leq__split,axiom,
! [X6: multiset @ ( product_prod @ nat @ nat ),Y7: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ X6 @ Y7 )
=> ? [A13: multiset @ ( product_prod @ nat @ nat ),B11: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat )] :
( ( X6
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ A13 @ Z10 ) )
& ( Y7
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ B11 @ 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 ) @ A13 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B11 ) ) @ fun_max_strict )
| ( ( B11
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( A13
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% pw_leq_split
thf(fact_7784_coinduct3__lemma,axiom,
! [A: $tType,X6: set @ A,F3: ( set @ A ) > ( set @ A )] :
( ( ord_less_eq @ ( set @ A ) @ X6
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X4: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ord_less_eq @ ( set @ A )
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X4: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) )
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X4: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ) ) ) ).
% coinduct3_lemma
thf(fact_7785_def__coinduct3,axiom,
! [A: $tType,A5: set @ A,F3: ( set @ A ) > ( set @ A ),A3: A,X6: set @ A] :
( ( A5
= ( complete_lattice_gfp @ ( set @ A ) @ F3 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X4: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) @ A5 ) ) ) )
=> ( member @ A @ A3 @ A5 ) ) ) ) ) ).
% def_coinduct3
thf(fact_7786_def__Collect__coinduct,axiom,
! [A: $tType,A5: set @ A,P: ( set @ A ) > A > $o,A3: A,X6: set @ A] :
( ( A5
= ( complete_lattice_gfp @ ( set @ A )
@ ^ [W3: set @ A] : ( collect @ A @ ( P @ W3 ) ) ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A )
@ ^ [W3: set @ A] : ( collect @ A @ ( P @ W3 ) ) )
=> ( ( member @ A @ A3 @ X6 )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ X6 )
=> ( P @ ( sup_sup @ ( set @ A ) @ X6 @ A5 ) @ Z3 ) )
=> ( member @ A @ A3 @ A5 ) ) ) ) ) ).
% def_Collect_coinduct
thf(fact_7787_gfp__fun__UnI2,axiom,
! [A: $tType,F3: ( set @ A ) > ( set @ A ),A3: A,X6: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) )
=> ( member @ A @ A3 @ ( F3 @ ( sup_sup @ ( set @ A ) @ X6 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_7788_gfp__const,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [T6: A] :
( ( complete_lattice_gfp @ A
@ ^ [X4: A] : T6 )
= T6 ) ) ).
% gfp_const
thf(fact_7789_gfp__rolling,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [G3: A > B,F3: B > A] :
( ( order_mono @ A @ B @ G3 )
=> ( ( order_mono @ B @ A @ F3 )
=> ( ( G3
@ ( complete_lattice_gfp @ A
@ ^ [X4: A] : ( F3 @ ( G3 @ X4 ) ) ) )
= ( complete_lattice_gfp @ B
@ ^ [X4: B] : ( G3 @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% gfp_rolling
thf(fact_7790_gfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_gfp @ A )
= ( ^ [F4: A > A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ U2 @ ( F4 @ U2 ) ) ) ) ) ) ) ).
% gfp_def
thf(fact_7791_weak__coinduct__image,axiom,
! [A: $tType,B: $tType,A3: A,X6: set @ A,G3: A > B,F3: ( set @ B ) > ( set @ B )] :
( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G3 @ X6 ) @ ( F3 @ ( image2 @ A @ B @ G3 @ X6 ) ) )
=> ( member @ B @ ( G3 @ A3 ) @ ( complete_lattice_gfp @ ( set @ B ) @ F3 ) ) ) ) ).
% weak_coinduct_image
thf(fact_7792_gfp__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A > A] :
( ! [X3: A,Y3: A,W: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ W @ Z3 )
=> ( ord_less_eq @ A @ ( F3 @ X3 @ W ) @ ( F3 @ Y3 @ Z3 ) ) ) )
=> ( ( complete_lattice_gfp @ A
@ ^ [X4: A] : ( complete_lattice_gfp @ A @ ( F3 @ X4 ) ) )
= ( complete_lattice_gfp @ A
@ ^ [X4: A] : ( F3 @ X4 @ X4 ) ) ) ) ) ).
% gfp_gfp
thf(fact_7793_weak__coinduct,axiom,
! [A: $tType,A3: A,X6: set @ A,F3: ( set @ A ) > ( set @ A )] :
( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( F3 @ X6 ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ).
% weak_coinduct
thf(fact_7794_gfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,G3: A > A] :
( ! [Z10: A] : ( ord_less_eq @ A @ ( F3 @ Z10 ) @ ( G3 @ Z10 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ ( complete_lattice_gfp @ A @ G3 ) ) ) ) ).
% gfp_mono
thf(fact_7795_gfp__upperbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X6: A,F3: A > A] :
( ( ord_less_eq @ A @ X6 @ ( F3 @ X6 ) )
=> ( ord_less_eq @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ).
% gfp_upperbound
thf(fact_7796_gfp__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,X6: A] :
( ! [U6: A] :
( ( ord_less_eq @ A @ U6 @ ( F3 @ U6 ) )
=> ( ord_less_eq @ A @ U6 @ X6 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ X6 ) ) ) ).
% gfp_least
thf(fact_7797_gfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F5: A > A,X: A] :
( ( order_mono @ A @ A @ F5 )
=> ( ( ( F5 @ X )
= X )
=> ( ! [Z3: A] :
( ( ( F5 @ Z3 )
= Z3 )
=> ( ord_less_eq @ A @ Z3 @ X ) )
=> ( ( complete_lattice_gfp @ A @ F5 )
= X ) ) ) ) ) ).
% gfp_eqI
thf(fact_7798_def__coinduct__set,axiom,
! [A: $tType,A5: set @ A,F3: ( set @ A ) > ( set @ A ),A3: A,X6: set @ A] :
( ( A5
= ( complete_lattice_gfp @ ( set @ A ) @ F3 ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( F3 @ ( sup_sup @ ( set @ A ) @ X6 @ A5 ) ) )
=> ( member @ A @ A3 @ A5 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_7799_coinduct__set,axiom,
! [A: $tType,F3: ( set @ A ) > ( set @ A ),A3: A,X6: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( F3 @ ( sup_sup @ ( set @ A ) @ X6 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ) ).
% coinduct_set
thf(fact_7800_coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,X6: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ X6 @ ( F3 @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) ) )
=> ( ord_less_eq @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ).
% coinduct
thf(fact_7801_def__coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: A,F3: A > A,X6: A] :
( ( A5
= ( complete_lattice_gfp @ A @ F3 ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ X6 @ ( F3 @ ( sup_sup @ A @ X6 @ A5 ) ) )
=> ( ord_less_eq @ A @ X6 @ A5 ) ) ) ) ) ).
% def_coinduct
thf(fact_7802_coinduct__lemma,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X6: A,F3: A > A] :
( ( ord_less_eq @ A @ X6 @ ( F3 @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) @ ( F3 @ ( sup_sup @ A @ X6 @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_7803_gfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F3 )
=> ( ! [S9: A] :
( ( P @ S9 )
=> ( ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F3 ) @ S9 )
=> ( P @ ( F3 @ S9 ) ) ) )
=> ( ! [M13: set @ A] :
( ! [X7: A] :
( ( member @ A @ X7 @ M13 )
=> ( P @ X7 ) )
=> ( P @ ( complete_Inf_Inf @ A @ M13 ) ) )
=> ( P @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_7804_gfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,N2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( complete_lattice_gfp @ A @ ( compow @ ( A > A ) @ ( suc @ N2 ) @ F3 ) )
= ( complete_lattice_gfp @ A @ F3 ) ) ) ) ).
% gfp_funpow
thf(fact_7805_lfp__le__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A] :
( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ ( complete_lattice_gfp @ A @ F3 ) ) ) ) ).
% lfp_le_gfp
thf(fact_7806_gfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,K: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F3 @ ( top_top @ A ) )
= ( compow @ ( A > A ) @ K @ F3 @ ( top_top @ A ) ) )
=> ( ( complete_lattice_gfp @ A @ F3 )
= ( compow @ ( A > A ) @ K @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_7807_coinduct3,axiom,
! [A: $tType,F3: ( set @ A ) > ( set @ A ),A3: A,X6: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F3 )
=> ( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6
@ ( F3
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X4: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F3 @ X4 ) @ X6 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F3 ) ) ) ) ) ).
% coinduct3
thf(fact_7808_AboveS__def,axiom,
! [A: $tType] :
( ( order_AboveS @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B6: A] :
( ( member @ A @ B6 @ ( field2 @ A @ R4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ( B6 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B6 ) @ R4 ) ) ) ) ) ) ) ).
% AboveS_def
thf(fact_7809_wfP__SUP,axiom,
! [B: $tType,A: $tType,R3: A > B > B > $o] :
( ! [I3: A] : ( wfP @ B @ ( R3 @ I3 ) )
=> ( ! [I3: A,J2: A] :
( ( ( R3 @ I3 )
!= ( R3 @ J2 ) )
=> ( ( inf_inf @ ( B > $o ) @ ( domainp @ B @ B @ ( R3 @ I3 ) ) @ ( rangep @ B @ B @ ( R3 @ J2 ) ) )
= ( bot_bot @ ( B > $o ) ) ) )
=> ( wfP @ B @ ( complete_Sup_Sup @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% wfP_SUP
thf(fact_7810_wfP__empty,axiom,
! [A: $tType] :
( wfP @ A
@ ^ [X4: A,Y4: A] : $false ) ).
% wfP_empty
thf(fact_7811_wfP__subset,axiom,
! [A: $tType,R3: A > A > $o,P3: A > A > $o] :
( ( wfP @ A @ R3 )
=> ( ( ord_less_eq @ ( A > A > $o ) @ P3 @ R3 )
=> ( wfP @ A @ P3 ) ) ) ).
% wfP_subset
thf(fact_7812_Rangep_Ocases,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,A3: B] :
( ( rangep @ A @ B @ R3 @ A3 )
=> ~ ! [A6: A] :
~ ( R3 @ A6 @ A3 ) ) ).
% Rangep.cases
thf(fact_7813_Rangep_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rangep @ A @ B )
= ( ^ [R4: A > B > $o,A7: B] :
? [B6: A,C5: B] :
( ( A7 = C5 )
& ( R4 @ B6 @ C5 ) ) ) ) ).
% Rangep.simps
thf(fact_7814_RangePI,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,A3: A,B2: B] :
( ( R3 @ A3 @ B2 )
=> ( rangep @ A @ B @ R3 @ B2 ) ) ).
% RangePI
thf(fact_7815_RangepE,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,B2: B] :
( ( rangep @ A @ B @ R3 @ B2 )
=> ~ ! [A6: A] :
~ ( R3 @ A6 @ B2 ) ) ).
% RangepE
thf(fact_7816_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_7817_wfP__less,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( wfP @ A @ ( ord_less @ A ) ) ) ).
% wfP_less
thf(fact_7818_wfP__wf__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( wfP @ A
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) )
= ( wf @ A @ R3 ) ) ).
% wfP_wf_eq
thf(fact_7819_wfP__def,axiom,
! [A: $tType] :
( ( wfP @ A )
= ( ^ [R4: A > A > $o] : ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ).
% wfP_def
thf(fact_7820_AboveS__disjoint,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( order_AboveS @ A @ R3 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% AboveS_disjoint
thf(fact_7821_AboveS__Field,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( order_AboveS @ A @ R3 @ A5 ) @ ( field2 @ A @ R3 ) ) ).
% AboveS_Field
thf(fact_7822_wo__rel_Osuc__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ B2 @ B4 )
=> ( ( ( bNF_Wellorder_wo_suc @ A @ R3 @ B4 )
!= B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B4 ) ) @ R3 ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_7823_wo__rel_Osuc__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B4 ) @ ( order_AboveS @ A @ R3 @ B4 ) ) ) ) ) ).
% wo_rel.suc_AboveS
thf(fact_7824_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R3 @ B4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B4 ) @ A3 ) @ R3 ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_7825_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R3 @ B4 ) )
=> ( ! [A11: A] :
( ( member @ A @ A11 @ ( order_AboveS @ A @ R3 @ B4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A11 ) @ R3 ) )
=> ( A3
= ( bNF_Wellorder_wo_suc @ A @ R3 @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_7826_wo__rel_Osuc__inField,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R3 ) )
=> ( ( ( order_AboveS @ A @ R3 @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( bNF_Wellorder_wo_suc @ A @ R3 @ B4 ) @ ( field2 @ A @ R3 ) ) ) ) ) ).
% wo_rel.suc_inField
thf(fact_7827_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( ( order_AboveS @ A @ R3 @ A5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R3 @ A5 ) ) @ R3 )
=> ( ( B2
!= ( bNF_Wellorder_wo_suc @ A @ R3 @ A5 ) )
=> ( member @ A @ B2 @ A5 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_7828_right__total__relcompp__transfer,axiom,
! [C: $tType,A: $tType,E: $tType,F: $tType,B: $tType,D: $tType,B4: A > B > $o,A5: C > D > $o,C4: E > F > $o] :
( ( right_total @ A @ B @ B4 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F > $o ) > D > F > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A5
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) )
@ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F > $o ) @ ( C > E > $o ) @ ( D > F > $o )
@ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F > $o ) @ B4
@ ( bNF_rel_fun @ E @ F @ $o @ $o @ C4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) )
@ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F > $o ) @ A5
@ ( bNF_rel_fun @ E @ F @ $o @ $o @ C4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) ) )
@ ^ [R6: C > A > $o,S5: A > E > $o,X4: C,Z7: E] :
? [Y4: A] :
( ( member @ A @ Y4 @ ( collect @ A @ ( domainp @ A @ B @ B4 ) ) )
& ( R6 @ X4 @ Y4 )
& ( S5 @ Y4 @ Z7 ) )
@ ( relcompp @ D @ B @ F ) ) ) ).
% right_total_relcompp_transfer
thf(fact_7829_relcompp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R2: A > C > $o,S: C > B > $o,T2: C > B > $o] :
( ( relcompp @ A @ C @ B @ R2 @ ( sup_sup @ ( C > B > $o ) @ S @ T2 ) )
= ( sup_sup @ ( A > B > $o ) @ ( relcompp @ A @ C @ B @ R2 @ S ) @ ( relcompp @ A @ C @ B @ R2 @ T2 ) ) ) ).
% relcompp_distrib
thf(fact_7830_relcompp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S: A > C > $o,T2: A > C > $o,R2: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( sup_sup @ ( A > C > $o ) @ S @ T2 ) @ R2 )
= ( sup_sup @ ( A > B > $o ) @ ( relcompp @ A @ C @ B @ S @ R2 ) @ ( relcompp @ A @ C @ B @ T2 @ R2 ) ) ) ).
% relcompp_distrib2
thf(fact_7831_relcompp__bot1,axiom,
! [C: $tType,B: $tType,A: $tType,R2: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( bot_bot @ ( A > C > $o ) ) @ R2 )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot1
thf(fact_7832_relcompp__bot2,axiom,
! [C: $tType,B: $tType,A: $tType,R2: A > C > $o] :
( ( relcompp @ A @ C @ B @ R2 @ ( bot_bot @ ( C > B > $o ) ) )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot2
thf(fact_7833_pos__fun__distr,axiom,
! [E: $tType,C: $tType,A: $tType,B: $tType,D: $tType,F: $tType,R2: A > E > $o,S: B > F > $o,R9: E > C > $o,S3: F > D > $o] : ( ord_less_eq @ ( ( A > B ) > ( C > D ) > $o ) @ ( relcompp @ ( A > B ) @ ( E > F ) @ ( C > D ) @ ( bNF_rel_fun @ A @ E @ B @ F @ R2 @ S ) @ ( bNF_rel_fun @ E @ C @ F @ D @ R9 @ S3 ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ ( relcompp @ A @ E @ C @ R2 @ R9 ) @ ( relcompp @ B @ F @ D @ S @ S3 ) ) ) ).
% pos_fun_distr
thf(fact_7834_leq__OOI,axiom,
! [A: $tType,R2: A > A > $o] :
( ( R2
= ( ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
=> ( ord_less_eq @ ( A > A > $o ) @ R2 @ ( relcompp @ A @ A @ A @ R2 @ R2 ) ) ) ).
% leq_OOI
thf(fact_7835_relcompp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R7: A > B > $o,R3: A > B > $o,S4: B > C > $o,S2: B > C > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R7 @ R3 )
=> ( ( ord_less_eq @ ( B > C > $o ) @ S4 @ S2 )
=> ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R7 @ S4 ) @ ( relcompp @ A @ B @ C @ R3 @ S2 ) ) ) ) ).
% relcompp_mono
thf(fact_7836_wo__rel_Oofilter__linord,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( order_ofilter @ A @ R3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
| ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ) ) ).
% wo_rel.ofilter_linord
thf(fact_7837_relcompp__SUP__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S2: A > C > $o,R3: D > C > B > $o,I5: set @ D] :
( ( relcompp @ A @ C @ B @ S2 @ ( complete_Sup_Sup @ ( C > B > $o ) @ ( image2 @ D @ ( C > B > $o ) @ R3 @ I5 ) ) )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I4: D] : ( relcompp @ A @ C @ B @ S2 @ ( R3 @ I4 ) )
@ I5 ) ) ) ).
% relcompp_SUP_distrib
thf(fact_7838_relcompp__SUP__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R3: D > A > C > $o,I5: set @ D,S2: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( complete_Sup_Sup @ ( A > C > $o ) @ ( image2 @ D @ ( A > C > $o ) @ R3 @ I5 ) ) @ S2 )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I4: D] : ( relcompp @ A @ C @ B @ ( R3 @ I4 ) @ S2 )
@ I5 ) ) ) ).
% relcompp_SUP_distrib2
thf(fact_7839_pcr__Domainp__par,axiom,
! [A: $tType,B: $tType,C: $tType,B4: A > B > $o,P24: A > $o,A5: C > A > $o,P1: C > $o,P23: C > $o] :
( ( ( domainp @ A @ B @ B4 )
= P24 )
=> ( ( ( domainp @ C @ A @ A5 )
= P1 )
=> ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A5
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P23
@ P24 )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A5 @ B4 ) )
= ( inf_inf @ ( C > $o ) @ P1 @ P23 ) ) ) ) ) ).
% pcr_Domainp_par
thf(fact_7840_nchotomy__relcomppE,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F3: B > A,R3: C > A > $o,S2: A > D > $o,A3: C,C2: D] :
( ! [Y3: A] :
? [X7: B] :
( Y3
= ( F3 @ X7 ) )
=> ( ( relcompp @ C @ A @ D @ R3 @ S2 @ A3 @ C2 )
=> ~ ! [B5: B] :
( ( R3 @ A3 @ ( F3 @ B5 ) )
=> ~ ( S2 @ ( F3 @ B5 ) @ C2 ) ) ) ) ).
% nchotomy_relcomppE
thf(fact_7841_relcompp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,R3: A > B > $o,S2: B > C > $o,A1: A,A22: C] :
( ( relcompp @ A @ B @ C @ R3 @ S2 @ A1 @ A22 )
=> ~ ! [B5: B] :
( ( R3 @ A1 @ B5 )
=> ~ ( S2 @ B5 @ A22 ) ) ) ).
% relcompp.cases
thf(fact_7842_relcompp_Osimps,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R4: A > B > $o,S6: B > C > $o,A12: A,A23: C] :
? [A7: A,B6: B,C5: C] :
( ( A12 = A7 )
& ( A23 = C5 )
& ( R4 @ A7 @ B6 )
& ( S6 @ B6 @ C5 ) ) ) ) ).
% relcompp.simps
thf(fact_7843_OO__eq,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ( relcompp @ A @ B @ B @ R2
@ ^ [Y5: B,Z4: B] : Y5 = Z4 )
= R2 ) ).
% OO_eq
thf(fact_7844_eq__OO,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ( relcompp @ A @ A @ B
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ R2 )
= R2 ) ).
% eq_OO
thf(fact_7845_relcomppE,axiom,
! [A: $tType,B: $tType,C: $tType,R3: A > B > $o,S2: B > C > $o,A3: A,C2: C] :
( ( relcompp @ A @ B @ C @ R3 @ S2 @ A3 @ C2 )
=> ~ ! [B5: B] :
( ( R3 @ A3 @ B5 )
=> ~ ( S2 @ B5 @ C2 ) ) ) ).
% relcomppE
thf(fact_7846_relcomppI,axiom,
! [A: $tType,B: $tType,C: $tType,R3: A > B > $o,A3: A,B2: B,S2: B > C > $o,C2: C] :
( ( R3 @ A3 @ B2 )
=> ( ( S2 @ B2 @ C2 )
=> ( relcompp @ A @ B @ C @ R3 @ S2 @ A3 @ C2 ) ) ) ).
% relcomppI
thf(fact_7847_relcompp__apply,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R6: A > B > $o,S5: B > C > $o,A7: A,C5: C] :
? [B6: B] :
( ( R6 @ A7 @ B6 )
& ( S5 @ B6 @ C5 ) ) ) ) ).
% relcompp_apply
thf(fact_7848_relcompp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R3: A > D > $o,S2: D > C > $o,T6: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R3 @ S2 ) @ T6 )
= ( relcompp @ A @ D @ B @ R3 @ ( relcompp @ D @ C @ B @ S2 @ T6 ) ) ) ).
% relcompp_assoc
thf(fact_7849_OO__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcompp @ A @ C @ B )
= ( ^ [R6: A > C > $o,S5: C > B > $o,X4: A,Z7: B] :
? [Y4: C] :
( ( R6 @ X4 @ Y4 )
& ( S5 @ Y4 @ Z7 ) ) ) ) ).
% OO_def
thf(fact_7850_pcr__Domainp,axiom,
! [B: $tType,A: $tType,C: $tType,B4: A > B > $o,P: A > $o,A5: C > A > $o] :
( ( ( domainp @ A @ B @ B4 )
= P )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A5 @ B4 ) )
= ( ^ [X4: C] :
? [Y4: A] :
( ( A5 @ X4 @ Y4 )
& ( P @ Y4 ) ) ) ) ) ).
% pcr_Domainp
thf(fact_7851_relcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcomp @ A @ B @ C )
= ( ^ [R4: set @ ( product_prod @ A @ B ),S6: set @ ( product_prod @ B @ C )] :
( collect @ ( product_prod @ A @ C )
@ ( product_case_prod @ A @ C @ $o
@ ( relcompp @ A @ B @ C
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R4 )
@ ^ [X4: B,Y4: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X4 @ Y4 ) @ S6 ) ) ) ) ) ) ).
% relcomp_def
thf(fact_7852_relcompp__relcomp__eq,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( relcompp @ A @ B @ C
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R3 )
@ ^ [X4: B,Y4: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X4 @ Y4 ) @ S2 ) )
= ( ^ [X4: A,Y4: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y4 ) @ ( relcomp @ A @ B @ C @ R3 @ S2 ) ) ) ) ).
% relcompp_relcomp_eq
thf(fact_7853_wo__rel_Oofilter__UNION,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),I5: set @ B,A5: B > ( set @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( order_ofilter @ A @ R3 @ ( A5 @ I3 ) ) )
=> ( order_ofilter @ A @ R3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) ) ) ) ).
% wo_rel.ofilter_UNION
thf(fact_7854_wo__rel_Oofilter__AboveS__Field,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ ( order_AboveS @ A @ R3 @ A5 ) )
= ( field2 @ A @ R3 ) ) ) ) ).
% wo_rel.ofilter_AboveS_Field
thf(fact_7855_ofilterIncl__def,axiom,
! [A: $tType] :
( ( bNF_We413866401316099525erIncl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A8: set @ A,B7: set @ A] :
( ( order_ofilter @ A @ R4 @ A8 )
& ( A8
!= ( field2 @ A @ R4 ) )
& ( order_ofilter @ A @ R4 @ B7 )
& ( B7
!= ( field2 @ A @ R4 ) )
& ( ord_less @ ( set @ A ) @ A8 @ B7 ) ) ) ) ) ) ).
% ofilterIncl_def
thf(fact_7856_neg__fun__distr1,axiom,
! [D: $tType,A: $tType,B: $tType,C: $tType,E: $tType,F: $tType,R2: A > B > $o,R9: B > C > $o,S: D > F > $o,S3: F > E > $o] :
( ( left_unique @ A @ B @ R2 )
=> ( ( right_total @ A @ B @ R2 )
=> ( ( right_unique @ B @ C @ R9 )
=> ( ( left_total @ B @ C @ R9 )
=> ( ord_less_eq @ ( ( A > D ) > ( C > E ) > $o ) @ ( bNF_rel_fun @ A @ C @ D @ E @ ( relcompp @ A @ B @ C @ R2 @ R9 ) @ ( relcompp @ D @ F @ E @ S @ S3 ) ) @ ( relcompp @ ( A > D ) @ ( B > F ) @ ( C > E ) @ ( bNF_rel_fun @ A @ B @ D @ F @ R2 @ S ) @ ( bNF_rel_fun @ B @ C @ F @ E @ R9 @ S3 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_7857_typedef__right__unique,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T2: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T2
= ( ^ [X4: A,Y4: B] :
( X4
= ( Rep @ Y4 ) ) ) )
=> ( right_unique @ A @ B @ T2 ) ) ) ).
% typedef_right_unique
thf(fact_7858_neg__fun__distr2,axiom,
! [F: $tType,E: $tType,A: $tType,B: $tType,D: $tType,C: $tType,R9: A > B > $o,S3: C > D > $o,R2: E > A > $o,S: F > C > $o] :
( ( right_unique @ A @ B @ R9 )
=> ( ( left_total @ A @ B @ R9 )
=> ( ( left_unique @ C @ D @ S3 )
=> ( ( right_total @ C @ D @ S3 )
=> ( ord_less_eq @ ( ( E > F ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ E @ B @ F @ D @ ( relcompp @ E @ A @ B @ R2 @ R9 ) @ ( relcompp @ F @ C @ D @ S @ S3 ) ) @ ( relcompp @ ( E > F ) @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ E @ A @ F @ C @ R2 @ S ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R9 @ S3 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_7859_composed__equiv__rel__eq__onp,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,P: A > $o,P8: B > $o,P10: A > $o] :
( ( left_unique @ A @ B @ R2 )
=> ( ( bNF_rel_fun @ A @ B @ $o @ $o @ R2
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ P
@ P8 )
=> ( ( ( domainp @ A @ B @ R2 )
= P10 )
=> ( ( relcompp @ A @ B @ A @ R2 @ ( relcompp @ B @ B @ A @ ( bNF_eq_onp @ B @ P8 ) @ ( conversep @ A @ B @ R2 ) ) )
= ( bNF_eq_onp @ A @ ( inf_inf @ ( A > $o ) @ P10 @ P ) ) ) ) ) ) ).
% composed_equiv_rel_eq_onp
thf(fact_7860_wo__rel_Oofilter__under__UNION,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( A5
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ A @ ( set @ A ) @ ( order_under @ A @ R3 ) @ A5 ) ) ) ) ) ).
% wo_rel.ofilter_under_UNION
thf(fact_7861_conversep__eq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [Y5: A,Z4: A] : Y5 = Z4 ) ) ).
% conversep_eq
thf(fact_7862_conversep__iff,axiom,
! [B: $tType,A: $tType] :
( ( conversep @ A @ B )
= ( ^ [R4: A > B > $o,A7: B,B6: A] : ( R4 @ B6 @ A7 ) ) ) ).
% conversep_iff
thf(fact_7863_conversep__inject,axiom,
! [A: $tType,B: $tType,R3: B > A > $o,S2: B > A > $o] :
( ( ( conversep @ B @ A @ R3 )
= ( conversep @ B @ A @ S2 ) )
= ( R3 = S2 ) ) ).
% conversep_inject
thf(fact_7864_conversep__conversep,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( conversep @ B @ A @ ( conversep @ A @ B @ R3 ) )
= R3 ) ).
% conversep_conversep
thf(fact_7865_conversep__noteq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [X4: A,Y4: A] : X4 != Y4 )
= ( ^ [X4: A,Y4: A] : X4 != Y4 ) ) ).
% conversep_noteq
thf(fact_7866_conversep__mono,axiom,
! [A: $tType,B: $tType,R3: B > A > $o,S2: B > A > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ ( conversep @ B @ A @ R3 ) @ ( conversep @ B @ A @ S2 ) )
= ( ord_less_eq @ ( B > A > $o ) @ R3 @ S2 ) ) ).
% conversep_mono
thf(fact_7867_left__unique__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( left_unique @ A @ B )
= ( ^ [R6: A > B > $o] :
( ord_less_eq @ ( A > A > $o ) @ ( relcompp @ A @ B @ A @ R6 @ ( conversep @ A @ B @ R6 ) )
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) ) ) ).
% left_unique_alt_def
thf(fact_7868_Quotient__composition__le__eq,axiom,
! [B: $tType,A: $tType,T2: A > B > $o,R2: B > B > $o] :
( ( left_unique @ A @ B @ T2 )
=> ( ( ord_less_eq @ ( B > B > $o ) @ R2
@ ^ [Y5: B,Z4: B] : Y5 = Z4 )
=> ( ord_less_eq @ ( A > A > $o ) @ ( relcompp @ A @ B @ A @ T2 @ ( relcompp @ B @ B @ A @ R2 @ ( conversep @ A @ B @ T2 ) ) )
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) ) ) ).
% Quotient_composition_le_eq
thf(fact_7869_left__total__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( left_total @ A @ B )
= ( ^ [R6: A > B > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( relcompp @ A @ B @ A @ R6 @ ( conversep @ A @ B @ R6 ) ) ) ) ) ).
% left_total_alt_def
thf(fact_7870_Quotient__composition__ge__eq,axiom,
! [B: $tType,A: $tType,T2: A > B > $o,R2: B > B > $o] :
( ( left_total @ A @ B @ T2 )
=> ( ( ord_less_eq @ ( B > B > $o )
@ ^ [Y5: B,Z4: B] : Y5 = Z4
@ R2 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( relcompp @ A @ B @ A @ T2 @ ( relcompp @ B @ B @ A @ R2 @ ( conversep @ A @ B @ T2 ) ) ) ) ) ) ).
% Quotient_composition_ge_eq
thf(fact_7871_right__unique__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( right_unique @ A @ B )
= ( ^ [R6: A > B > $o] :
( ord_less_eq @ ( B > B > $o ) @ ( relcompp @ B @ A @ B @ ( conversep @ A @ B @ R6 ) @ R6 )
@ ^ [Y5: B,Z4: B] : Y5 = Z4 ) ) ) ).
% right_unique_alt_def
thf(fact_7872_right__total__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( right_total @ A @ B )
= ( ^ [R6: A > B > $o] :
( ord_less_eq @ ( B > B > $o )
@ ^ [Y5: B,Z4: B] : Y5 = Z4
@ ( relcompp @ B @ A @ B @ ( conversep @ A @ B @ R6 ) @ R6 ) ) ) ) ).
% right_total_alt_def
thf(fact_7873_converse__relcompp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: B > C > $o,S2: C > A > $o] :
( ( conversep @ B @ A @ ( relcompp @ B @ C @ A @ R3 @ S2 ) )
= ( relcompp @ A @ C @ B @ ( conversep @ C @ A @ S2 ) @ ( conversep @ B @ C @ R3 ) ) ) ).
% converse_relcompp
thf(fact_7874_leq__conversepI,axiom,
! [A: $tType,R2: A > A > $o] :
( ( R2
= ( ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
=> ( ord_less_eq @ ( A > A > $o ) @ R2 @ ( conversep @ A @ A @ R2 ) ) ) ).
% leq_conversepI
thf(fact_7875_conversep__le__swap,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,S2: B > A > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R3 @ ( conversep @ B @ A @ S2 ) )
= ( ord_less_eq @ ( B > A > $o ) @ ( conversep @ A @ B @ R3 ) @ S2 ) ) ).
% conversep_le_swap
thf(fact_7876_under__Field,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] : ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R3 @ A3 ) @ ( field2 @ A @ R3 ) ) ).
% under_Field
thf(fact_7877_converse__meet,axiom,
! [A: $tType,B: $tType,R3: B > A > $o,S2: B > A > $o] :
( ( conversep @ B @ A @ ( inf_inf @ ( B > A > $o ) @ R3 @ S2 ) )
= ( inf_inf @ ( A > B > $o ) @ ( conversep @ B @ A @ R3 ) @ ( conversep @ B @ A @ S2 ) ) ) ).
% converse_meet
thf(fact_7878_conversep_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( conversep @ A @ B )
= ( ^ [R4: A > B > $o,A12: B,A23: A] :
? [A7: A,B6: B] :
( ( A12 = B6 )
& ( A23 = A7 )
& ( R4 @ A7 @ B6 ) ) ) ) ).
% conversep.simps
thf(fact_7879_conversepD,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,B2: B,A3: A] :
( ( conversep @ A @ B @ R3 @ B2 @ A3 )
=> ( R3 @ A3 @ B2 ) ) ).
% conversepD
thf(fact_7880_conversepE,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,A1: B,A22: A] :
( ( conversep @ A @ B @ R3 @ A1 @ A22 )
=> ( R3 @ A22 @ A1 ) ) ).
% conversepE
thf(fact_7881_conversepI,axiom,
! [B: $tType,A: $tType,R3: A > B > $o,A3: A,B2: B] :
( ( R3 @ A3 @ B2 )
=> ( conversep @ A @ B @ R3 @ B2 @ A3 ) ) ).
% conversepI
thf(fact_7882_converse__join,axiom,
! [A: $tType,B: $tType,R3: B > A > $o,S2: B > A > $o] :
( ( conversep @ B @ A @ ( sup_sup @ ( B > A > $o ) @ R3 @ S2 ) )
= ( sup_sup @ ( A > B > $o ) @ ( conversep @ B @ A @ R3 ) @ ( conversep @ B @ A @ S2 ) ) ) ).
% converse_join
thf(fact_7883_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( conversep @ A @ B
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R3 ) )
= ( ^ [X4: B,Y4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y4 ) @ ( converse @ A @ B @ R3 ) ) ) ) ).
% conversep_converse_eq
thf(fact_7884_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ A7 ) @ R4 ) ) ) ) ).
% under_def
thf(fact_7885_converse__def,axiom,
! [B: $tType,A: $tType] :
( ( converse @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ( conversep @ A @ B
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R4 ) ) ) ) ) ) ).
% converse_def
thf(fact_7886_under__incr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R3 @ A3 ) @ ( order_under @ A @ R3 @ B2 ) ) ) ) ).
% under_incr
thf(fact_7887_ofilter__def,axiom,
! [A: $tType] :
( ( order_ofilter @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R4 @ X4 ) @ A8 ) ) ) ) ) ).
% ofilter_def
thf(fact_7888_flip__pred,axiom,
! [A: $tType,B: $tType,A5: set @ ( product_prod @ A @ B ),R2: B > A > $o] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( conversep @ B @ A @ R2 ) ) ) )
=> ( 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,Y4: B] : ( product_Pair @ B @ A @ Y4 @ X4 ) )
@ A5 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R2 ) ) ) ) ).
% flip_pred
thf(fact_7889_wo__rel_Oofilter__def,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R3 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R3 @ X4 ) @ A5 ) ) ) ) ) ).
% wo_rel.ofilter_def
thf(fact_7890_multiset_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( rel_mset @ A @ B )
= ( ^ [R6: A > B > $o] :
( relcompp @ ( multiset @ A ) @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ B )
@ ( conversep @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ A )
@ ( bNF_Grp @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ A )
@ ( collect @ ( multiset @ ( product_prod @ A @ B ) )
@ ^ [X4: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( image_mset @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( multiset @ ( product_prod @ A @ B ) ) @ ( multiset @ B )
@ ( collect @ ( multiset @ ( product_prod @ A @ B ) )
@ ^ [X4: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( image_mset @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% multiset.rel_compp_Grp
thf(fact_7891_fun_Orel__compp__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,R2: A > B > $o] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R2 )
= ( relcompp @ ( D > A ) @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
@ ( conversep @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
@ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X4: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X4 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X4: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X4 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R2 ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_7892_fun_Orel__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ( bNF_Grp @ A @ B @ A5 @ F3 ) )
= ( bNF_Grp @ ( D > A ) @ ( D > B )
@ ( collect @ ( D > A )
@ ^ [X4: D > A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ X4 @ ( top_top @ ( set @ D ) ) ) @ A5 ) )
@ ( comp @ A @ B @ D @ F3 ) ) ) ).
% fun.rel_Grp
thf(fact_7893_list_Orel__Grp,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( list_all2 @ A @ B @ ( bNF_Grp @ A @ B @ A5 @ F3 ) )
= ( bNF_Grp @ ( list @ A ) @ ( list @ B )
@ ( collect @ ( list @ A )
@ ^ [X4: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ X4 ) @ A5 ) )
@ ( map @ A @ B @ F3 ) ) ) ).
% list.rel_Grp
thf(fact_7894_Grp__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Grp @ A @ B )
= ( ^ [A8: set @ A,F4: A > B,A7: A,B6: B] :
( ( B6
= ( F4 @ A7 ) )
& ( member @ A @ A7 @ A8 ) ) ) ) ).
% Grp_def
thf(fact_7895_Grp__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less_eq @ ( A > B > $o ) @ ( bNF_Grp @ A @ B @ A5 @ F3 ) @ ( bNF_Grp @ A @ B @ B4 @ F3 ) ) ) ).
% Grp_mono
thf(fact_7896_multiset_Orel__Grp,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( rel_mset @ A @ B @ ( bNF_Grp @ A @ B @ A5 @ F3 ) )
= ( bNF_Grp @ ( multiset @ A ) @ ( multiset @ B )
@ ( collect @ ( multiset @ A )
@ ^ [X4: multiset @ A] : ( ord_less_eq @ ( set @ A ) @ ( set_mset @ A @ X4 ) @ A5 ) )
@ ( image_mset @ A @ B @ F3 ) ) ) ).
% multiset.rel_Grp
thf(fact_7897_eq__le__Grp__id__iff,axiom,
! [A: $tType,R2: A > $o] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ( bNF_Grp @ A @ A @ ( collect @ A @ R2 ) @ ( id @ A ) ) )
= ( ! [X11: A] : ( R2 @ X11 ) ) ) ).
% eq_le_Grp_id_iff
thf(fact_7898_OO__Grp__alt,axiom,
! [B: $tType,C: $tType,A: $tType,A5: set @ C,F3: C > A,G3: C > B] :
( ( relcompp @ A @ C @ B @ ( conversep @ C @ A @ ( bNF_Grp @ C @ A @ A5 @ F3 ) ) @ ( bNF_Grp @ C @ B @ A5 @ G3 ) )
= ( ^ [X4: A,Y4: B] :
? [Z7: C] :
( ( member @ C @ Z7 @ A5 )
& ( ( F3 @ Z7 )
= X4 )
& ( ( G3 @ Z7 )
= Y4 ) ) ) ) ).
% OO_Grp_alt
thf(fact_7899_list_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R6: 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 @ R6 ) ) ) )
@ ( 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 @ R6 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% list.rel_compp_Grp
thf(fact_7900_bsqr__ofilter,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),D4: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ ( product_prod @ A @ A ) @ ( bNF_Wellorder_bsqr @ A @ R3 ) @ D4 )
=> ( ( ord_less @ ( set @ ( product_prod @ A @ A ) ) @ D4
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R3 )
@ ^ [Uu2: A] : ( field2 @ A @ R3 ) ) )
=> ( ~ ? [A6: A] :
( ( field2 @ A @ R3 )
= ( order_under @ A @ R3 @ A6 ) )
=> ? [A13: set @ A] :
( ( order_ofilter @ A @ R3 @ A13 )
& ( ord_less @ ( set @ A ) @ A13 @ ( field2 @ A @ R3 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ D4
@ ( product_Sigma @ A @ A @ A13
@ ^ [Uu2: A] : A13 ) ) ) ) ) ) ) ).
% bsqr_ofilter
thf(fact_7901_Well__order__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( order_well_order_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ) ).
% Well_order_Restr
thf(fact_7902_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_7903_well__order__on__domain,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_well_order_on @ A @ A5 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ A @ A3 @ A5 )
& ( member @ A @ B2 @ A5 ) ) ) ) ).
% well_order_on_domain
thf(fact_7904_natLeq__on__well__order__on,axiom,
! [N2: nat] :
( order_well_order_on @ nat
@ ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N2 ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y4: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y4 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_7905_well__order__on__Restr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R3 ) )
=> ( order_well_order_on @ A @ A5
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) ) ) ) ).
% well_order_on_Restr
thf(fact_7906_Field__Restr__ofilter,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
= A5 ) ) ) ).
% Field_Restr_ofilter
thf(fact_7907_natLeq__on__Well__order,axiom,
! [N2: nat] :
( order_well_order_on @ nat
@ ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y4: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y4 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y4: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y4 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_7908_Linear__order__Well__order__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R3 ) )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R3 ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_7909_ofilter__Restr__subset,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( order_ofilter @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu2: A] : B4 ) )
@ A5 ) ) ) ) ).
% ofilter_Restr_subset
thf(fact_7910_ofilter__Restr__Int,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( order_ofilter @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu2: A] : B4 ) )
@ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% ofilter_Restr_Int
thf(fact_7911_bsqr__max2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A1: A,A22: A,B1: A,B22: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( product_Pair @ A @ A @ B1 @ B22 ) ) @ ( bNF_Wellorder_bsqr @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A1 @ A22 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B1 @ B22 ) ) @ R3 ) ) ) ).
% bsqr_max2
thf(fact_7912_ofilter__Restr__under,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ( order_under @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
@ A3 )
= ( order_under @ A @ R3 @ A3 ) ) ) ) ) ).
% ofilter_Restr_under
thf(fact_7913_UNION__inj__on__ofilter,axiom,
! [C: $tType,A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),I5: set @ B,A5: B > ( set @ A ),F3: A > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( order_ofilter @ A @ R3 @ ( A5 @ I3 ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( inj_on @ A @ C @ F3 @ ( A5 @ I3 ) ) )
=> ( inj_on @ A @ C @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) ) ) ) ) ).
% UNION_inj_on_ofilter
thf(fact_7914_ofilter__subset__embedS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( order_ofilter @ A @ R3 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu2: A] : B4 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embedS
thf(fact_7915_embedS__Field,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R3 @ R7 @ F3 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( field2 @ A @ R3 ) ) @ ( field2 @ B @ R7 ) ) ) ) ).
% embedS_Field
thf(fact_7916_ofilter__subset__embedS__iso,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( order_ofilter @ A @ R3 @ B4 )
=> ( ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu2: A] : B4 ) )
@ ( id @ A ) ) )
& ( ( A5 = B4 )
= ( bNF_Wellorder_iso @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu2: A] : B4 ) )
@ ( id @ A ) ) ) ) ) ) ) ).
% ofilter_subset_embedS_iso
thf(fact_7917_ofilter__subset__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( order_ofilter @ A @ R3 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu2: A] : B4 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_7918_iso__forward,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( bNF_Wellorder_iso @ A @ B @ R3 @ R7 @ F3 )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) @ R7 ) ) ) ).
% iso_forward
thf(fact_7919_ordLess__irreflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ).
% ordLess_irreflexive
thf(fact_7920_ordLess__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R11: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_transitive
thf(fact_7921_iso__iff2,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_iso @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R14: set @ ( product_prod @ B @ B ),F4: A > B] :
( ( bij_betw @ A @ B @ F4 @ ( field2 @ A @ R4 ) @ ( field2 @ B @ R14 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R4 ) )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ ( field2 @ A @ R4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R4 )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) @ R14 ) ) ) ) ) ) ) ).
% iso_iff2
thf(fact_7922_finite__ordLess__infinite,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ~ ( 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 ) ) @ R3 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ).
% finite_ordLess_infinite
thf(fact_7923_ordLess__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We4044943003108391690rdLess @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),R14: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R4 ) @ R4 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R14 ) @ R14 )
& ? [X11: A > A2] : ( bNF_Wellorder_embedS @ A @ A2 @ R4 @ R14 @ X11 ) ) ) ) ) ).
% ordLess_def
thf(fact_7924_ofilter__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ ( field2 @ A @ R3 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
@ R3 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% ofilter_ordLess
thf(fact_7925_underS__Restr__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ A3 )
@ ^ [Uu2: A] : ( order_underS @ A @ R3 @ A3 ) ) )
@ R3 )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% underS_Restr_ordLess
thf(fact_7926_ofilter__embed,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R3 ) )
& ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
@ R3
@ ( id @ A ) ) ) ) ) ).
% ofilter_embed
thf(fact_7927_ordLess__not__embed,axiom,
! [A: $tType,B: $tType,R3: 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 ) ) @ R3 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ? [X_12: B > A] : ( bNF_Wellorder_embed @ B @ A @ R7 @ R3 @ X_12 ) ) ).
% ordLess_not_embed
thf(fact_7928_underS__Field2,axiom,
! [A: $tType,A3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( field2 @ A @ R3 ) ) ) ).
% underS_Field2
thf(fact_7929_underS__subset__under,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] : ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( order_under @ A @ R3 @ A3 ) ) ).
% underS_subset_under
thf(fact_7930_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B6: A] :
( ( B6 != A7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ A7 ) @ R4 ) ) ) ) ) ).
% underS_def
thf(fact_7931_underS__E,axiom,
! [A: $tType,I2: A,R2: set @ ( product_prod @ A @ A ),J: A] :
( ( member @ A @ I2 @ ( order_underS @ A @ R2 @ J ) )
=> ( ( I2 != J )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R2 ) ) ) ).
% underS_E
thf(fact_7932_underS__I,axiom,
! [A: $tType,I2: A,J: A,R2: set @ ( product_prod @ A @ A )] :
( ( I2 != J )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R2 )
=> ( member @ A @ I2 @ ( order_underS @ A @ R2 @ J ) ) ) ) ).
% underS_I
thf(fact_7933_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [A: $tType,I2: A,R2: set @ ( product_prod @ A @ A ),J: A] :
( ( member @ A @ I2 @ ( order_underS @ A @ R2 @ J ) )
=> ( member @ A @ I2 @ ( field2 @ A @ R2 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_7934_Order__Relation_OunderS__Field,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] : ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( field2 @ A @ R3 ) ) ).
% Order_Relation.underS_Field
thf(fact_7935_underS__empty,axiom,
! [A: $tType,A3: A,R3: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( order_underS @ A @ R3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% underS_empty
thf(fact_7936_embed__Field,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( bNF_Wellorder_embed @ A @ B @ R3 @ R7 @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( field2 @ A @ R3 ) ) @ ( field2 @ B @ R7 ) ) ) ).
% embed_Field
thf(fact_7937_underS__Field3,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( field2 @ A @ R3 ) ) ) ).
% underS_Field3
thf(fact_7938_iso__defs_I2_J,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_iso @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R14: set @ ( product_prod @ A2 @ A2 ),F4: A > A2] :
( ( bNF_Wellorder_embed @ A @ A2 @ R4 @ R14 @ F4 )
& ( bij_betw @ A @ A2 @ F4 @ ( field2 @ A @ R4 ) @ ( field2 @ A2 @ R14 ) ) ) ) ) ).
% iso_defs(2)
thf(fact_7939_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_7940_embedS__defs_I2_J,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_embedS @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R14: set @ ( product_prod @ A2 @ A2 ),F4: A > A2] :
( ( bNF_Wellorder_embed @ A @ A2 @ R4 @ R14 @ F4 )
& ~ ( bij_betw @ A @ A2 @ F4 @ ( field2 @ A @ R4 ) @ ( field2 @ A2 @ R14 ) ) ) ) ) ).
% embedS_defs(2)
thf(fact_7941_ordLess__iff,axiom,
! [B: $tType,A: $tType,R3: 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 ) ) @ R3 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
& ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 )
& ~ ? [X11: B > A] : ( bNF_Wellorder_embed @ B @ A @ R7 @ R3 @ X11 ) ) ) ).
% ordLess_iff
thf(fact_7942_underS__incl__iff,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( order_underS @ A @ R3 @ B2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_7943_underS__incr,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R3 )
=> ( ( antisym @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( order_underS @ A @ R3 @ B2 ) ) ) ) ) ).
% underS_incr
thf(fact_7944_embed__defs_I2_J,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_embed @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R14: set @ ( product_prod @ A2 @ A2 ),F4: A > A2] :
! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R4 ) )
=> ( bij_betw @ A @ A2 @ F4 @ ( order_under @ A @ R4 @ X4 ) @ ( order_under @ A2 @ R14 @ ( F4 @ X4 ) ) ) ) ) ) ).
% embed_defs(2)
thf(fact_7945_embedS__iff,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R3 @ R7 @ F3 )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R3 @ R7 @ F3 )
= ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( field2 @ A @ R3 ) ) @ ( field2 @ B @ R7 ) ) ) ) ) ).
% embedS_iff
thf(fact_7946_embed__implies__iso__Restr,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),F3: B > A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 )
=> ( ( bNF_Wellorder_embed @ B @ A @ R7 @ R3 @ F3 )
=> ( bNF_Wellorder_iso @ B @ A @ R7
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F3 @ ( field2 @ B @ R7 ) )
@ ^ [Uu2: A] : ( image2 @ B @ A @ F3 @ ( field2 @ B @ R7 ) ) ) )
@ F3 ) ) ) ) ).
% embed_implies_iso_Restr
thf(fact_7947_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_7948_Refl__under__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( ( order_under @ A @ R3 @ A3 )
= ( sup_sup @ ( set @ A ) @ ( order_underS @ A @ R3 @ A3 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Refl_under_underS
thf(fact_7949_ofilter__subset__embed,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( order_ofilter @ A @ R3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
= ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu2: A] : B4 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embed
thf(fact_7950_ordLess__iff__ordIso__Restr,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R3 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ X4 )
@ ^ [Uu2: A] : ( order_underS @ A @ R3 @ X4 ) ) ) )
@ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
thf(fact_7951_ordLeq__iff__ordLess__Restr,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ X4 )
@ ^ [Uu2: A] : ( order_underS @ A @ R3 @ X4 ) ) )
@ R7 )
@ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
thf(fact_7952_exists__minim__Well__order,axiom,
! [A: $tType,R2: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R2
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R2 )
=> ( order_well_order_on @ A @ ( field2 @ A @ X3 ) @ X3 ) )
=> ? [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R2 )
& ! [Xa2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ 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 ) ) @ X3 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Well_order
thf(fact_7953_Well__order__iso__copy,axiom,
! [B: $tType,A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),F3: A > B,A17: set @ B] :
( ( order_well_order_on @ A @ A5 @ R3 )
=> ( ( bij_betw @ A @ B @ F3 @ A5 @ A17 )
=> ? [R15: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ B @ A17 @ R15 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R15 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% Well_order_iso_copy
thf(fact_7954_finite__well__order__on__ordIso,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( order_well_order_on @ A @ A5 @ R3 )
=> ( ( order_well_order_on @ A @ A5 @ 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 ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% finite_well_order_on_ordIso
thf(fact_7955_ordLeq__total,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) @ R3 @ 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 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLeq_total
thf(fact_7956_ordIso__reflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_reflexive
thf(fact_7957_ordLeq__reflexive,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_reflexive
thf(fact_7958_ordLeq__Well__order__simp,axiom,
! [A: $tType,B: $tType,R3: 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 ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
& ( order_well_order_on @ B @ ( field2 @ B @ R7 ) @ R7 ) ) ) ).
% ordLeq_Well_order_simp
thf(fact_7959_ordLeq__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R11: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R11 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R11 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_ordIso_trans
thf(fact_7960_ordIso__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R11: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R11 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R11 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordIso_ordLeq_trans
thf(fact_7961_ordLeq__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R11: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R11 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R11 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_transitive
thf(fact_7962_ordIso__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R11: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R11 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R11 ) @ ( bNF_Wellorder_ordIso @ A @ C ) ) ) ) ).
% ordIso_transitive
thf(fact_7963_ordIso__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R3: 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 ) ) @ R3 @ 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 ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordIso_imp_ordLeq
thf(fact_7964_ordIso__iff__ordLeq,axiom,
! [A: $tType,B: $tType,R3: 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 ) ) @ R3 @ 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 ) ) @ R3 @ 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 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% ordIso_iff_ordLeq
thf(fact_7965_ordIso__symmetric,axiom,
! [A: $tType,B: $tType,R3: 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 ) ) @ R3 @ 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 @ R3 ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% ordIso_symmetric
thf(fact_7966_internalize__ordLeq,axiom,
! [A: $tType,B: $tType,R7: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R7 @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [P6: set @ ( product_prod @ B @ B )] :
( ( ord_less_eq @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R3 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R7 @ P6 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P6 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_ordLeq
thf(fact_7967_not__ordLess__ordIso,axiom,
! [B: $tType,A: $tType,R3: 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 ) ) @ R3 @ 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 ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% not_ordLess_ordIso
thf(fact_7968_not__ordLess__ordLeq,axiom,
! [A: $tType,B: $tType,R3: 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 ) ) @ R3 @ 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 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% not_ordLess_ordLeq
thf(fact_7969_ordLess__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R3: 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 ) ) @ R3 @ 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 ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordLess_imp_ordLeq
thf(fact_7970_ordIso__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R11: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordIso_ordLess_trans
thf(fact_7971_ordLeq__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R11: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLeq_ordLess_trans
thf(fact_7972_ordLess__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R11: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R11 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordIso_trans
thf(fact_7973_ordLess__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),R11: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R11 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordLeq_trans
thf(fact_7974_ordLeq__iff__ordLess__or__ordIso,axiom,
! [B: $tType,A: $tType,R3: 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 ) ) @ R3 @ 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 ) ) @ R3 @ 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 ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% ordLeq_iff_ordLess_or_ordIso
thf(fact_7975_internalize__ordLess,axiom,
! [A: $tType,B: $tType,R7: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R7 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ? [P6: set @ ( product_prod @ B @ B )] :
( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R3 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R7 @ P6 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P6 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ B ) ) ) ) ) ).
% internalize_ordLess
thf(fact_7976_ordLess__or__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) @ R3 @ 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 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLess_or_ordLeq
thf(fact_7977_not__ordLeq__iff__ordLess,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ).
% not_ordLeq_iff_ordLess
thf(fact_7978_not__ordLess__iff__ordLeq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% not_ordLess_iff_ordLeq
thf(fact_7979_ordLeq__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_ordLeq @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),R14: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R4 ) @ R4 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R14 ) @ R14 )
& ? [X11: A > A2] : ( bNF_Wellorder_embed @ A @ A2 @ R4 @ R14 @ X11 ) ) ) ) ) ).
% ordLeq_def
thf(fact_7980_ordIso__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_ordIso @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),R14: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R4 ) @ R4 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R14 ) @ R14 )
& ? [X11: A > A2] : ( bNF_Wellorder_iso @ A @ A2 @ R4 @ R14 @ X11 ) ) ) ) ) ).
% ordIso_def
thf(fact_7981_ofilter__subset__ordLeq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( order_ofilter @ A @ R3 @ A5 )
=> ( ( order_ofilter @ A @ R3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu2: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLeq
thf(fact_7982_comp__set__bd__Union__o__collect,axiom,
! [C: $tType,B: $tType,A: $tType,X: C,X6: set @ ( C > ( set @ ( set @ A ) ) ),Hbd: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
@ ( bNF_Ca6860139660246222851ard_of @ A
@ ( complete_Sup_Sup @ ( set @ A )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ ( C > ( set @ ( set @ A ) ) ) @ ( set @ ( set @ A ) )
@ ^ [F4: C > ( set @ ( set @ A ) )] : ( F4 @ X )
@ X6 ) ) ) )
@ Hbd )
@ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( bNF_collect @ C @ ( set @ A ) @ X6 ) @ X ) ) @ Hbd ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% comp_set_bd_Union_o_collect
thf(fact_7983_dir__image__ordIso,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F3: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( inj_on @ A @ B @ F3 @ ( field2 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_We2720479622203943262_image @ A @ B @ R3 @ F3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image_ordIso
thf(fact_7984_card__of__least,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A5 @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_least
thf(fact_7985_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( 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 @ A5
@ ^ [Uu2: A] : B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( 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 @ B4
@ ^ [Uu2: B] : A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% card_of_Times_infinite
thf(fact_7986_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( 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 @ A5
@ ^ [Uu2: A] : B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(1)
thf(fact_7987_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( 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 @ A5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(2)
thf(fact_7988_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( 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 @ B4
@ ^ [Uu2: B] : A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(3)
thf(fact_7989_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( 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 @ A5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu2: B] : A5 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(4)
thf(fact_7990_card__of__Times1,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( A5
!= ( 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 @ B4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu2: B] : A5 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_7991_card__of__Times2,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( A5
!= ( 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 @ B4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_7992_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 @ ( insert @ A @ A1 @ ( insert @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ) ).
% card_of_bool
thf(fact_7993_card__of__empty,axiom,
! [B: $tType,A: $tType,A5: 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 @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_empty
thf(fact_7994_card__of__empty2,axiom,
! [B: $tType,A: $tType,A5: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty2
thf(fact_7995_card__of__empty3,axiom,
! [B: $tType,A: $tType,A5: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty3
thf(fact_7996_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_7997_card__of__ordIso,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( ? [F4: A > B] : ( bij_betw @ A @ B @ F4 @ A5 @ B4 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIso
thf(fact_7998_card__of__ordIsoI,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A5 @ B4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIsoI
thf(fact_7999_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A5: set @ A,R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ? [F2: B > A,B11: set @ B] : ( bij_betw @ B @ A @ F2 @ B11 @ A5 ) ) ).
% ex_bij_betw
thf(fact_8000_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,B4: set @ A,I5: set @ B,A5: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B4 )
=> ( ( 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 @ I5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( 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 @ ( A5 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( 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 @ I5 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
thf(fact_8001_card__of__Times__same__infinite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( 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 @ A5
@ ^ [Uu2: A] : A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% card_of_Times_same_infinite
thf(fact_8002_card__of__Times3,axiom,
! [A: $tType,A5: 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 @ A5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu2: A] : A5 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ A ) ) ) ).
% card_of_Times3
thf(fact_8003_card__of__Sigma__mono1,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B4: A > ( set @ C )] :
( ! [X3: A] :
( ( member @ A @ X3 @ I5 )
=> ( 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 @ ( A5 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( B4 @ X3 ) ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C ) @ ( product_Sigma @ A @ C @ I5 @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Sigma_mono1
thf(fact_8004_card__of__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A5: set @ A,B4: set @ B,C4: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( 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 @ A5
@ ^ [Uu2: A] : C4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ B4
@ ^ [Uu2: B] : C4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% card_of_Times_mono1
thf(fact_8005_card__of__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A5: set @ A,B4: set @ B,C4: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( 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 @ C4
@ ^ [Uu2: C] : A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ C4
@ ^ [Uu2: C] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% card_of_Times_mono2
thf(fact_8006_card__of__Times__commute,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: 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 @ A5
@ ^ [Uu2: A] : B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu2: B] : A5 ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% card_of_Times_commute
thf(fact_8007_card__of__refl,axiom,
! [A: $tType,A5: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ).
% card_of_refl
thf(fact_8008_card__of__ordLeqI,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B4: set @ B] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A5 )
=> ( member @ B @ ( F3 @ A6 ) @ B4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeqI
thf(fact_8009_infinite__iff__card__of__nat,axiom,
! [A: $tType,A5: set @ A] :
( ( ~ ( finite_finite2 @ A @ A5 ) )
= ( 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 @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_card_of_nat
thf(fact_8010_card__of__ordIso__finite,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ A5 )
= ( finite_finite2 @ B @ B4 ) ) ) ).
% card_of_ordIso_finite
thf(fact_8011_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( finite_finite2 @ B @ B4 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% card_of_ordLeq_finite
thf(fact_8012_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ~ ( finite_finite2 @ B @ B4 ) ) ) ).
% card_of_ordLeq_infinite
thf(fact_8013_card__of__cong,axiom,
! [B: $tType,A: $tType,R3: 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 ) ) @ R3 @ 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 @ R3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R7 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_cong
thf(fact_8014_card__of__mono2,axiom,
! [B: $tType,A: $tType,R3: 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 ) ) @ R3 @ 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 @ R3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R7 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_mono2
thf(fact_8015_card__of__UNION__Sigma,axiom,
! [B: $tType,A: $tType,A5: B > ( set @ A ),I5: 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 ) @ A5 @ I5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A ) @ ( product_Sigma @ B @ A @ I5 @ A5 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ).
% card_of_UNION_Sigma
thf(fact_8016_card__of__image,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: 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 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_image
thf(fact_8017_type__copy__set__bd,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,S: A > ( set @ B ),Bd: set @ ( product_prod @ C @ C ),Rep: D > A,X: D] :
( ! [Y3: A] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( S @ Y3 ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( comp @ A @ ( set @ B ) @ D @ S @ Rep @ X ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) ) ).
% type_copy_set_bd
thf(fact_8018_card__of__Pow__Func,axiom,
! [A: $tType,A5: 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 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > $o ) @ ( bNF_Wellorder_Func @ A @ $o @ A5 @ ( top_top @ ( set @ $o ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( A > $o ) ) ) ).
% card_of_Pow_Func
thf(fact_8019_Func__Times__Range,axiom,
! [C: $tType,B: $tType,A: $tType,A5: set @ A,B4: set @ B,C4: 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 ) @ A5
@ ( product_Sigma @ B @ C @ B4
@ ^ [Uu2: B] : C4 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ ( A > B ) @ ( A > C ) )
@ ( product_Sigma @ ( A > B ) @ ( A > C ) @ ( bNF_Wellorder_Func @ A @ B @ A5 @ B4 )
@ ^ [Uu2: A > B] : ( bNF_Wellorder_Func @ A @ C @ A5 @ C4 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% Func_Times_Range
thf(fact_8020_card__of__Func__Times,axiom,
! [C: $tType,B: $tType,A: $tType,A5: set @ A,B4: set @ B,C4: 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 @ A5
@ ^ [Uu2: A] : B4 )
@ C4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B > C ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A5 @ ( bNF_Wellorder_Func @ B @ C @ B4 @ C4 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) ) ) ).
% card_of_Func_Times
thf(fact_8021_card__of__mono1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_mono1
thf(fact_8022_card__of__Pow,axiom,
! [A: $tType,A5: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ A5 ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ).
% card_of_Pow
thf(fact_8023_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
! [B: $tType,A: $tType,R3: 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 ) ) @ R3 @ 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 @ R3 ) ) @ R7 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ).
% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_8024_dir__image__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We2720479622203943262_image @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),F4: A > A2] :
( collect @ ( product_prod @ A2 @ A2 )
@ ^ [Uu2: product_prod @ A2 @ A2] :
? [A7: A,B6: A] :
( ( Uu2
= ( product_Pair @ A2 @ A2 @ ( F4 @ A7 ) @ ( F4 @ B6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B6 ) @ R4 ) ) ) ) ) ).
% dir_image_def
thf(fact_8025_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ B4 )
= A5 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_8026_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B4: set @ A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( member @ ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% surj_imp_ordLeq
thf(fact_8027_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A5: set @ A,B2: B] :
( ( A5
!= ( 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 @ ( insert @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_singl_ordLeq
thf(fact_8028_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: set @ B] :
( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F4: B > A] :
( ( image2 @ B @ A @ F4 @ A5 )
= B4 ) )
= ( 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_8029_internalize__card__of__ordLeq2,axiom,
! [A: $tType,B: $tType,A5: set @ A,C4: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ C4 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ 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 @ C4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq2
thf(fact_8030_card__of__Field__ordLess,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_Field_ordLess
thf(fact_8031_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,B4: 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 ) ) @ B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F4: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) @ B4 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% card_of_Func_UNIV
thf(fact_8032_ordLeq__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),A5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ A5
@ ^ [Uu2: C] : ( field2 @ A @ R3 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ A5
@ ^ [Uu2: C] : ( field2 @ B @ R7 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% ordLeq_Times_mono2
thf(fact_8033_ordLeq__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),C4: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R3 )
@ ^ [Uu2: A] : C4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ ( field2 @ B @ R7 )
@ ^ [Uu2: B] : C4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% ordLeq_Times_mono1
thf(fact_8034_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A5 ) @ B4 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_ordLeq
thf(fact_8035_internalize__card__of__ordLeq,axiom,
! [A: $tType,B: $tType,A5: set @ A,R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ ( field2 @ B @ R3 ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq
thf(fact_8036_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( ~ ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A5 ) @ B4 ) ) )
= ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% card_of_ordLess
thf(fact_8037_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B4: set @ A,I5: set @ B,A5: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B4 )
=> ( ( 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 @ I5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( 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 @ ( A5 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( 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 ) @ A5 @ I5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_8038_regularCard__def,axiom,
! [A: $tType] :
( ( bNF_Ca7133664381575040944arCard @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [K7: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ K7 @ ( field2 @ A @ R4 ) )
& ( bNF_Ca7293521722713021262ofinal @ A @ K7 @ R4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ K7 ) @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% regularCard_def
thf(fact_8039_card__of__ordIso__subst,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_ordIso_subst
thf(fact_8040_cone__def,axiom,
( bNF_Cardinal_cone
= ( bNF_Ca6860139660246222851ard_of @ product_unit @ ( insert @ product_unit @ product_Unity @ ( bot_bot @ ( set @ product_unit ) ) ) ) ) ).
% cone_def
thf(fact_8041_SIGMA__CSUM,axiom,
! [B: $tType,A: $tType,I5: set @ A,As10: A > ( set @ B )] :
( ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ I5 @ As10 ) )
= ( bNF_Cardinal_Csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ I5 )
@ ^ [I4: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( As10 @ I4 ) ) ) ) ).
% SIGMA_CSUM
thf(fact_8042_Csum__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Cardinal_Csum @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),Rs: A > ( set @ ( product_prod @ B @ B ) )] :
( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R4 )
@ ^ [I4: A] : ( field2 @ B @ ( Rs @ I4 ) ) ) ) ) ) ).
% Csum_def
thf(fact_8043_card__of__Times__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ B,B4: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu2: B] : B4 ) )
@ R3 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Times_ordLeq_infinite_Field
thf(fact_8044_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),I5: set @ B,A5: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( 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 @ ( A5 @ X3 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I5 @ A5 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
thf(fact_8045_card__of__unique,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A5 @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_unique
thf(fact_8046_Card__order__wo__rel,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( bNF_Wellorder_wo_rel @ A @ R3 ) ) ).
% Card_order_wo_rel
thf(fact_8047_card__order__on__def,axiom,
! [A: $tType] :
( ( bNF_Ca8970107618336181345der_on @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A8 @ R4 )
& ! [R14: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A8 @ R14 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R14 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% card_order_on_def
thf(fact_8048_card__order__on__ordIso,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A5 @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ A5 @ 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 ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ).
% card_order_on_ordIso
thf(fact_8049_Card__order__ordIso,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R3 ) @ ( bNF_Wellorder_ordIso @ B @ A ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R7 ) @ R7 ) ) ) ).
% Card_order_ordIso
thf(fact_8050_Card__order__ordIso2,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R7 ) @ R7 ) ) ) ).
% Card_order_ordIso2
thf(fact_8051_ordLeq__refl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_refl
thf(fact_8052_ordIso__refl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_refl
thf(fact_8053_Card__order__trans,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( X != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R3 )
=> ( ( Y != Z2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R3 )
=> ( ( X != Z2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R3 ) ) ) ) ) ) ) ).
% Card_order_trans
thf(fact_8054_infinite__Card__order__limit,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( A3 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X3 ) @ R3 ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_8055_Card__order__iff__ordLeq__card__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% Card_order_iff_ordLeq_card_of
thf(fact_8056_ordIso__card__of__imp__Card__order,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) ) ).
% ordIso_card_of_imp_Card_order
thf(fact_8057_Card__order__iff__ordIso__card__of,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% Card_order_iff_ordIso_card_of
thf(fact_8058_card__of__Field__ordIso,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_Field_ordIso
thf(fact_8059_dir__image,axiom,
! [B: $tType,A: $tType,F3: A > B,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A] :
( ( ( F3 @ X3 )
= ( F3 @ Y3 ) )
= ( X3 = Y3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_We2720479622203943262_image @ A @ B @ R3 @ F3 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image
thf(fact_8060_exists__minim__Card__order,axiom,
! [A: $tType,R2: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R2
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R2 )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X3 ) @ X3 ) )
=> ? [X3: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X3 @ R2 )
& ! [Xa2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ 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 ) ) @ X3 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Card_order
thf(fact_8061_Card__order__empty,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% Card_order_empty
thf(fact_8062_card__of__ordIso__finite__Field,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
= ( finite_finite2 @ B @ A5 ) ) ) ) ).
% card_of_ordIso_finite_Field
thf(fact_8063_card__of__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( order_underS @ A @ R3 @ A3 ) ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% card_of_underS
thf(fact_8064_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),B2: B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% Card_order_singl_ordLeq
thf(fact_8065_card__of__Un__ordLeq__infinite__Field,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ B,B4: set @ B] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ) ).
% card_of_Un_ordLeq_infinite_Field
thf(fact_8066_card__of__empty1,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
| ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_empty1
thf(fact_8067_Card__order__Pow,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ ( field2 @ A @ R3 ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% Card_order_Pow
thf(fact_8068_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ R3
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R3 )
@ ^ [Uu2: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% Card_order_Times1
thf(fact_8069_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ R3
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu2: B] : ( field2 @ A @ R3 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% Card_order_Times2
thf(fact_8070_Card__order__Times__same__infinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R3 )
@ ^ [Uu2: A] : ( field2 @ A @ R3 ) ) )
@ R3 )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ A ) @ A ) ) ) ) ).
% Card_order_Times_same_infinite
thf(fact_8071_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),I5: set @ B,A5: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( 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 @ ( A5 @ X3 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
thf(fact_8072_Card__order__iff__Restr__underS,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R3 @ X4 )
@ ^ [Uu2: A] : ( order_underS @ A @ R3 @ X4 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R3 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% Card_order_iff_Restr_underS
thf(fact_8073_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),As10: A > ( set @ B ),B4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( bNF_Ca7133664381575040944arCard @ A @ R3 )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R3 @ As10 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As10 @ ( field2 @ A @ R3 ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( ord_less_eq @ ( set @ B ) @ B4 @ ( As10 @ X3 ) ) ) ) ) ) ) ) ).
% regularCard_UNION
thf(fact_8074_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P3: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ B @ P3 )
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P3 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R3 )
@ ^ [Uu2: A] : ( field2 @ B @ P3 ) ) )
@ R3 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ ( field2 @ B @ P3 )
@ ^ [Uu2: B] : ( field2 @ A @ R3 ) ) )
@ R3 )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% Card_order_Times_infinite
thf(fact_8075_ex__toCard__pred,axiom,
! [B: $tType,A: $tType,A5: set @ A,R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ? [X_1: A > B] : ( bNF_Gr1419584066657907630d_pred @ A @ B @ A5 @ R3 @ X_1 ) ) ) ).
% ex_toCard_pred
thf(fact_8076_toCard__pred__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr1419584066657907630d_pred @ A @ B )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ B @ B ),F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A8 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A8 ) @ ( field2 @ B @ R4 ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ) ).
% toCard_pred_def
thf(fact_8077_toCard__pred__toCard,axiom,
! [A: $tType,B: $tType,A5: set @ A,R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( bNF_Gr1419584066657907630d_pred @ A @ B @ A5 @ R3 @ ( bNF_Greatest_toCard @ A @ B @ A5 @ R3 ) ) ) ) ).
% toCard_pred_toCard
thf(fact_8078_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),As10: ( set @ A ) > ( set @ B ),B4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R3 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) @ As10 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As10 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B4 @ ( As10 @ X3 ) ) ) ) ) ) ) ) ).
% cardSuc_UNION
thf(fact_8079_cardSuc__ordLeq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R3 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( set @ A ) ) ) ) ).
% cardSuc_ordLeq
thf(fact_8080_cardSuc__greater,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R3 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% cardSuc_greater
thf(fact_8081_cardSuc__ordLess__ordLeq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) @ R3 @ 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 @ R3 ) @ R7 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_ordLess_ordLeq
thf(fact_8082_cardSuc__least,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) @ R3 @ 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 @ R3 ) @ R7 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_least
thf(fact_8083_cardSuc__mono__ordLeq,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 @ R3 ) @ ( 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 ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% cardSuc_mono_ordLeq
thf(fact_8084_cardSuc__invar__ordIso,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 @ R3 ) @ ( 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 ) ) @ R3 @ R7 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% cardSuc_invar_ordIso
thf(fact_8085_cardSuc__least__aux,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 ) ) ) @ R3 @ 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 @ R3 ) @ R7 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ).
% cardSuc_least_aux
thf(fact_8086_cardSuc__ordLeq__ordLess,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( 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 @ R3 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ ( set @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R7 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% cardSuc_ordLeq_ordLess
thf(fact_8087_toCard__inj,axiom,
! [B: $tType,A: $tType,A5: set @ A,R3: set @ ( product_prod @ B @ B ),X: A,Y: A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y @ A5 )
=> ( ( ( bNF_Greatest_toCard @ A @ B @ A5 @ R3 @ X )
= ( bNF_Greatest_toCard @ A @ B @ A5 @ R3 @ Y ) )
= ( X = Y ) ) ) ) ) ) ).
% toCard_inj
thf(fact_8088_fromCard__toCard,axiom,
! [B: $tType,A: $tType,A5: set @ A,R3: set @ ( product_prod @ B @ B ),B2: A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ A @ B2 @ A5 )
=> ( ( bNF_Gr5436034075474128252omCard @ A @ B @ A5 @ R3 @ ( bNF_Greatest_toCard @ A @ B @ A5 @ R3 @ B2 ) )
= B2 ) ) ) ) ).
% fromCard_toCard
thf(fact_8089_isCardSuc__def,axiom,
! [A: $tType] :
( ( bNF_Ca6246979054910435723ardSuc @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R14: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R14 ) @ R14 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R4 @ R14 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
& ! [R16: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R16 ) @ R16 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R4 @ R16 ) @ ( 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 ) ) ) @ R14 @ R16 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) ).
% isCardSuc_def
thf(fact_8090_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),As10: ( set @ A ) > ( set @ B ),B4: set @ B] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) @ As10 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As10 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R3 ) ) )
& ( ord_less_eq @ ( set @ B ) @ B4 @ ( As10 @ X3 ) ) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
thf(fact_8091_card__of__Csum__Times_H,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),I5: set @ B,A5: B > ( set @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ I5 )
=> ( 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 @ ( A5 @ X3 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
@ ( bNF_Cardinal_Csum @ B @ C @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 )
@ ^ [I4: B] : ( bNF_Ca6860139660246222851ard_of @ C @ ( A5 @ I4 ) ) )
@ ( bNF_Cardinal_cprod @ B @ A @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ R3 ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ ( product_prod @ B @ A ) ) ) ) ) ).
% card_of_Csum_Times'
thf(fact_8092_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R3: set @ ( product_prod @ A @ A ),X2: A] :
( ( member @ A @ X1 @ ( field2 @ A @ R3 ) )
=> ( ( member @ A @ X2 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( X1 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X3 ) @ R3 )
& ( X2 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X3 ) @ R3 ) ) ) ) ) ).
% Cinfinite_limit2
thf(fact_8093_Cinfinite__limit,axiom,
! [A: $tType,X: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ( X != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X3 ) @ R3 ) ) ) ) ).
% Cinfinite_limit
thf(fact_8094_cprod__infinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ A @ R3 @ R3 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% cprod_infinite
thf(fact_8095_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_8096_cprod__com,axiom,
! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P25: 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 @ P25 ) @ ( bNF_Cardinal_cprod @ B @ A @ P25 @ P13 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% cprod_com
thf(fact_8097_cprod__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P3: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),Q4: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P3 @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q4 @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P3 ) @ P3 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P3 @ Q4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ B ) ) ) ) ) ) ) ).
% cprod_cinfinite_bound
thf(fact_8098_cprod__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),P3: set @ ( product_prod @ B @ B ),P11: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca4139267488887388095finite @ A @ R3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P3 @ P11 ) @ ( bNF_Cardinal_cprod @ A @ A @ R3 @ R3 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ ( product_prod @ A @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P3 @ P11 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% cprod_dup
thf(fact_8099_cprod__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Cardinal_cprod @ A @ B )
= ( ^ [R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B )] :
( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R13 )
@ ^ [Uu2: A] : ( field2 @ B @ R24 ) ) ) ) ) ).
% cprod_def
thf(fact_8100_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_8101_Cinfinite__limit__finite,axiom,
! [A: $tType,X6: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( field2 @ A @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ X6 )
=> ( ( Xa2 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
thf(fact_8102_cprod__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P25: 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 ) ) @ P25 @ 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 @ P25 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_mono2
thf(fact_8103_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_8104_cprod__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P25: 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 ) ) @ P25 @ 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 @ P25 ) @ ( bNF_Cardinal_cprod @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_mono
thf(fact_8105_cprod__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P25: 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 ) ) @ P25 @ 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 @ P25 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_cong2
thf(fact_8106_cprod__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P25: 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 @ P25 ) @ ( bNF_Cardinal_cprod @ B @ C @ R12 @ P25 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_cong1
thf(fact_8107_cprod__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P25: 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 ) ) @ P25 @ 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 @ P25 ) @ ( bNF_Cardinal_cprod @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_cong
thf(fact_8108_Un__Cinfinite__bound,axiom,
! [B: $tType,A: $tType,A5: set @ A,R3: set @ ( product_prod @ B @ B ),B4: 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 @ A5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% Un_Cinfinite_bound
thf(fact_8109_UNION__Cinfinite__bound,axiom,
! [A: $tType,B: $tType,C: $tType,I5: set @ A,R3: set @ ( product_prod @ B @ B ),A5: 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 @ I5 ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ I5 )
=> ( 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 @ ( A5 @ X3 ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A5 @ I5 ) ) ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) ) ) ) ).
% UNION_Cinfinite_bound
thf(fact_8110_card__of__Csum__Times,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B4: set @ C] :
( ! [X3: A] :
( ( member @ A @ X3 @ I5 )
=> ( 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 @ ( A5 @ X3 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B4 ) ) @ ( 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 @ I5 )
@ ^ [I4: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( A5 @ I4 ) ) )
@ ( bNF_Cardinal_cprod @ A @ C @ ( bNF_Ca6860139660246222851ard_of @ A @ I5 ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Csum_Times
thf(fact_8111_comp__single__set__bd,axiom,
! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set @ ( product_prod @ A @ A ),Fset: B > ( set @ C ),Gset: D > ( set @ B ),Gbd: set @ ( product_prod @ E @ E ),X: D] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ Fbd ) @ Fbd )
=> ( ! [X3: B] : ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( Fset @ X3 ) ) @ Fbd ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ! [X3: D] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( Gset @ X3 ) ) @ Gbd ) @ ( bNF_Wellorder_ordLeq @ B @ E ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ Fset @ ( Gset @ X ) ) ) ) @ ( bNF_Cardinal_cprod @ E @ A @ Gbd @ Fbd ) ) @ ( bNF_Wellorder_ordLeq @ C @ ( product_prod @ E @ A ) ) ) ) ) ) ).
% comp_single_set_bd
thf(fact_8112_Cfinite__cprod__Cinfinite,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S2 ) @ S2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R3 @ S2 ) @ S2 ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ B ) ) ) ) ).
% Cfinite_cprod_Cinfinite
thf(fact_8113_cprod__infinite1_H,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),P3: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P3 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P3 ) @ P3 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P3 @ R3 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R3 @ P3 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% cprod_infinite1'
thf(fact_8114_czero__def,axiom,
! [A: $tType] :
( ( bNF_Cardinal_czero @ A )
= ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% czero_def
thf(fact_8115_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_8116_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_8117_cinfinite__not__czero,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca4139267488887388095finite @ B @ R3 )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% cinfinite_not_czero
thf(fact_8118_czeroE,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( field2 @ A @ R3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% czeroE
thf(fact_8119_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A5: 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 @ A5 ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_ordIso_czero_iff_empty
thf(fact_8120_czeroI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( ( ( field2 @ A @ R3 )
= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% czeroI
thf(fact_8121_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( field2 @ A @ R3 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% Cnotzero_imp_not_empty
thf(fact_8122_Cnotzero__mono,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),Q4: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q4 ) @ Q4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ Q4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q4 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q4 ) @ Q4 ) ) ) ) ) ).
% Cnotzero_mono
thf(fact_8123_Cinfinite__Cnotzero,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) ) ) ).
% Cinfinite_Cnotzero
thf(fact_8124_cone__ordLeq__Cnotzero,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ R3 ) @ ( bNF_Wellorder_ordLeq @ product_unit @ A ) ) ) ).
% cone_ordLeq_Cnotzero
thf(fact_8125_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_8126_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_8127_ordLeq__cprod2,axiom,
! [A: $tType,B: $tType,P13: set @ ( product_prod @ A @ A ),P25: 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 @ P25 ) @ P25 )
=> ( 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 ) ) ) @ P25 @ ( bNF_Cardinal_cprod @ A @ B @ P13 @ P25 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ) ).
% ordLeq_cprod2
thf(fact_8128_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_8129_Cfinite__ordLess__Cinfinite,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S2 ) @ S2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ S2 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ).
% Cfinite_ordLess_Cinfinite
thf(fact_8130_cexp__mono,axiom,
! [E: $tType,F: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P25: 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 ) ) @ P25 @ 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 ) ) @ P25 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ C @ E ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F @ F ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F @ F ) ) @ R23 @ ( bNF_Cardinal_czero @ F ) ) @ ( bNF_Wellorder_ordIso @ D @ F ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P25 ) @ P25 )
=> ( 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 @ P25 ) @ ( bNF_Cardinal_cexp @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_mono
thf(fact_8131_cexp__mono2,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P25: 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 ) ) @ P25 @ 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 ) ) @ P25 @ ( 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 @ P25 ) @ P25 )
=> ( 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 @ P25 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ) ).
% cexp_mono2
thf(fact_8132_cprod__cexp,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set @ ( product_prod @ B @ B ),S2: set @ ( product_prod @ C @ C ),T6: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R3 @ S2 ) @ T6 ) @ ( bNF_Cardinal_cprod @ ( A > B ) @ ( A > C ) @ ( bNF_Cardinal_cexp @ B @ A @ R3 @ T6 ) @ ( bNF_Cardinal_cexp @ C @ A @ S2 @ T6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% cprod_cexp
thf(fact_8133_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_8134_cexp__cone,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cexp @ A @ product_unit @ R3 @ bNF_Cardinal_cone ) @ R3 ) @ ( bNF_Wellorder_ordIso @ ( product_unit > A ) @ A ) ) ) ).
% cexp_cone
thf(fact_8135_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_8136_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P25: 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 ) ) @ P25 @ R23 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( ( field2 @ C @ P25 )
= ( 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 @ P25 ) @ ( bNF_Cardinal_cexp @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% cexp_mono'
thf(fact_8137_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_8138_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P25: 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 ) ) @ P25 @ R23 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( ( ( field2 @ A @ P25 )
= ( 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 @ P25 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2'
thf(fact_8139_ordLeq__cexp1,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),Q4: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q4 ) @ Q4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ Q4 @ ( bNF_Cardinal_cexp @ B @ A @ Q4 @ R3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( A > B ) ) ) ) ) ).
% ordLeq_cexp1
thf(fact_8140_cexp__cong2,axiom,
! [B: $tType,C: $tType,A: $tType,P25: 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 ) ) @ P25 @ R23 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q4 ) @ Q4 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P25 ) @ P25 )
=> ( 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 @ P25 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordIso @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_cong2
thf(fact_8141_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_8142_cexp__cong,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R12: set @ ( product_prod @ B @ B ),P25: 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 ) ) @ P25 @ R23 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( ( bNF_Ca8970107618336181345der_on @ D @ ( field2 @ D @ R23 ) @ R23 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P25 ) @ P25 )
=> ( 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 @ P25 ) @ ( bNF_Cardinal_cexp @ B @ D @ R12 @ R23 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_cong
thf(fact_8143_cexp__mono2__Cnotzero,axiom,
! [B: $tType,C: $tType,A: $tType,P25: 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 ) ) @ P25 @ 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 ) ) @ P25 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P25 ) @ P25 ) )
=> ( 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 @ P25 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q4 @ R23 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2_Cnotzero
thf(fact_8144_ordLeq__cexp2,axiom,
! [A: $tType,B: $tType,Q4: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q4 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ R3 @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R3 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( B > A ) ) ) ) ) ).
% ordLeq_cexp2
thf(fact_8145_Cfinite__cexp__Cinfinite,axiom,
! [A: $tType,B: $tType,S2: set @ ( product_prod @ A @ A ),T6: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ S2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ S2 ) @ S2 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ T6 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ T6 ) @ T6 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) @ ( bNF_Cardinal_cexp @ A @ B @ S2 @ T6 ) @ ( bNF_Cardinal_cexp @ $o @ B @ bNF_Cardinal_ctwo @ T6 ) ) @ ( bNF_Wellorder_ordLeq @ ( B > A ) @ ( B > $o ) ) ) ) ) ).
% Cfinite_cexp_Cinfinite
thf(fact_8146_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_8147_ordLess__ctwo__cexp,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ R3 @ ( bNF_Cardinal_cexp @ $o @ A @ bNF_Cardinal_ctwo @ R3 ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( A > $o ) ) ) ) ).
% ordLess_ctwo_cexp
thf(fact_8148_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_8149_ctwo__ordLeq__Cinfinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) ) ) ).
% ctwo_ordLeq_Cinfinite
thf(fact_8150_ctwo__ordLess__Cinfinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R3 ) @ ( bNF_We4044943003108391690rdLess @ $o @ A ) ) ) ).
% ctwo_ordLess_Cinfinite
thf(fact_8151_Cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q4: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q4 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R3 ) )
& ( bNF_Ca8970107618336181345der_on @ ( B > A ) @ ( field2 @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R3 ) ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R3 ) ) ) ) ) ).
% Cinfinite_cexp
thf(fact_8152_cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q4: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q4 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R3 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) )
=> ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q4 @ R3 ) ) ) ) ).
% cinfinite_cexp
thf(fact_8153_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A5: set @ A,B4: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A1 @ ( insert @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( 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 @ A5 @ B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu2: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_8154_natLeq__ordLeq__cinfinite,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R3 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ R3 ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% natLeq_ordLeq_cinfinite
thf(fact_8155_ordIso__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),P3: set @ ( product_prod @ C @ C ),P11: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 ) ) @ P3 @ P11 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R3 ) @ ( field2 @ C @ P3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R7 ) @ ( field2 @ D @ P11 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordIso_Plus_cong
thf(fact_8156_ordLeq__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),P3: set @ ( product_prod @ C @ C ),P11: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 ) ) @ P3 @ P11 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R3 ) @ ( field2 @ C @ P3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R7 ) @ ( field2 @ D @ P11 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordLeq_Plus_mono
thf(fact_8157_ordIso__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),C4: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R3 ) @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R7 ) @ C4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordIso_Plus_cong1
thf(fact_8158_ordIso__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),A5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ A5 @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A5 @ ( field2 @ B @ R7 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordIso_Plus_cong2
thf(fact_8159_ordLeq__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),C4: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ R3 ) @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R7 ) @ C4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordLeq_Plus_mono1
thf(fact_8160_ordLeq__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R3: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ B @ B ),A5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ 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 @ A5 @ ( field2 @ A @ R3 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A5 @ ( field2 @ B @ R7 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordLeq_Plus_mono2
thf(fact_8161_Card__order__Plus2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A5 @ ( field2 @ A @ R3 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% Card_order_Plus2
thf(fact_8162_Card__order__Plus1,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),B4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R3 ) @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ R3 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R3 ) @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% Card_order_Plus1
thf(fact_8163_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_8164_card__of__Plus1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus1
thf(fact_8165_card__of__Plus2,axiom,
! [B: $tType,A: $tType,B4: set @ A,A5: 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A5 @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus2
thf(fact_8166_card__of__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A5: set @ A,B4: set @ B,C4: set @ C,D4: set @ D] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( 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 @ C4 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D4 ) ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A5 @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B4 @ D4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_cong
thf(fact_8167_card__of__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A5: set @ A,B4: set @ B,C4: set @ C,D4: set @ D] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( 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 @ C4 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D4 ) ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A5 @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B4 @ D4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_mono
thf(fact_8168_card__of__Plus__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,A5: set @ A,B4: set @ B,C4: 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 @ A5 @ B4 ) @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_Plus @ A @ ( sum_sum @ B @ C ) @ A5 @ ( sum_Plus @ B @ C @ B4 @ C4 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_assoc
thf(fact_8169_card__of__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,A5: set @ A,B4: set @ B,C4: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( 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 @ A5 @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B4 @ C4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_cong1
thf(fact_8170_card__of__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,A5: set @ A,B4: set @ B,C4: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( 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 @ C4 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C4 @ B4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_cong2
thf(fact_8171_card__of__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A5: set @ A,B4: set @ B,C4: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( 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 @ A5 @ C4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B4 @ C4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_mono1
thf(fact_8172_card__of__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A5: set @ A,B4: set @ B,C4: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( 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 @ C4 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C4 @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_mono2
thf(fact_8173_card__of__Plus__commute,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: 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 @ A5 @ B4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B4 @ A5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_commute
thf(fact_8174_card__of__Plus__Times__bool,axiom,
! [A: $tType,A5: 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 @ A5 @ A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ $o )
@ ( product_Sigma @ A @ $o @ A5
@ ^ [Uu2: A] : ( top_top @ ( set @ $o ) ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ A ) @ ( product_prod @ A @ $o ) ) ) ).
% card_of_Plus_Times_bool
thf(fact_8175_card__of__Times__Plus__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,A5: set @ A,B4: set @ B,C4: 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 ) @ A5
@ ^ [Uu2: A] : ( sum_Plus @ B @ C @ B4 @ C4 ) ) )
@ ( 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 @ A5
@ ^ [Uu2: A] : B4 )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu2: A] : C4 ) ) ) )
@ ( 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_8176_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A5: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( bot_bot @ ( set @ B ) ) @ A5 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_empty2
thf(fact_8177_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A5: 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ ( bot_bot @ ( set @ B ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus_empty1
thf(fact_8178_card__of__Un__Plus__ordLeq,axiom,
! [A: $tType,A5: set @ A,B4: 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 ) @ A5 @ B4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A5 @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% card_of_Un_Plus_ordLeq
thf(fact_8179_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_8180_natLeq__underS__less,axiom,
! [N2: nat] :
( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N2 )
= ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N2 ) ) ) ).
% natLeq_underS_less
thf(fact_8181_card__of__Plus__infinite2,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( 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 @ B4 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ).
% card_of_Plus_infinite2
thf(fact_8182_card__of__Plus__infinite1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( 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 @ A5 @ B4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% card_of_Plus_infinite1
thf(fact_8183_card__of__Plus__infinite,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( 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 @ A5 @ B4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( 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 @ B4 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Plus_infinite
thf(fact_8184_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,C4: set @ A,A5: set @ B,B4: set @ C] :
( ~ ( finite_finite2 @ A @ C4 )
=> ( ( 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 @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C4 ) ) @ ( 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 @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C4 ) ) @ ( 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 @ A5 @ B4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C4 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite
thf(fact_8185_finite__iff__ordLess__natLeq,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A8: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A8 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% finite_iff_ordLess_natLeq
thf(fact_8186_Restr__natLeq,axiom,
! [N2: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N2 ) )
@ ^ [Uu2: nat] :
( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N2 ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y4: nat] :
( ( ord_less @ nat @ X4 @ N2 )
& ( ord_less @ nat @ Y4 @ N2 )
& ( ord_less_eq @ nat @ X4 @ Y4 ) ) ) ) ) ).
% Restr_natLeq
% Type constructors (794)
thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( code_term_of @ ( heap_Time_Heap @ A21 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( typerep @ ( heap_Time_Heap @ A21 ) ) ) ).
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,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( code_term_of @ ( heap_ext @ A21 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_8,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( typerep @ ( heap_ext @ A21 ) ) ) ).
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,
! [A21: $tType,A26: $tType] :
( ( ( typerep @ A21 )
& ( typerep @ A26 ) )
=> ( code_term_of @ ( product_prod @ A21 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___Enum_Oenum_12,axiom,
! [A21: $tType,A26: $tType] :
( ( ( enum @ A21 )
& ( enum @ A26 ) )
=> ( enum @ ( product_prod @ A21 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_13,axiom,
! [A21: $tType,A26: $tType] :
( ( ( typerep @ A21 )
& ( typerep @ A26 ) )
=> ( typerep @ ( product_prod @ A21 @ A26 ) ) ) ).
thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_14,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( code_term_of @ ( multiset @ A21 ) ) ) ).
thf(tcon_Multiset_Omultiset___Typerep_Otyperep_15,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( typerep @ ( multiset @ A21 ) ) ) ).
thf(tcon_Assertions_Oassn___Typerep_Otyperep_16,axiom,
typerep @ assn ).
thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_17,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( code_term_of @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Enum_Oenum_18,axiom,
! [A21: $tType] :
( ( enum @ A21 )
=> ( enum @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Typerep_Otyperep_19,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( typerep @ ( option @ A21 ) ) ) ).
thf(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_20,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( code_term_of @ ( filter @ A21 ) ) ) ).
thf(tcon_Filter_Ofilter___Typerep_Otyperep_21,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( typerep @ ( filter @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_22,axiom,
! [A21: $tType,A26: $tType] :
( ( ( typerep @ A21 )
& ( typerep @ A26 ) )
=> ( code_term_of @ ( sum_sum @ A21 @ A26 ) ) ) ).
thf(tcon_Sum__Type_Osum___Enum_Oenum_23,axiom,
! [A21: $tType,A26: $tType] :
( ( ( enum @ A21 )
& ( enum @ A26 ) )
=> ( enum @ ( sum_sum @ A21 @ A26 ) ) ) ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_24,axiom,
! [A21: $tType,A26: $tType] :
( ( ( typerep @ A21 )
& ( typerep @ A26 ) )
=> ( typerep @ ( sum_sum @ A21 @ A26 ) ) ) ).
thf(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_25,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( code_term_of @ ( array @ A21 ) ) ) ).
thf(tcon_Heap_Oarray___Typerep_Otyperep_26,axiom,
! [A21: $tType] : ( typerep @ ( array @ A21 ) ) ).
thf(tcon_List_Olist___Code__Evaluation_Oterm__of_27,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( code_term_of @ ( list @ A21 ) ) ) ).
thf(tcon_List_Olist___Typerep_Otyperep_28,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( typerep @ ( list @ A21 ) ) ) ).
thf(tcon_Heap_Oref___Code__Evaluation_Oterm__of_29,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( code_term_of @ ( ref @ A21 ) ) ) ).
thf(tcon_Heap_Oref___Typerep_Otyperep_30,axiom,
! [A21: $tType] : ( typerep @ ( ref @ A21 ) ) ).
thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_31,axiom,
code_term_of @ $o ).
thf(tcon_HOL_Obool___Enum_Oenum_32,axiom,
enum @ $o ).
thf(tcon_HOL_Obool___Typerep_Otyperep_33,axiom,
typerep @ $o ).
thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_34,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( code_term_of @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Enum_Oenum_35,axiom,
! [A21: $tType] :
( ( enum @ A21 )
=> ( enum @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Typerep_Otyperep_36,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( typerep @ ( set @ A21 ) ) ) ).
thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_37,axiom,
code_term_of @ rat ).
thf(tcon_Rat_Orat___Typerep_Otyperep_38,axiom,
typerep @ rat ).
thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_39,axiom,
code_term_of @ num ).
thf(tcon_Num_Onum___Typerep_Otyperep_40,axiom,
typerep @ num ).
thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_41,axiom,
code_term_of @ nat ).
thf(tcon_Nat_Onat___Typerep_Otyperep_42,axiom,
typerep @ nat ).
thf(tcon_Int_Oint___Code__Evaluation_Oterm__of_43,axiom,
code_term_of @ int ).
thf(tcon_Int_Oint___Typerep_Otyperep_44,axiom,
typerep @ int ).
thf(tcon_itself___Code__Evaluation_Oterm__of_45,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( code_term_of @ ( itself @ A21 ) ) ) ).
thf(tcon_itself___Typerep_Otyperep_46,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( typerep @ ( itself @ A21 ) ) ) ).
thf(tcon_fun___Code__Evaluation_Oterm__of_47,axiom,
! [A21: $tType,A26: $tType] :
( ( ( typerep @ A21 )
& ( typerep @ A26 ) )
=> ( code_term_of @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Enum_Oenum_48,axiom,
! [A21: $tType,A26: $tType] :
( ( ( enum @ A21 )
& ( enum @ A26 ) )
=> ( enum @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Typerep_Otyperep_49,axiom,
! [A21: $tType,A26: $tType] :
( ( ( typerep @ A21 )
& ( typerep @ A26 ) )
=> ( typerep @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A21: $tType,A26: $tType] :
( ( comple6319245703460814977attice @ A26 )
=> ( condit1219197933456340205attice @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A21: $tType,A26: $tType] :
( ( comple592849572758109894attice @ A26 )
=> ( comple592849572758109894attice @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
! [A21: $tType,A26: $tType] :
( ( comple489889107523837845lgebra @ A26 )
=> ( comple489889107523837845lgebra @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
! [A21: $tType,A26: $tType] :
( ( ( cl_HOL_Oequal @ A21 )
& ( quickc3360725361186068524ustive @ A21 )
& ( quickc3360725361186068524ustive @ A26 ) )
=> ( quickc3360725361186068524ustive @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A21: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounde4967611905675639751up_bot @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A21: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounde4346867609351753570nf_top @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A21: $tType,A26: $tType] :
( ( comple6319245703460814977attice @ A26 )
=> ( comple6319245703460814977attice @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
! [A21: $tType,A26: $tType] :
( ( ( cl_HOL_Oequal @ A21 )
& ( quickc658316121487927005ustive @ A21 )
& ( quickc658316121487927005ustive @ A26 ) )
=> ( quickc658316121487927005ustive @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A21: $tType,A26: $tType] :
( ( boolea8198339166811842893lgebra @ A26 )
=> ( boolea8198339166811842893lgebra @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A21: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounded_lattice_top @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A21: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounded_lattice_bot @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A21: $tType,A26: $tType] :
( ( comple6319245703460814977attice @ A26 )
=> ( comple9053668089753744459l_ccpo @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Quickcheck__Random_Orandom,axiom,
! [A21: $tType,A26: $tType] :
( ( ( code_term_of @ A21 )
& ( cl_HOL_Oequal @ A21 )
& ( quickcheck_random @ A26 ) )
=> ( quickcheck_random @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A21: $tType,A26: $tType] :
( ( semilattice_sup @ A26 )
=> ( semilattice_sup @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A21: $tType,A26: $tType] :
( ( semilattice_inf @ A26 )
=> ( semilattice_inf @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A21: $tType,A26: $tType] :
( ( distrib_lattice @ A26 )
=> ( distrib_lattice @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A21: $tType,A26: $tType] :
( ( bounded_lattice @ A26 )
=> ( bounded_lattice @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A21: $tType,A26: $tType] :
( ( complete_Sup @ A26 )
=> ( complete_Sup @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OInf,axiom,
! [A21: $tType,A26: $tType] :
( ( complete_Inf @ A26 )
=> ( complete_Inf @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A21: $tType,A26: $tType] :
( ( order_top @ A26 )
=> ( order_top @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A21: $tType,A26: $tType] :
( ( order_bot @ A26 )
=> ( order_bot @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A21: $tType,A26: $tType] :
( ( preorder @ A26 )
=> ( preorder @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A21: $tType,A26: $tType] :
( ( ( finite_finite @ A21 )
& ( finite_finite @ A26 ) )
=> ( finite_finite @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A21: $tType,A26: $tType] :
( ( lattice @ A26 )
=> ( lattice @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A21: $tType,A26: $tType] :
( ( order @ A26 )
=> ( order @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A21: $tType,A26: $tType] :
( ( top @ A26 )
=> ( top @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A21: $tType,A26: $tType] :
( ( ord @ A26 )
=> ( ord @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A21: $tType,A26: $tType] :
( ( bot @ A26 )
=> ( bot @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A21: $tType,A26: $tType] :
( ( uminus @ A26 )
=> ( uminus @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Osup,axiom,
! [A21: $tType,A26: $tType] :
( ( semilattice_sup @ A26 )
=> ( sup @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Lattices_Oinf,axiom,
! [A21: $tType,A26: $tType] :
( ( semilattice_inf @ A26 )
=> ( inf @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A21: $tType,A26: $tType] :
( ( minus @ A26 )
=> ( minus @ ( A21 > A26 ) ) ) ).
thf(tcon_fun___HOL_Oequal,axiom,
! [A21: $tType,A26: $tType] :
( ( ( enum @ A21 )
& ( cl_HOL_Oequal @ A26 ) )
=> ( cl_HOL_Oequal @ ( A21 > A26 ) ) ) ).
thf(tcon_itself___Quickcheck__Random_Orandom_50,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( quickcheck_random @ ( itself @ A21 ) ) ) ).
thf(tcon_itself___HOL_Oequal_51,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( itself @ A21 ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_52,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___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_53,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___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Quickcheck__Exhaustive_Oexhaustive_54,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___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___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ 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_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_55,axiom,
quickcheck_random @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_56,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_57,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_58,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___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OSup_59,axiom,
complete_Sup @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OInf_60,axiom,
complete_Inf @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_61,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_62,axiom,
lattice @ 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_63,axiom,
order @ 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_64,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_65,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___Lattices_Osup_66,axiom,
sup @ int ).
thf(tcon_Int_Oint___Lattices_Oinf_67,axiom,
inf @ int ).
thf(tcon_Int_Oint___Groups_Otimes,axiom,
times @ int ).
thf(tcon_Int_Oint___Groups_Ominus_68,axiom,
minus @ 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___Divides_Ounique__euclidean__semiring__numeral_78,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_79,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_80,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_81,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_82,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_83,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_84,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_85,axiom,
quickc3360725361186068524ustive @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_86,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_87,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_88,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_89,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_90,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_91,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_92,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_93,axiom,
quickc658316121487927005ustive @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_94,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_95,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_96,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_97,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_98,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_99,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_100,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_101,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_102,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_103,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Random_Orandom_104,axiom,
quickcheck_random @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_105,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_106,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_107,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_108,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_109,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_110,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_111,axiom,
comm_monoid_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_112,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_113,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_114,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_115,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_116,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_117,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_118,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_119,axiom,
complete_Sup @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OInf_120,axiom,
complete_Inf @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_121,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_122,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_123,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_124,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_125,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_126,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_127,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_128,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_129,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_130,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_131,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_132,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_133,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_134,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_135,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_136,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_137,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_138,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_139,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_140,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_141,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_142,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_143,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Lattices_Osup_144,axiom,
sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Oinf_145,axiom,
inf @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes_146,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_147,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_148,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_149,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_150,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_151,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_152,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_153,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_154,axiom,
heap @ nat ).
thf(tcon_Nat_Onat___HOL_Oequal_155,axiom,
cl_HOL_Oequal @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_156,axiom,
quickc3360725361186068524ustive @ num ).
thf(tcon_Num_Onum___Quickcheck__Random_Orandom_157,axiom,
quickcheck_random @ num ).
thf(tcon_Num_Onum___Orderings_Opreorder_158,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_159,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_160,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_161,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Otimes_162,axiom,
times @ num ).
thf(tcon_Num_Onum___Groups_Oplus_163,axiom,
plus @ num ).
thf(tcon_Num_Onum___HOL_Oequal_164,axiom,
cl_HOL_Oequal @ num ).
thf(tcon_Num_Onum___Nat_Osize_165,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_166,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_167,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_168,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_169,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_170,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_171,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_172,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_173,axiom,
quickc3360725361186068524ustive @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_174,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_175,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_176,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_177,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_178,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_179,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_180,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_181,axiom,
quickc658316121487927005ustive @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_182,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_183,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_184,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_185,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_186,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_187,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_188,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_189,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_190,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_191,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_192,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_193,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_194,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_195,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_196,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Random_Orandom_197,axiom,
quickcheck_random @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_198,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_199,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_200,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_201,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_202,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_203,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_204,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_205,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_206,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_207,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_208,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_209,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_210,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_211,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_212,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_213,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_214,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_215,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_216,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_217,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_218,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_219,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_220,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_221,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_222,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_223,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_224,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_225,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_226,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_227,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_228,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_229,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_230,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_231,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_232,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_233,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_234,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_235,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_236,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_237,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_238,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_239,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Lattices_Osup_240,axiom,
sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Oinf_241,axiom,
inf @ rat ).
thf(tcon_Rat_Orat___Groups_Otimes_242,axiom,
times @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_243,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_244,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_245,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_246,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_247,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_248,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_249,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_250,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_251,axiom,
dvd @ rat ).
thf(tcon_Rat_Orat___HOL_Oequal_252,axiom,
cl_HOL_Oequal @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_253,axiom,
! [A21: $tType] : ( condit1219197933456340205attice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_254,axiom,
! [A21: $tType] : ( comple592849572758109894attice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_255,axiom,
! [A21: $tType] : ( comple489889107523837845lgebra @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_256,axiom,
! [A21: $tType] :
( ( quickc3360725361186068524ustive @ A21 )
=> ( quickc3360725361186068524ustive @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_257,axiom,
! [A21: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_258,axiom,
! [A21: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_259,axiom,
! [A21: $tType] : ( comple6319245703460814977attice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_260,axiom,
! [A21: $tType] :
( ( quickc658316121487927005ustive @ A21 )
=> ( quickc658316121487927005ustive @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_261,axiom,
! [A21: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_262,axiom,
! [A21: $tType] : ( bounded_lattice_top @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_263,axiom,
! [A21: $tType] : ( bounded_lattice_bot @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_264,axiom,
! [A21: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Random_Orandom_265,axiom,
! [A21: $tType] :
( ( quickcheck_random @ A21 )
=> ( quickcheck_random @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_266,axiom,
! [A21: $tType] : ( semilattice_sup @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_267,axiom,
! [A21: $tType] : ( semilattice_inf @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_268,axiom,
! [A21: $tType] : ( distrib_lattice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_269,axiom,
! [A21: $tType] : ( bounded_lattice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_270,axiom,
! [A21: $tType] : ( complete_Sup @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OInf_271,axiom,
! [A21: $tType] : ( complete_Inf @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_272,axiom,
! [A21: $tType] : ( order_top @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_273,axiom,
! [A21: $tType] : ( order_bot @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_274,axiom,
! [A21: $tType] : ( preorder @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_275,axiom,
! [A21: $tType] :
( ( finite_finite @ A21 )
=> ( finite_finite @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_276,axiom,
! [A21: $tType] : ( lattice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_277,axiom,
! [A21: $tType] : ( order @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_278,axiom,
! [A21: $tType] : ( top @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_279,axiom,
! [A21: $tType] : ( ord @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_280,axiom,
! [A21: $tType] : ( bot @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_281,axiom,
! [A21: $tType] : ( uminus @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Osup_282,axiom,
! [A21: $tType] : ( sup @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Oinf_283,axiom,
! [A21: $tType] : ( inf @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_284,axiom,
! [A21: $tType] : ( minus @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___HOL_Oequal_285,axiom,
! [A21: $tType] :
( ( cl_HOL_Oequal @ A21 )
=> ( cl_HOL_Oequal @ ( set @ A21 ) ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_286,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_287,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_288,axiom,
comple489889107523837845lgebra @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_289,axiom,
quickc3360725361186068524ustive @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_290,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_291,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_292,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_293,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_294,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_295,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_296,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Random_Orandom_297,axiom,
quickcheck_random @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_298,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_299,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_300,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_301,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_302,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OInf_303,axiom,
complete_Inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_304,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_305,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_306,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_307,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_308,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_309,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_310,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_311,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_312,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_313,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_314,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Lattices_Osup_315,axiom,
sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Oinf_316,axiom,
inf @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_317,axiom,
minus @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_318,axiom,
heap @ $o ).
thf(tcon_HOL_Obool___HOL_Oequal_319,axiom,
cl_HOL_Oequal @ $o ).
thf(tcon_Heap_Oref___Quickcheck__Exhaustive_Ofull__exhaustive_320,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( quickc3360725361186068524ustive @ ( ref @ A21 ) ) ) ).
thf(tcon_Heap_Oref___Quickcheck__Random_Orandom_321,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( quickcheck_random @ ( ref @ A21 ) ) ) ).
thf(tcon_Heap_Oref___Heap_Oheap_322,axiom,
! [A21: $tType] : ( heap @ ( ref @ A21 ) ) ).
thf(tcon_Heap_Oref___HOL_Oequal_323,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( ref @ A21 ) ) ).
thf(tcon_Heap_Oref___Nat_Osize_324,axiom,
! [A21: $tType] : ( size @ ( ref @ A21 ) ) ).
thf(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_325,axiom,
! [A21: $tType] :
( ( quickc3360725361186068524ustive @ A21 )
=> ( quickc3360725361186068524ustive @ ( list @ A21 ) ) ) ).
thf(tcon_List_Olist___Quickcheck__Random_Orandom_326,axiom,
! [A21: $tType] :
( ( quickcheck_random @ A21 )
=> ( quickcheck_random @ ( list @ A21 ) ) ) ).
thf(tcon_List_Olist___Heap_Oheap_327,axiom,
! [A21: $tType] :
( ( heap @ A21 )
=> ( heap @ ( list @ A21 ) ) ) ).
thf(tcon_List_Olist___HOL_Oequal_328,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( list @ A21 ) ) ).
thf(tcon_List_Olist___Nat_Osize_329,axiom,
! [A21: $tType] : ( size @ ( list @ A21 ) ) ).
thf(tcon_Heap_Oarray___Quickcheck__Exhaustive_Ofull__exhaustive_330,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( quickc3360725361186068524ustive @ ( array @ A21 ) ) ) ).
thf(tcon_Heap_Oarray___Quickcheck__Random_Orandom_331,axiom,
! [A21: $tType] :
( ( typerep @ A21 )
=> ( quickcheck_random @ ( array @ A21 ) ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_332,axiom,
! [A21: $tType] : ( heap @ ( array @ A21 ) ) ).
thf(tcon_Heap_Oarray___HOL_Oequal_333,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( array @ A21 ) ) ).
thf(tcon_Heap_Oarray___Nat_Osize_334,axiom,
! [A21: $tType] : ( size @ ( array @ A21 ) ) ).
thf(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_335,axiom,
! [A21: $tType,A26: $tType] :
( ( ( quickc3360725361186068524ustive @ A21 )
& ( quickc3360725361186068524ustive @ A26 ) )
=> ( quickc3360725361186068524ustive @ ( sum_sum @ A21 @ A26 ) ) ) ).
thf(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_336,axiom,
! [A21: $tType,A26: $tType] :
( ( ( quickcheck_random @ A21 )
& ( quickcheck_random @ A26 ) )
=> ( quickcheck_random @ ( sum_sum @ A21 @ A26 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_337,axiom,
! [A21: $tType,A26: $tType] :
( ( ( finite_finite @ A21 )
& ( finite_finite @ A26 ) )
=> ( finite_finite @ ( sum_sum @ A21 @ A26 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_338,axiom,
! [A21: $tType,A26: $tType] :
( ( ( heap @ A21 )
& ( heap @ A26 ) )
=> ( heap @ ( sum_sum @ A21 @ A26 ) ) ) ).
thf(tcon_Sum__Type_Osum___HOL_Oequal_339,axiom,
! [A21: $tType,A26: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A21 @ A26 ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_340,axiom,
! [A21: $tType,A26: $tType] : ( size @ ( sum_sum @ A21 @ A26 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_341,axiom,
! [A21: $tType] : ( condit1219197933456340205attice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_342,axiom,
! [A21: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_343,axiom,
! [A21: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_344,axiom,
! [A21: $tType] : ( comple6319245703460814977attice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_345,axiom,
! [A21: $tType] : ( bounded_lattice_top @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_346,axiom,
! [A21: $tType] : ( bounded_lattice_bot @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_347,axiom,
! [A21: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_348,axiom,
! [A21: $tType] : ( semilattice_sup @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_349,axiom,
! [A21: $tType] : ( semilattice_inf @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_350,axiom,
! [A21: $tType] : ( distrib_lattice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_351,axiom,
! [A21: $tType] : ( bounded_lattice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OSup_352,axiom,
! [A21: $tType] : ( complete_Sup @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OInf_353,axiom,
! [A21: $tType] : ( complete_Inf @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_354,axiom,
! [A21: $tType] : ( order_top @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_355,axiom,
! [A21: $tType] : ( order_bot @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_356,axiom,
! [A21: $tType] : ( preorder @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_357,axiom,
! [A21: $tType] : ( lattice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_358,axiom,
! [A21: $tType] : ( order @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_359,axiom,
! [A21: $tType] : ( top @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_360,axiom,
! [A21: $tType] : ( ord @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_361,axiom,
! [A21: $tType] : ( bot @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osup_362,axiom,
! [A21: $tType] : ( sup @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Oinf_363,axiom,
! [A21: $tType] : ( inf @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___HOL_Oequal_364,axiom,
! [A21: $tType] :
( ( cl_HOL_Oequal @ A21 )
=> ( cl_HOL_Oequal @ ( filter @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_365,axiom,
! [A21: $tType] :
( ( comple5582772986160207858norder @ A21 )
=> ( condit6923001295902523014norder @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_366,axiom,
! [A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( condit1219197933456340205attice @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_367,axiom,
! [A21: $tType] :
( ( comple592849572758109894attice @ A21 )
=> ( comple592849572758109894attice @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_368,axiom,
! [A21: $tType] :
( ( quickc3360725361186068524ustive @ A21 )
=> ( quickc3360725361186068524ustive @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_369,axiom,
! [A21: $tType] :
( ( lattice @ A21 )
=> ( bounde4967611905675639751up_bot @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_370,axiom,
! [A21: $tType] :
( ( bounded_lattice_top @ A21 )
=> ( bounde4346867609351753570nf_top @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
! [A21: $tType] :
( ( comple5582772986160207858norder @ A21 )
=> ( comple5582772986160207858norder @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_371,axiom,
! [A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( comple6319245703460814977attice @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_372,axiom,
! [A21: $tType] :
( ( bounded_lattice_top @ A21 )
=> ( bounded_lattice_top @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_373,axiom,
! [A21: $tType] :
( ( lattice @ A21 )
=> ( bounded_lattice_bot @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_374,axiom,
! [A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( comple9053668089753744459l_ccpo @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Random_Orandom_375,axiom,
! [A21: $tType] :
( ( quickcheck_random @ A21 )
=> ( quickcheck_random @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_376,axiom,
! [A21: $tType] :
( ( semilattice_sup @ A21 )
=> ( semilattice_sup @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_377,axiom,
! [A21: $tType] :
( ( semilattice_inf @ A21 )
=> ( semilattice_inf @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Odistrib__lattice_378,axiom,
! [A21: $tType] :
( ( distrib_lattice @ A21 )
=> ( distrib_lattice @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice_379,axiom,
! [A21: $tType] :
( ( bounded_lattice_top @ A21 )
=> ( bounded_lattice @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_OSup_380,axiom,
! [A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( complete_Sup @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_OInf_381,axiom,
! [A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( complete_Inf @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_382,axiom,
! [A21: $tType] :
( ( wellorder @ A21 )
=> ( wellorder @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_383,axiom,
! [A21: $tType] :
( ( order_top @ A21 )
=> ( order_top @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_384,axiom,
! [A21: $tType] :
( ( order @ A21 )
=> ( order_bot @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_385,axiom,
! [A21: $tType] :
( ( preorder @ A21 )
=> ( preorder @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_386,axiom,
! [A21: $tType] :
( ( linorder @ A21 )
=> ( linorder @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_387,axiom,
! [A21: $tType] :
( ( finite_finite @ A21 )
=> ( finite_finite @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Olattice_388,axiom,
! [A21: $tType] :
( ( lattice @ A21 )
=> ( lattice @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_389,axiom,
! [A21: $tType] :
( ( order @ A21 )
=> ( order @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Otop_390,axiom,
! [A21: $tType] :
( ( order_top @ A21 )
=> ( top @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_391,axiom,
! [A21: $tType] :
( ( preorder @ A21 )
=> ( ord @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Obot_392,axiom,
! [A21: $tType] :
( ( order @ A21 )
=> ( bot @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osup_393,axiom,
! [A21: $tType] :
( ( sup @ A21 )
=> ( sup @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Oinf_394,axiom,
! [A21: $tType] :
( ( inf @ A21 )
=> ( inf @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_395,axiom,
! [A21: $tType] :
( ( heap @ A21 )
=> ( heap @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___HOL_Oequal_396,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( option @ A21 ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_397,axiom,
! [A21: $tType] : ( size @ ( option @ A21 ) ) ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_398,axiom,
bounde4967611905675639751up_bot @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_399,axiom,
bounde4346867609351753570nf_top @ assn ).
thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_400,axiom,
boolea8198339166811842893lgebra @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_401,axiom,
bounded_lattice_top @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_402,axiom,
bounded_lattice_bot @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_403,axiom,
semilattice_sup @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_404,axiom,
semilattice_inf @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_405,axiom,
distrib_lattice @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice_406,axiom,
bounded_lattice @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_407,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_408,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_409,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__top_410,axiom,
order_top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_411,axiom,
order_bot @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Opreorder_412,axiom,
preorder @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_413,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Olattice_414,axiom,
lattice @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder_415,axiom,
order @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Otop_416,axiom,
top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oord_417,axiom,
ord @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Obot_418,axiom,
bot @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ouminus_419,axiom,
uminus @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osup_420,axiom,
sup @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Oinf_421,axiom,
inf @ assn ).
thf(tcon_Assertions_Oassn___Groups_Otimes_422,axiom,
times @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ominus_423,axiom,
minus @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_424,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_425,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_426,axiom,
dvd @ assn ).
thf(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_427,axiom,
! [A21: $tType] :
( ( quickc3360725361186068524ustive @ A21 )
=> ( quickc3360725361186068524ustive @ ( multiset @ A21 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_428,axiom,
! [A21: $tType] :
( ( preorder @ A21 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A21 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_429,axiom,
! [A21: $tType] : ( cancel_semigroup_add @ ( multiset @ A21 ) ) ).
thf(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_430,axiom,
! [A21: $tType] :
( ( quickcheck_random @ A21 )
=> ( quickcheck_random @ ( multiset @ A21 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_431,axiom,
! [A21: $tType] : ( comm_monoid_add @ ( multiset @ A21 ) ) ).
thf(tcon_Multiset_Omultiset___Complete__Lattices_OSup_432,axiom,
! [A21: $tType] : ( complete_Sup @ ( multiset @ A21 ) ) ).
thf(tcon_Multiset_Omultiset___Complete__Lattices_OInf_433,axiom,
! [A21: $tType] : ( complete_Inf @ ( multiset @ A21 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_434,axiom,
! [A21: $tType] :
( ( preorder @ A21 )
=> ( preorder @ ( multiset @ A21 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_435,axiom,
! [A21: $tType] : ( monoid_add @ ( multiset @ A21 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_436,axiom,
! [A21: $tType] :
( ( preorder @ A21 )
=> ( order @ ( multiset @ A21 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_437,axiom,
! [A21: $tType] :
( ( preorder @ A21 )
=> ( ord @ ( multiset @ A21 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ominus_438,axiom,
! [A21: $tType] : ( minus @ ( multiset @ A21 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_439,axiom,
! [A21: $tType] : ( zero @ ( multiset @ A21 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oplus_440,axiom,
! [A21: $tType] : ( plus @ ( multiset @ A21 ) ) ).
thf(tcon_Multiset_Omultiset___HOL_Oequal_441,axiom,
! [A21: $tType] :
( ( cl_HOL_Oequal @ A21 )
=> ( cl_HOL_Oequal @ ( multiset @ A21 ) ) ) ).
thf(tcon_Multiset_Omultiset___Nat_Osize_442,axiom,
! [A21: $tType] : ( size @ ( multiset @ A21 ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_443,axiom,
! [A21: $tType,A26: $tType] :
( ( ( quickc3360725361186068524ustive @ A21 )
& ( quickc3360725361186068524ustive @ A26 ) )
=> ( quickc3360725361186068524ustive @ ( product_prod @ A21 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_444,axiom,
! [A21: $tType,A26: $tType] :
( ( ( quickc658316121487927005ustive @ A21 )
& ( quickc658316121487927005ustive @ A26 ) )
=> ( quickc658316121487927005ustive @ ( product_prod @ A21 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_445,axiom,
! [A21: $tType,A26: $tType] :
( ( ( quickcheck_random @ A21 )
& ( quickcheck_random @ A26 ) )
=> ( quickcheck_random @ ( product_prod @ A21 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_446,axiom,
! [A21: $tType,A26: $tType] :
( ( ( finite_finite @ A21 )
& ( finite_finite @ A26 ) )
=> ( finite_finite @ ( product_prod @ A21 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_447,axiom,
! [A21: $tType,A26: $tType] :
( ( ( heap @ A21 )
& ( heap @ A26 ) )
=> ( heap @ ( product_prod @ A21 @ A26 ) ) ) ).
thf(tcon_Product__Type_Oprod___HOL_Oequal_448,axiom,
! [A21: $tType,A26: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A21 @ A26 ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_449,axiom,
! [A21: $tType,A26: $tType] : ( size @ ( product_prod @ A21 @ A26 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_450,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_451,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_452,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_453,axiom,
comple489889107523837845lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_454,axiom,
quickc3360725361186068524ustive @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_455,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_456,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_457,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_458,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_459,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_460,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_461,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_462,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_463,axiom,
quickcheck_random @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_464,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_465,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_466,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_467,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OSup_468,axiom,
complete_Sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OInf_469,axiom,
complete_Inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_470,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_471,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_472,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_473,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_474,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_475,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_476,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_477,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_478,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_479,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_480,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_481,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osup_482,axiom,
sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Oinf_483,axiom,
inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_484,axiom,
minus @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_485,axiom,
heap @ product_unit ).
thf(tcon_Product__Type_Ounit___HOL_Oequal_486,axiom,
cl_HOL_Oequal @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_487,axiom,
! [A21: $tType] :
( ( quickcheck_random @ A21 )
=> ( quickcheck_random @ ( heap_ext @ A21 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_488,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( heap_ext @ A21 ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_489,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_490,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_491,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_492,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_493,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_494,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_495,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_496,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_497,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_498,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_499,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_500,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_501,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_502,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_503,axiom,
quickc3360725361186068524ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_504,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_505,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_506,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_507,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_508,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_509,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_510,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_511,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_512,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_513,axiom,
quickc658316121487927005ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_514,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_515,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_516,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_517,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_518,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_519,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_520,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_521,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_522,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_523,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_524,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_525,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_526,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_527,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_528,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_529,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_530,axiom,
quickcheck_random @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_531,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_532,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_533,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_534,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_535,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_536,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_537,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_538,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_539,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_540,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_541,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_542,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_543,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_544,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_545,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_546,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_547,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_548,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_549,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_550,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_551,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_552,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_553,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_554,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_555,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_556,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_557,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_558,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_559,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_560,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_561,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_562,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_563,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_564,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_565,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_566,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_567,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_568,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_569,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_570,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Otimes_571,axiom,
times @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_572,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_573,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_574,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_575,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_576,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_577,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_578,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_579,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_580,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_581,axiom,
cl_HOL_Oequal @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_582,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_583,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_584,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_585,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_586,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_587,axiom,
semiri6575147826004484403cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_588,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_589,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_590,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_591,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_592,axiom,
linord2810124833399127020strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_593,axiom,
quickc3360725361186068524ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_594,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_595,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_596,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_597,axiom,
linord4140545234300271783up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_598,axiom,
semiri2026040879449505780visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_599,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_600,axiom,
linord8928482502909563296strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_601,axiom,
quickc658316121487927005ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_602,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_603,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_604,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_605,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_606,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_607,axiom,
ordere2520102378445227354miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_608,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_609,axiom,
linordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_610,axiom,
ordered_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_611,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_612,axiom,
quickcheck_random @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_613,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_614,axiom,
semiring_1_cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_615,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_616,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_617,axiom,
ordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_618,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_619,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_620,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_621,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_622,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_623,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_624,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_625,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_626,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_627,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_628,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_629,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_630,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_631,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_632,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_633,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_634,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_635,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_636,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_637,axiom,
mult_zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_638,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_639,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_640,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Otimes_641,axiom,
times @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ominus_642,axiom,
minus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_643,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_644,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_645,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oplus_646,axiom,
plus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_647,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_648,axiom,
dvd @ code_natural ).
thf(tcon_Code__Numeral_Onatural___HOL_Oequal_649,axiom,
cl_HOL_Oequal @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_650,axiom,
size @ code_natural ).
thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Random_Orandom_651,axiom,
! [A21: $tType] :
( ( quickcheck_random @ A21 )
=> ( quickcheck_random @ ( heap_Time_Heap @ A21 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_652,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( heap_Time_Heap @ A21 ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_653,axiom,
! [A21: $tType] : ( size @ ( heap_Time_Heap @ A21 ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
rep_assn @ ( times_times @ assn @ ( q @ r2 ) @ r ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ h @ ( hoare_new_addrs @ h2 @ as @ h ) ) ).
%------------------------------------------------------------------------------